![Introduction
The Backbone of a Complex System
Imagine you are invited to a party; you observe what happens in the room when the other
guests arrive. They start to talk in small groups, usually of two people, then the groups grow
in size, they split, merge again, change shape. Some of the people move from one group
to another. Some of them know each other already, while others are introduced by mutual
friends at the party. Suppose you are also able to track all of the guests and their movements
in space; their head and body gestures, the content of their discussions. Each person is
different from the others. Some are more lively and act as the centre of the social gathering:
they tell good stories, attract the attention of the others and lead the group conversation.
Other individuals are more shy: they stay in smaller groups and prefer to listen to the
others. It is also interesting to notice how different genders and ages vary between groups.
For instance, there may be groups which are mostly male, others which are mostly female,
and groups with a similar proportion of both men and women. The topic of each discussion
might even depend on the group composition. Then, when food and beverages arrive, the
people move towards the main table. They organise into more or less regular queues, so
that the shape of the newly formed groups is different. The individuals rearrange again into
new groups sitting at the various tables. Old friends, but also those who have just met at
the party, will tend to sit at the same tables. Then, discussions will start again during the
dinner, on the same topics as before, or on some new topics. After dinner, when the music
begins, we again observe a change in the shape and size of the groups, with the formation
of couples and the emergence of collective motion as everybody starts to dance.
The social system we have just considered is a typical example of what is known today
as a complex system [16, 44]. The study of complex systems is a new science, and so a
commonly accepted formal definition of a complex system is still missing. We can roughly
say that a complex system is a system made by a large number of single units (individuals,
components or agents) interacting in such a way that the behaviour of the system is not
a simple combination of the behaviours of the single units. In particular, some collective
behaviours emerge without the need for any central control. This is exactly what we have
observed by monitoring the evolution of our party with the formation of social groups, and
the emergence of discussions on some particular topics. This kind of behaviour is what we
find in human societies at various levels, where the interactions of many individuals give
rise to the emergence of civilisation, urban forms, cultures and economies. Analogously,
animal societies such as, for instance, ant colonies, accomplish a variety of different tasks,
xii](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-17-320.jpg)
![xv Introduction
white waterfall cascading down a cliff, and a stream flowing quietly through a green valley.
There is no need to say that “lake”, “waterfall”, “white”, “stream”, “cliff”, “valley” and
“green” form a network of words associations, in which a link exists between two words
if these words are often associated with each other in our minds. Before leaving the office,
you book a flight to go to Prague for a conference. Obviously, also the air transportation
system is a network, whose nodes are airports and links are airline routes.
When you drive back home you feel a bit tired and you think of the various networks
in our body, from the network of blood vessels which transports blood to our organs to the
intricate set of relationships among genes and proteins which allow the perfect functioning
of the cells of our body. Examples of these genetic networks are the transcription regula-
tion networks in which the nodes are genes and links represent transcription regulation of
a gene by the transcription factor produced by another gene, protein interaction networks
whose nodes are protein and there is a link between two proteins if they bind together to
perform complex cellular functions, and metabolic networks where nodes are chemicals,
and links represent chemical reactions.
During dinner you hear on the news that the total export for your country has decreased
by 2.3% this year; the system of commercial relationships among countries can be seen
as a network, in which links indicate import/export activities. Then you watch a movie on
your sofa: you can construct an actor collaboration network where nodes represent movie
actors and links are formed if two actors have appeared in the same movie. Exhausted, you
go to bed and fall asleep while images of networks of all kinds still twist and dance in your
mind, which is, after all, the marvellous combination of the activity of billions of neurons
and trillions of synapses in your brain network. Yet another network.
Why Study Complex Networks?
In the late 1990s two research papers radically changed our view on complex systems,
moving the attention of the scientific community to the study of the architecture of a com-
plex system and creating an entire new research field known today as network science. The
first paper, authored by Duncan Watts and Steven Strogatz, was published in the journal
Nature in 1998 and was about small-world networks [311]. The second one, on scale-free
networks, appeared one year later in Science and was authored by Albert-László Barabási
and Réka Albert [19]. The two papers provided clear indications, from different angles,
that:
• the networks of real-world complex systems have non-trivial structures and are very
different from lattices or random graphs, which were instead the standard networks
commonly used in all the current models of a complex system.
• some structural properties are universal, i.e. are common to networks as diverse as those
of biological, social and man-made systems.
• the structure of the network plays a major role in the dynamics of a complex system and
characterises both the emergence and the properties of its collective behaviours.](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-20-320.jpg)
![xviii Introduction
The standard tools to study complex networks are a mixture of mathematical and com-
putational methods. They require some basic knowledge of graph theory, probability,
differential equations, data structures and algorithms, which will be introduced in this
book from scratch and in a friendly way. Also, network theory has found many interest-
ing applications in several different fields, including social sciences, biology, neuroscience
and technology. In the book we have therefore included a large variety of examples to
emphasise the power of network science. This book is essentially on the structure of com-
plex networks, since we have decided that the detailed treatment of the different types of
dynamical processes that can take place over a complex network should be left to another
book, which will follow this one.
The book is organised into ten chapters. The first six chapters (Chapters 1–6) form the
core of the book. They introduce the main concepts of network science and the basic
measures and models used to characterise and reproduce the structure of various com-
plex networks. The remaining four chapters (Chapters 7–10) cover more advanced topics
that could be skipped by a lecturer who wants to teach a short course based on the book.
In Chapter 1 we introduce some basic definitions from graph theory, setting up the lan-
guage we will need for the remainder of the book. The aim of the chapter is to show
that complex network theory is deeply grounded in a much older mathematical discipline,
namely graph theory.
In Chapter 2 we focus on the concept of centrality, along with some of the related mea-
sures originally introduced in the context of social network analysis, which are today used
extensively in the identification of the key components of any complex system, not only
of social networks. We will see some of the measures at work, using them to quantify the
centrality of movie actors in the actor collaboration network.
Chapter 3 is where we first discuss network models. In this chapter we introduce the
classical random graph models proposed by Erdős and Rényi (ER) in the late 1950s, in
which the edges are randomly distributed among the nodes with a uniform probability.
This allows us to analytically derive some important properties such as, for instance, the
number and order of graph components in a random graph, and to use ER models as term
of comparison to investigate scientific collaboration networks. We will also show that the
average distance between two nodes in ER random graphs increases only logarithmically
with the number of nodes.
In Chapter 4 we see that in real-world systems, such as the neural network of the C. ele-
gans or the movie actor collaboration network, the neighbours of a randomly chosen node
are directly linked to each other much more frequently than would occur in a purely ran-
dom network, giving rise to the presence of many triangles. In order to quantify this, we
introduce the so-called clustering coefficient. We then discuss the Watts and Strogatz (WS)
small-world model to construct networks with both a small average distance between nodes
and a high clustering coefficient.
In Chapter 5 the focus is on how the degree k is distributed among the nodes of a network.
We start by considering the graph of the World Wide Web and by showing that it is a
scale-free network, i.e. it has a power–law degree distribution pk ∼ k−γ with an exponent
γ ∈ [2, 3]. This is a property shared by many other networks, while neither ER random
graphs nor the WS model can reproduce such a feature. Hence, we introduce the so-called](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-23-320.jpg)
![1 Graphs and Graph Theory
Graphs are the mathematical objects used to represent networks, and graph theory is the
branch of mathematics that deals with the study of graphs. Graph theory has a long his-
tory. The notion of the graph was introduced for the first time in 1763 by Euler, to settle
a famous unsolved problem of his time: the so-called Königsberg bridge problem. It is no
coincidence that the first paper on graph theory arose from the need to solve a problem from
the real world. Also subsequent work in graph theory by Kirchhoff and Cayley had its root
in the physical world. For instance, Kirchhoff’s investigations into electric circuits led to
his development of a set of basic concepts and theorems concerning trees in graphs. Nowa-
days, graph theory is a well-established discipline which is commonly used in areas as
diverse as computer science, sociology and biology. To give some examples, graph theory
helps us to schedule airplane routing and has solved problems such as finding the max-
imum flow per unit time from a source to a sink in a network of pipes, or colouring the
regions of a map using the minimum number of different colours so that no neighbouring
regions are coloured the same way. In this chapter we introduce the basic definitions, set-
ting up the language we will need in the rest of the book. We also present the first data set
of a real network in this book, namely Elisa’s kindergarten network. The two final sections
are devoted to, respectively, the proof of the Euler theorem and the description of a graph
as an array of numbers.
1.1 What Is a Graph?
The natural framework for the exact mathematical treatment of a complex network is a
branch of discrete mathematics known as graph theory [48, 47, 313, 150, 272, 144]. Dis-
crete mathematics, also called finite mathematics, is the study of mathematical structures
that are fundamentally discrete, i.e. made up of distinct parts, not supporting or requiring
the notion of continuity. Most of the objects studied in discrete mathematics are count-
able sets, such as integers and finite graphs. Discrete mathematics has become popular in
recent decades because of its applications to computer science. In fact, concepts and nota-
tions from discrete mathematics are often useful to study or describe objects or problems
in computer algorithms and programming languages. The concept of the graph is better
introduced by the two following examples.
1](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-30-320.jpg)
![3 1.1 What Is a Graph?
the image we can easily distinguish the borders between any two nations. Let us suppose
now that we are interested not in the precise shape and geographical position of each coun-
try, but simply in which nations have common borders. We can thus transform the map into
a much simpler representation that preserves entirely that information. In order to do so we
need, with a little bit of abstraction, to transform each nation into a point. We can then
place the points in the plane as we want, although it can be convenient to maintain similar
positions to those of the corresponding nations in the map. Finally, we connect two points
with a line if there is a common boundary between the corresponding two nations. Notice
that in this particular case, due to the placement of the points in the plane, it is possible to
draw all the connections with no line intersections.
The mathematical entity used to represent the existence or the absence of links among
various objects is called the graph. A graph is defined by giving a set of elements, the
graph nodes, and a set of links that join some (or all) pairs of nodes. In Example 1.1 we
are using a graph to represent a network of social acquaintances. The people invited at a
party are the nodes of the graph, while the existence of acquaintances between two persons
defines the links in the graph. In Example 1.1 the nodes of the graph are the countries of
the European Union, while a link between two countries indicates that there is a common
boundary between them. A graph is defined in mathematical terms in the following way:
Definition 1.1 (Undirected graph) A graph, more specifically an undirected graph, G ≡
(N, L), consists of two sets, N = ∅ and L. The elements of N ≡ {n1, n2, . . . , nN} are
distinct and are called the nodes (or vertices, or points) of the graph G. The elements
of L ≡ {l1, l2, . . . , lK} are distinct unordered pairs of distinct elements of N, and are
called links (or edges, or lines).
The number of vertices N ≡ N[G] = |N|, where the symbol | · | denotes the cardinality
of a set, is usually referred as the order of G, while the number of edges K ≡ K[G] = |L|
is the size of G.[1] A node is usually referred to by a label that identifies it. The label is often
an integer index from 1 to N, representing the order of the node in the set N. We shall use
this labelling throughout the book, unless otherwise stated. In an undirected graph, each of
the links is defined by a pair of nodes, i and j, and is denoted as (i, j) or (j, i). In some cases
we also denote the link as lij or lji. The link is said to be incident in nodes i and j, or to
join the two nodes; the two nodes i and j are referred to as the end-nodes of link (i, j). Two
nodes joined by a link are referred to as adjacent or neighbouring.
As shown in Example 1.1, the usual way to picture a graph is by drawing a dot or a
small circle for each node, and joining two dots by a line if the two corresponding nodes
are connected by an edge. How these dots and lines are drawn in the page is in principle
irrelevant, as is the length of the lines. The only thing that matters in a graph is which pairs
of nodes form a link and which ones do not. However, the choice of a clear drawing can be
[1] Sometimes, especially in the physical literature, the word size is associated with the number of nodes, rather
than with the number of links. We prefer to consider K as the size of the graph. However, in many cases of
interest, the number of links K is proportional to the number of nodes N, and therefore the concept of size of
a graph can equally well be represented by the number of its nodes N or by the number of its edges K.](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-32-320.jpg)
![4 Graphs and Graph Theory
very important in making the properties of the graph easy to read. Of course, the quality
and usefulness of a particular way to draw a graph depends on the type of graph and on the
purpose for which the drawing is generated and, although there is no general prescription,
there are various standard drawing setups and different algorithms for drawing graphs that
can be used and compared. Some of them are illustrated in Box 1.1.
Figure 1.1 shows four examples of small undirected graphs. Graph G1 is made of N = 5
nodes and K = 4 edges. Notice that any pair of nodes of this graph can be connected in
only one way. As we shall see later in detail, such a graph is called a tree. Graphs G2 has
N = K = 4. By starting from one node, say node 1, one can go to all the other nodes 2,
3, 4, and back again to 1, by visiting each node and each link just once, except of course
node 1, which is visited twice, being both the starting and ending node. As we shall see,
we say that the graph G2 contains a cycle. The same can be said about graph G3. Graph G3
contains an isolated node and three nodes connected by three links. We say that graphs G1
and G2 are connected, in the sense that any node can be reached, starting from any other
node, by “walking” on the graph, while graph G3 is not.
Notice that, in the definition of graph given above, we deliberately avoided loops, i.e.
links from a node to itself, and multiple edges, i.e. pairs of nodes connected by more than
one link. Graphs with either of these elements are called multigraphs [48, 47, 308]. An
example of multigraph is G4 in Figure 1.1. In such a multigraph, node 1 is connected to
itself by a loop, and it is connected to node 3 by two links. In this book, we will deal with
graphs rather than multigraphs, unless otherwise stated.
For a graph G of order N, the number of edges K is at least 0, in which case the graph
is formed by N isolated nodes, and at most N(N − 1)/2, when all the nodes are pairwise
adjacent. The ratio between the actual number of edges K and its maximum possible num-
ber N(N − 1)/2 is known as the density of G. A graph with N nodes and no edges has zero
t
Fig. 1.1 Some examples of undirected graphs, namely a tree, G1; two graphs containing cycles, G2 and G3; and an undirected
multigraph, G4.](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-33-320.jpg)
![5 1.1 What Is a Graph?
Box 1.1 Graph Drawing
A good drawing can be very helpful to highlight the properties of a graph. In one standard setup, the so
called circular layout, the nodes are placed on a circle and the edges are drawn across the circle. In another
set-up, known as the spring model, the nodes and links are positioned in the plane by assuming the graph
is a physical system of unit masses (the nodes) connected by springs (the links). An example is shown in the
figure below, where the same graph is drawn using a circular layout (left) and a spring-based layout (right)
based on the Kamada–Kawaialgorithm [173].
By nature, springs attract their endpoints when stretched and repel their endpoints when compressed.
In this way, adjacent nodes on the graph are moved closer in space and, by looking for the equilibrium
conditions, we get a layout where edges are short lines, and edge crossings with other nodes and edges
are minimised. There are many software packages specifically focused on graph visualisation, including
Pajek (http://mrvar.fdv.uni-lj.si/pajek/), Gephi (https://gephi.org/) and
GraphViz (http://www.graphviz.org/). Moreover, most of the libraries for network analysis,
including NetworkX (https://networkx.github.io/), iGraph (http://igraph.org/)
and SNAP (Stanford Network Analysis Platform, http://snap.stanford.edu/), support differ-
ent algorithms for network visualisation.
density and is said to be empty, while a graph with K = N(N −1)/2 edges, denoted as KN,
has density equal to 1 and is said to be complete. The complete graphs with N = 3, N = 4
and N = 5 respectively, are illustrated in Figure 1.2. In particular, K3 is called a triangle,
and in the rest of this book will also be indicated by the symbol . As we shall see, we are
often interested in the asymptotic properties of graphs when the order N becomes larger
and larger. The maximum number of edges in a graph scales as N2. If the actual number of
edges in a sequence of graphs of increasing number of nodes scales as N2, then the graphs
of the sequence are called dense. It is often the case that the number of edges in a graph of
a given sequence scales much more slowly than N2. In this case we say that the graphs are
sparse.
We will now focus on how to compare graphs with the same order and size. Two graphs
G1 = (N1, L1) and G2 = (N2, L2) are the same graph if N1 = N2 and L1 = L2; that is, if
both their node sets and their edge sets (i.e. the sets of unordered pairs of nodes defining L)
are the same. In this case, we write G1 = G2. For example, graphs (a) and (b) in Figure 1.3](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-34-320.jpg)
![8 Graphs and Graph Theory
t
Fig. 1.5 All possible automorphisms of graph C4 in Figure 1.4.
Example 1.3 Consider the star graph S4 with a central node and three links shown in Figure
1.4. There are six automorphisms, corresponding to the following transformations: identity,
rotation by 120◦ counterclockwise, rotation by 120◦ clockwise and three specular reflec-
tions, respectively around edge (1, 2), (1, 3), (1, 4). There are no more automorphisms,
because in all permutations, node 1 has to remain fixed. Therefore, the number aG of pos-
sible automorphisms is given by the number of permutations of the three remaining labels,
that is, 3! = 6.
Finally, we consider some basic operations to produce new graphs from old ones, for
instance, by merging together two graphs or by considering only a portion of a given graph.
Let us start by introducing the definition of the union of two graphs. Let G1 = (N1, L1)
and G2 = (N2, L2) be two graphs. We define graph G = (N, L), where N = N1 ∪N2 and
L = L1 ∪ L2, as the union of G1 and G2, and we denote it as G = G1 + G2. A concept
that will be very useful in the following is that of subgraph of a given graph.
Definition 1.4 (Subgraph) A subgraph of G = (N, L) is a graph G = (N, L) such that
N ⊆ N and L ⊆ L. If G contains all links of G that join two nodes in N, then G is
said to be the subgraph induced or generated by N, and is denoted as G = G[N].
Figure 1.6 shows some examples of subgraphs. A subgraph is said to be maximal with
respect to a given property if it cannot be extended without losing that property. For
example, the subgraph induced by nodes 2, 3, 4, 6 in Figure 1.6 is the maximal complete
subgraph of order four of graph G. Of particular relevance for some of the definitions given
in the following is the subgraph of the neighbours of a given node i, denoted as Gi. Gi is
defined as the subgraph induced by Ni, the set of nodes adjacent to i, i.e. Gi = G[Ni]. In
Figure 1.6, graph (c) represents the graph G6, induced by the neighbours of node 6.
Let G = (N, L), and let s ∈ L. If we remove edge s from G we shall denote the new
graph as G = (N, L − s), or simply G = G − s. Analogously, let L ⊆ L. We denote as](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-37-320.jpg)
![10 Graphs and Graph Theory
where the N = 6 nodes represent the terminals, while the links indicate the presence of a
shuttle connecting one terminal to another. Notice, however, that in this case it is neces-
sary to associate a direction with each link. A directed link is usually called an arc. The
graph shown in the right-hand side of the figure has indeed K = 8 arcs. Notice that there
can be two arcs between the same pair of nodes. For instance, arc (A, D) is different from
arc (D, A).
We lose important information if we represent the system in the example as a graph accord-
ing to Definition 1.1. We need therefore to extend the mathematical concept of graph, to
make it better suited to describe real situations. We introduce the following definition of
the directed graph.
Definition 1.5 (Directed graph) A directed graph G ≡ (N, L) consists of two sets, N = ∅
and L. The elements of N ≡ {n1, n2, . . . , nN} are the nodes of the graph G. The elements
of L ≡ {l1, l2, . . . , lK} are distinct ordered pairs of distinct elements of N, and are called
directed links, or arcs.
In a directed graph, an arc between node i and node j is denoted by the ordered pair
(i, j), and we say that the link is ingoing in j and outgoing from i. Such an arc may still be
denoted as lij. However, at variance with undirected graphs, this time the order of the two
nodes is important. Namely, lij ≡ (i, j) stands for an arc from i to j, and lij = lji, or in other
terms the arc (i, j) is different from the arc (j, i).
As another example of a directed network we introduce here the first data set of this
book, namely DATA SET 1. As with all the other data sets that will be provided and studied
in this book, this refers to the network of a real system. In this case, the network describes
friendships between children at the kindergarten of Elisa, the daughter of the first author of
this book. The choice of this system as an example of a directed network is not accidental.
Friendship networks of children are, in fact, among social systems, cases in which the
directionality of a link can be extremely important. In the case under study, friendships have
been determined by interviewing the children. As an outcome of the interview, friendship
relations are directed, since it often happens that child A indicates B as his/her friend,
without B saying that A is his/her friend. The basic properties of Elisa’s kindergarten
network are illustrated in the DATA SET Box 1.2, and the network can be downloaded
from the book’s webpage. Of course, one of the first things that catches our eye in the
directed graph shown in Box 1.2 is that many of the relations are not reciprocated. This
property can be quantified mathematically. A traditional measure of graph reciprocity is the
ratio r between the number of arcs in the network pointing in both directions and the total
number of arcs [308] (see Problem 1.2 for a mathematical expression of r, and the work by
Diego Garlaschelli and Maria Loffredo for alternative measures of the reciprocity [128]).
The reciprocity r takes the value r = 0 for a purely unidirectional graph, while r = 1
for a purely bidirectional one. For Elisa’s kindergarten we get a value r = 34/57 ≈ 0.6,
since the number of arcs between reciprocating pairs is 34 while we have 57 arcs in total.
This means that only 60 per cent of the relations are reciprocated in this network, or, more
precisely, if there is an arc pointing from node i to node j, then there is a 60 per cent
probability that there will also be an arc from j to i.](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-39-320.jpg)
![11 1.2 Directed, Weighted and Bipartite Graphs
Box 1.2 DATA SET 1: Elisa’s Kindergarten Network
Elisa’s kindergarten network describes N = 16 children between three and five years old, and their
declaredfriendshiprelations.ThenetworkgiveninthisdatasetisadirectedgraphwithK = 57arcsandis
shown in the figure. The nine girls are represented as circles, while the seven boys are squares. Bidirectional
relations are indicated as full-line double arrows, while purely unidirectional ones as dashed-line arrows.
Notice that only a certain percentage of the relations are reciprocated.
Itisinterestingtonoticethat,withtheexceptionofElvis,theyoungestboyintheclass,thereisalmostasplit
between two groups, the boys and the girls. You certainly would not observe this in a network of friendship
in a high school. In the kindergarten network, Matteo is the child connecting the two communities.
Summing up, the most basic definition is that of undirected graph, which describes
systems in which the links have no directionality. In the case, instead, in which the
directionality of the connections is important, the directed graph definition is more appro-
priate. Examples of an undirected graph and of a directed graph, with N = 7 nodes, and
K = 8 links and K = 11 arcs respectively, are shown in Figure 1.7 (a) and (b). The directed
graph in panel (b) does not contain loops, nor multiple arcs, since these elements are not
allowed by the standard definition of directed graph given above. Directed graphs with
either of these elements are called directed multigraphs [48, 47, 308].
Also, we often need to deal with networks displaying a large heterogeneity in the rel-
evance of the connections. Typical examples are social systems where it is possible to
measure the strength of the interactions between individuals, or cases such as the one
discussed in the following example.
Example 1.5 Suppose we have to construct a network of roads to connect N towns, so
that it is possible to go from each town to any other. A natural question is: what is the](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-40-320.jpg)
![16 Graphs and Graph Theory
Box 1.4 Path-Finding Behaviours in Animals
Findingtheshortestrouteisextremelyimportantalsoforanimalsmovingregularlybetweendifferentpoints.
How can animals, with only limited local information, achieve this? Ants, for instance, find the shortest path
betweentheirnestandtheirfoodsourcebycommunicatingwitheachotherviatheirpheromone,achemical
substancethatattractsotherants.Initially,antsexploreallthepossiblepathstothefoodsource.Antstaking
shorter paths will take a shorter time to arrive at the food. This causes the quantity of pheromone on the
shorterpathstogrowfasterthanonthelongerones,andthereforetheprobabilitywithwhichanysingleant
choosesthepathtofollowisquicklybiasedtowardstheshorterones.Thefinalresultisthat,duetothesocial
cooperative behaviour of the individuals, very quickly all ants will choose the shortest path [141].
Even more striking is the fact that unicellular organisms can also exhibit similar path-finding behaviours.
A well-studied case is the plasmodium of a slime mould, the Physarum polycephalum, a large amoeba-like
cell. The body of the plasmodium contains a network of tubes, which enables nutrients and chemical sig-
nals to circulate through the organism. When food sources are presented to a starved plasmodium that has
spread over the entire surface of an agar plate, parts of the organism concentrate over the food sources and
are connected by only a few tubes. It has been shown in a series of experiments that the path connecting
these parts of the plasmodium is the shortest possible, even in a maze [224]. Check Ref. [296] if you want to
see path-finding algorithms inspired by the remarkable process of cellular computation exhibited by the P.
polycephalum.
In a directed graph, the situation is more complex than in an undirected graph. In fact,
as observed before, a directed path may exist through the network from vertex i to vertex
j, but that does not guarantee that one exists from j to i. Consequently, we have various
definitions of connectedness between two nodes, and we can define weakly and strongly
connected components as below.
Definition 1.12 (Connectedness and components in directed graphs) Two nodes i and j of a
directed graph G are said to be strongly connected if there exists a path from i to j and a
path from j to i. A directed graph G is said to be strongly connected if all pairs of nodes
(i, j) are strongly connected. A strongly connected component of G associated with node
i is the maximal strongly connected induced subgraph containing node i, i.e. it is the
subgraph which is induced by all nodes which are strongly connected to node i.
The undirected graph Gu obtained by removing all directions in the arcs of G is called
the underlying undirected graph of G. A directed graph G is said to be weakly connected
if the underlying undirected graph Gu is connected. A weakly connected component of
G is a component of its underlying undirected graph Gu.
Example 1.9 Most graphs shown in the previous figures are connected. Examples of dis-
connected graphs are graph G3 in Figure 1.1 and graph (b) in Figure 1.6. Graph G3 in
Figure 1.1 has two components, one given by node 1 and the other given by the subgraph
induced by nodes {2, 3, 4}. Graph (b) in Figure 1.6 has also two components, one given by](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-45-320.jpg)
![21 1.5 Graph Theory and the Bridges of Königsberg
Königsberg, having k = 3.5. This number is of the same order of magnitude as the exact
value (see Problem 1.5). Notice that this number grows exponentially with K, so that in a
city with a larger number of bridges it can become impossible to explore all the different
trips. For instance, in the case of the historical part of Venice, with its 428 bridges, even
by assuming a small value k = 2, we get a number of N · 2428 walks to check. Thus, an
exhaustive search for an Eulerian path over the graph represented by the islands of Venice
and its bridges will be far beyond the computational capabilities of modern computers. In
fact, even assuming that a computer can check 1015 walks per second, it would be able to
check about 1032 walks in a timespan equal to the age of the universe; this number is much
smaller than N · 2428 ≈ 7N · 10128.
After having shown that the problem can be rephrased in terms of a graph, Euler gave a
general theorem on the conditions for a graph to be Eulerian.
Theorem1.1(Eulertheorem) A connected graph is Eulerian iff each vertex has even degree.
It has a Eulerian trail from vertex i to vertex j, i = j, iff i and j are the only vertices of odd
degree.
To be more precise, Euler himself actually proved only a necessary condition for the
existence of an Eulerian circuit, i.e. he proved that if some nodes have an odd degree, then
an Eulerian trail cannot exist. The proof given by Euler can be summarised as follows.
Proof Suppose that there exists an Euler circuit. This means that each node i is a crossing
point, therefore if we denote by pi the number of times the node i is traversed by the
circuit, its degree has to be ki = 2pi, and therefore it has to be even. If we only assume the
existence of an Eulerian trail, then there is no guarantee that the starting point coincides
with the ending point, and therefore the degree of such two points may be odd.
Euler believed that the converse was also true, i.e. that if all nodes have an even degree
then there exists an Eulerian circuit, and he gave some argument about this, but he never
rigorously proved the sufficient condition [105]. The proof that the condition that all nodes
have even degree is sufficient for the existence of an Eulerian trail appeared more than
a century later, and was due to the German mathematician Carl Hierholzer, who pub-
lished the first characterisation of Eulerian graphs in 1873 [152]. The early history of graph
theory, including the work of Euler and Hierholzer, is illustrated in [245].
Here we shall give a complete proof of the Euler theorem based on the concept of par-
tition of a graph into cycles. Consider the set of edges L of a graph G. We say that a
subset L1 ⊆ L is a cycle if there exists a cycle, Z1, that contains all and only the edges
L1. We say that the set L is partitioned if there exists a certain number s of subsets of L,
L1, L2, . . . , Ls, such that:
Li ∩ Lj = ∅, ∀i, j ∈ [1, . . . , s], ∪s
i=1Li = L
Now we can state the characterisation of Eulerian graphs in the form of equivalence of the
following three statements [150]:](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-50-320.jpg)
![22 Graphs and Graph Theory
t
Fig. 1.12 The reduction process to prove that a graph with all nodes with even degree can be partitioned into cycles.
(1) G is an Eulerian graph (i.e. it contains at least one Eulerian circuit)
(2) ∀i ∈ N, ki is even
(3) there exists a partition of L into cycles.
Proof Proof (1) =⇒ (2). It is clear that the existence of an Eulerian circuit implies that
every node i is a crossing point, i.e. every node can be considered both a “starting point”
and an “ending point”, therefore its degree ki has to be even.
Proof Proof (2) =⇒ (3). From property (2) it follows that in G there exists at least one
cycle, Z1, otherwise G would be a tree, and it would therefore have vertices of degree one
(see Section 1.4). If G Z1,[2] then (3) is proved. If G = Z1, let G2 = G1 − L1, i.e. G2
is the graph obtained from G after removing all edges of Z1. It is clear that all vertices of
G2 have even degree, because the degree of each node belonging to Z1 has been decreased
by 2. Let G
2 be the graph obtained from G2 by eliminating all isolated vertices of G2.
Since all vertices of G
2 have even degree, this means that G2 contains at least a cycle, Z2,
and the argument repeats. Proceeding with the reduction, one will at last reach a graph
G
Z, therefore the set of links L is partitioned in cycles L1, L2, . . . , L. The procedure
is illustrated in Figure 1.12.
Proof Proof of (3) =⇒ (1). We now assume that the set of edges L can be partitioned
into a certain number s of cycles, L1, L2, . . . , Ls. Let us denote by Z1, Z2, . . . , Zs the cor-
responding graphs. If Z1 G then (1) is proved. Otherwise, let Z2 be a cycle with a vertex
i in common with Z1. The circuit that starts in i and passes through all edges of Z1 and Z2
contains all edges of Z1 and Z2 exactly once. Hence, it is an Eulerian circuit for Z1 ∪ Z2.
If G Z1 ∪ Z2 the assert is proved, otherwise let Z3 be another cycle with a vertex in
common with Z1 ∪ Z2, and so on. By iterating the procedure, one can construct in G an
Eulerian circuit.
The Euler theorem provides a general solution to the bridge problem: the request to
pass over every bridge exactly once can be satisfied if and only if the vertices with odd
degree are zero (starting and ending point coincide) or two (starting and ending point do
not coincide). Now, if we go back to the graph of Königsberg we see that the conditions of
the theorem are not verified. Actually, all the four vertices in the graph in Figure 1.11 have
[2] The symbol indicates that the two graphs are isomorphic. See Section 1.1](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-51-320.jpg)
![24 Graphs and Graph Theory
Au =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 1 0 0 1
1 0 1 0 1
0 1 0 0 0
0 0 0 0 0
1 1 0 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
, Ad =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 1 0 0 1
0 0 1 0 0
0 0 0 1 1
0 1 0 0 0
1 0 0 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
Au is symmetric and contains 2K non-zero entries. The number of ones in row i, or equiv-
alently in column i, is equal to the degree of vertex i. The adjacency matrix of a directed
graph is in general not symmetric. This is the case of Ad. This matrix has K elements dif-
ferent from zero, and the number of ones in row i is equal to the number of outgoing links
kout
i , while the number of ones in column i is equal to the number of ingoing links kin
i .
Clearly, Definition 1.15 refers only to undirected and directed graphs without loops or
multiple links, which will be our main interest in this section and in the rest of the book.
While weighted graphs will be treated in detail in Chapter 10, here we want to add that it is
also possible to describe a bipartite graph by means of a slightly different definition of the
adjacency matrix than that given above, and which in general is not a square matrix. In fact,
a bipartite graph such as that shown in Figure 1.8 can be described by an N × V adjacency
matrix A, such that entry aiα, with i = 1, . . . , N and α = 1, . . . , V, is equal to 1 if node i
of the first set and node α of the second set are connected, while it is 0 otherwise. Notice
that we used Roman and Greek letters to avoid confusion between the two types of nodes.
Using this representation of a bipartite graph in terms of an adjacency matrix, we show in
the next example how to formally describe a commonly used method to recommend a set
of objects to a set of users (see also Box 1.3).
Example 1.13 (Recommendation systems: collaborative filtering) Consider a bipartite
graph of users and objects such as that shown in Figure 1.8, in which the existence of
the link between node i and node α denotes that user ui has selected object oα. A famous
personalised recommendation system, known as collaborative filtering (CF), is based on
the construction of a N × N user similarity matrix S = {sij}. The similarity between two
users ui and uj can be expressed in terms of the adjacency matrix of the graph as:
sij =
V
α=1 aiαajα
min{kui , kuj }
, (1.1)
where kui =
V
α=1 aiα is the degree of user ui, i.e. the number of objects chosen by
ui [324]. Based on the similarity matrix S, we can then construct an N × V recommen-
dation matrix R = {riα}. In fact, for any user–object pair ui, oα, if ui has not yet chosen oα,
i.e. if aiα = 0, we can define a recommendation score riα measuring to what extent ui may
like oα, as:
riα =
N
j=1,j=i sijajα
N
j=1,j=i sij
. (1.2)](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-53-320.jpg)
![25 1.6 How to Represent a Graph
At the numerator, we sum the similarity between user ui and all the other users that have
chosen object oα. In practice, we count the number of users that chose object oα, weight-
ing each of them with the similarity with user ui. The normalisation at the denominator
guarantees that riα ranges in the interval [0, 1]. Finally, in order to recommend items to
a user ui, we need to compute the values of the recommendation score riα for all objects
oα, α = 1, . . . , V, such that aiα = 0. Then, all the non-zero values of riα are sorted in
decreasing order, and the objects in the top of the list are recommended to user ui.
Let us now come back to the main issue of this section, namely how to represent an
undirected or directed graph. An alternative possibility to the adjacency matrix is a N × K
matrix called the incidence matrix, in which the rows represent different nodes, while the
columns stand for the links.
Definition 1.16 (Incidence matrix) The incidence matrix B of an undirected graph is an
N × K matrix whose entry bik is equal to 1 whenever the node i is incident with the link
lk, and is zero otherwise. If the graph is directed, the adopted convention is that the entry
bik of B is equal to 1 if arc k points to node i, it is equal to −1 if the arc leaves node i,
and is zero otherwise.
Example 1.14 (Incidence matrix) The incidence matrices respectively associated with
the undirected and directed graphs considered in Example 1.12 are:
Bu =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1 1 0 0
1 0 1 1
0 0 0 1
0 0 0 0
0 1 1 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
, Bd =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
−1 1 −1 0 0 0 0
1 0 0 −1 0 1 0
0 0 0 1 −1 0 −1
0 0 0 0 1 −1 0
0 −1 1 0 0 0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
Notice that Bu is a 5 × 4 matrix because the first graph has N = 5 nodes and K = 4 links,
while Bd is a 5×7 matrix because the second graph has N = 5 nodes and K = 7 arcs. Also,
notice that there are only two non-zero entries in each column of an incidence matrix.
Observe now that many elements of the adjacency and of the incidence matrix are zero.
In particular, if a graph is sparse, then the adjacency matrix is sparse: for an undirected
(directed) graph the number 2K (K) of non-zero elements is proportional to N, while the
total number of elements of the matrix is N2. When the adjacency matrix is sparse, it is
more convenient to use a different representation, in which only the non-zero elements
are stored. The most commonly used representation of sparse adjacency matrices is the
so-called ij-form, also known as edge list form.
Definition1.17(Edgelist) The edge list of a graph, also known as ij-form of the adjacency
matrix of the graph, consists of two vectors i and j of integer numbers storing the posi-
tions, i.e. respectively the row and column indices of the ones of the adjacency matrix](https://image.slidesharecdn.com/25355227-250519102959-4e72489d/85/Complex-Networks-Principles-Methods-And-Applications-Vito-Latora-54-320.jpg)
Complex Networks Principles Methods And Applications Vito Latora
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- 6. Complex Networks Principles, Methods and Applications Networks constitute the backbone of complex systems, from the human brain to computer communications, transport infrastructures to online social systems, metabolic reactions to financial markets. Characterising their structure improves our understanding of the physical, biological, economic and social phenomena that shape our world. Rigorous and thorough, this textbook presents a detailed overview of the new theory and methods of network science. Covering algorithms for graph exploration, node ranking and network generation, among the others, the book allows students to experiment with network models and real-world data sets, providing them with a deep understanding of the basics of network theory and its practical applications. Systems of growing complexity are examined in detail, challenging students to increase their level of skill. An engaging pre- sentation of the important principles of network science makes this the perfect reference for researchers and undergraduate and graduate students in physics, mathematics, engineering, biology, neuroscience and social sciences. Vito Latora is Professor of Applied Mathematics and Chair of Complex Systems at Queen Mary University of London. Noted for his research in statistical physics and in complex networks, his current interests include time-varying and multiplex networks, and their applications to socio-economic systems and to the human brain. Vincenzo Nicosia is Lecturer in Networks and Data Analysis at the School of Mathematical Sciences at Queen Mary University of London. His research spans several aspects of net- work structure and dynamics, and his recent interests include multi-layer networks and their applications to big data modelling. Giovanni Russo is Professor of Numerical Analysis in the Department of Mathematics and Computer Science at the University of Catania, Italy, focusing on numerical methods for partial differential equations, with particular application to hyperbolic and kinetic problems.
- 8. Complex Networks Principles, Methods and Applications VITO LATORA Queen Mary University of London VINCENZO NICOSIA Queen Mary University of London GIOVANNI RUSSO University of Catania, Italy
- 9. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107103184 DOI: 10.1017/9781316216002 © Vito Latora, Vincenzo Nicosia and Giovanni Russo 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Latora, Vito, author. | Nicosia, Vincenzo, author. | Russo, Giovanni, author. Title: Complex networks : principles, methods and applications / Vito Latora, Queen Mary University of London, Vincenzo Nicosia, Queen Mary University of London, Giovanni Russo, Università degli Studi di Catania, Italy. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2017. | Includes bibliographical references and index. Identifiers: LCCN 2017026029 | ISBN 9781107103184 (hardback) Subjects: LCSH: Network analysis (Planning) Classification: LCC T57.85 .L36 2017 | DDC 003/.72–dc23 LC record available at https://lccn.loc.gov/2017026029 ISBN 978-1-107-10318-4 Hardback Additional resources for this publication at www.cambridge.org/9781107103184. Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 10. To Giusi, Francesca and Alessandra
- 12. Contents Preface page xi Introduction xii The Backbone of a Complex System xii Complex Networks Are All Around Us xiv Why Study Complex Networks? xv Overview of the Book xvii Acknowledgements xx 1 Graphs and Graph Theory 1 1.1 What Is a Graph? 1 1.2 Directed, Weighted and Bipartite Graphs 9 1.3 Basic Definitions 13 1.4 Trees 17 1.5 Graph Theory and the Bridges of Königsberg 19 1.6 How to Represent a Graph 23 1.7 What We Have Learned and Further Readings 28 Problems 28 2 Centrality Measures 31 2.1 The Importance of Being Central 31 2.2 Connected Graphs and Irreducible Matrices 34 2.3 Degree and Eigenvector Centrality 39 2.4 Measures Based on Shortest Paths 47 2.5 Movie Actors 56 2.6 Group Centrality 62 2.7 What We Have Learned and Further Readings 64 Problems 65 3 Random Graphs 69 3.1 Erdős and Rényi (ER) Models 69 3.2 Degree Distribution 76 3.3 Trees, Cycles and Complete Subgraphs 79 3.4 Giant Connected Component 84 3.5 Scientific Collaboration Networks 90 3.6 Characteristic Path Length 94 vii
- 13. viii Contents 3.7 What We Have Learned and Further Readings 103 Problems 104 4 Small-World Networks 107 4.1 Six Degrees of Separation 107 4.2 The Brain of a Worm 112 4.3 Clustering Coefficient 116 4.4 The Watts–Strogatz (WS) Model 127 4.5 Variations to the Theme 135 4.6 Navigating Small-World Networks 144 4.7 What We Have Learned and Further Readings 148 Problems 148 5 Generalised Random Graphs 151 5.1 The World Wide Web 151 5.2 Power-Law Degree Distributions 161 5.3 The Configuration Model 171 5.4 Random Graphs with Arbitrary Degree Distribution 178 5.5 Scale-Free Random Graphs 184 5.6 Probability Generating Functions 188 5.7 What We Have Learned and Further Readings 202 Problems 204 6 Models of Growing Graphs 206 6.1 Citation Networks and the Linear Preferential Attachment 206 6.2 The Barabási–Albert (BA) Model 215 6.3 The Importance of Being Preferential and Linear 224 6.4 Variations to the Theme 230 6.5 Can Latecomers Make It? The Fitness Model 241 6.6 Optimisation Models 248 6.7 What We Have Learned and Further Readings 252 Problems 253 7 Degree Correlations 257 7.1 The Internet and Other Correlated Networks 257 7.2 Dealing with Correlated Networks 262 7.3 Assortative and Disassortative Networks 268 7.4 Newman’s Correlation Coefficient 275 7.5 Models of Networks with Degree–Degree Correlations 285 7.6 What We Have Learned and Further Readings 290 Problems 291 8 Cycles and Motifs 294 8.1 Counting Cycles 294 8.2 Cycles in Scale-Free Networks 303 8.3 Spatial Networks of Urban Streets 307
- 14. ix Contents 8.4 Transcription Regulation Networks 316 8.5 Motif Analysis 324 8.6 What We Have Learned and Further Readings 329 Problems 330 9 Community Structure 332 9.1 Zachary’s Karate Club 332 9.2 The Spectral Bisection Method 336 9.3 Hierarchical Clustering 342 9.4 The Girvan–Newman Method 349 9.5 Computer Generated Benchmarks 354 9.6 The Modularity 357 9.7 A Local Method 365 9.8 What We Have Learned and Further Readings 369 Problems 371 10 Weighted Networks 374 10.1 Tuning the Interactions 374 10.2 Basic Measures 381 10.3 Motifs and Communities 387 10.4 Growing Weighted Networks 393 10.5 Networks of Stocks in a Financial Market 401 10.6 What We Have Learned and Further Readings 407 Problems 408 Appendices 410 A.1 Problems, Algorithms and Time Complexity 410 A.2 A Simple Introduction to Computational Complexity 420 A.3 Elementary Data Structures 425 A.4 Basic Operations with Sparse Matrices 440 A.5 Eigenvalue and Eigenvector Computation 444 A.6 Computation of Shortest Paths 452 A.7 Computation of Node Betweenness 462 A.8 Component Analysis 467 A.9 Random Sampling 474 A.10 Erdős and Rényi Random Graph Models 485 A.11 The Watts–Strogatz Small-World Model 489 A.12 The Configuration Model 492 A.13 Growing Unweighted Graphs 499 A.14 Random Graphs with Degree–Degree Correlations 506 A.15 Johnson’s Algorithm to Enumerate Cycles 508 A.16 Motifs Analysis 511 A.17 Girvan–Newman Algorithm 515 A.18 Greedy Modularity Optimisation 519 A.19 Label Propagation 524
- 15. x Contents A.20 Kruskal’s Algorithm for Minimum Spanning Tree 528 A.21 Models for Weighted Networks 531 List of Programs 533 References 535 Author Index 550 Index 552
- 16. Preface Social systems, the human brain, the Internet and the World Wide Web are all examples of complex networks, i.e. systems composed of a large number of units interconnected through highly non-trivial patterns of interactions. This book is an introduction to the beau- tiful and multidisciplinary world of complex networks. The readers of the book will be exposed to the fundamental principles, methods and applications of a novel discipline: net- work science. They will learn how to characterise the architecture of a network and model its growth, and will uncover the principles common to networks from different fields. The book covers a large variety of topics including elements of graph theory, social networks and centrality measures, random graphs, small-world and scale-free networks, models of growing graphs and degree–degree correlations, as well as more advanced topics such as motif analysis, community structure and weighted networks. Each chapter presents its main ideas together with the related mathematical definitions, models and algorithms, and makes extensive use of network data sets to explore these ideas. The book contains several practical applications that range from determining the role of an individual in a social network or the importance of a player in a football team, to iden- tifying the sub-areas of a nervous systems or understanding correlations between stocks in a financial market. Thanks to its colloquial style, the extensive use of examples and the accompanying soft- ware tools and network data sets, this book is the ideal university-level textbook for a first module on complex networks. It can also be used as a comprehensive reference for researchers in mathematics, physics, engineering, biology and social sciences, or as a his- torical introduction to the main findings of one of the most active interdisciplinary research fields of the moment. This book is fundamentally on the structure of complex networks, and we hope it will be followed soon by a second book on the different types of dynamical processes that can take place over a complex network. Vito Latora Vincenzo Nicosia Giovanni Russo xi
- 17. Introduction The Backbone of a Complex System Imagine you are invited to a party; you observe what happens in the room when the other guests arrive. They start to talk in small groups, usually of two people, then the groups grow in size, they split, merge again, change shape. Some of the people move from one group to another. Some of them know each other already, while others are introduced by mutual friends at the party. Suppose you are also able to track all of the guests and their movements in space; their head and body gestures, the content of their discussions. Each person is different from the others. Some are more lively and act as the centre of the social gathering: they tell good stories, attract the attention of the others and lead the group conversation. Other individuals are more shy: they stay in smaller groups and prefer to listen to the others. It is also interesting to notice how different genders and ages vary between groups. For instance, there may be groups which are mostly male, others which are mostly female, and groups with a similar proportion of both men and women. The topic of each discussion might even depend on the group composition. Then, when food and beverages arrive, the people move towards the main table. They organise into more or less regular queues, so that the shape of the newly formed groups is different. The individuals rearrange again into new groups sitting at the various tables. Old friends, but also those who have just met at the party, will tend to sit at the same tables. Then, discussions will start again during the dinner, on the same topics as before, or on some new topics. After dinner, when the music begins, we again observe a change in the shape and size of the groups, with the formation of couples and the emergence of collective motion as everybody starts to dance. The social system we have just considered is a typical example of what is known today as a complex system [16, 44]. The study of complex systems is a new science, and so a commonly accepted formal definition of a complex system is still missing. We can roughly say that a complex system is a system made by a large number of single units (individuals, components or agents) interacting in such a way that the behaviour of the system is not a simple combination of the behaviours of the single units. In particular, some collective behaviours emerge without the need for any central control. This is exactly what we have observed by monitoring the evolution of our party with the formation of social groups, and the emergence of discussions on some particular topics. This kind of behaviour is what we find in human societies at various levels, where the interactions of many individuals give rise to the emergence of civilisation, urban forms, cultures and economies. Analogously, animal societies such as, for instance, ant colonies, accomplish a variety of different tasks, xii
- 18. xiii Introduction from nest maintenance to the organisation of food search, without the need for any central control. Let us consider another example of a complex system, certainly the most representative and beautiful one: the human brain. With around 102 billion neurons, each connected by synapses to several thousand other neurons, this is the most complicated organ in our body. Neurons are cells which process and transmit information through electrochemical signals. Although neurons are of different types and shapes, the “integrate-and-fire” mechanism at the core of their dynamics is relatively simple. Each neuron receives synaptic signals, which can be either excitatory or inhibitory, from other neurons. These signals are then integrated and, provided the combined excitation received is larger than a certain threshold, the neuron fires. This firing generates an electric signal, called an action potential, which propagates through synapses to other neurons. Notwithstanding the extreme simplicity of the interactions, the brain self-organises collective behaviours which are difficult to pre- dict from our knowledge of the dynamics of its individual elements. From an avalanche of simple integrate-and-fire interactions, the neurons of the brain are capable of organising a large variety of wonderful emerging behaviours. For instance, sensory neurons coordinate the response of the body to touch, light, sounds and other external stimuli. Motor neurons are in charge of the body’s movement by controlling the contraction or relaxation of the muscles. Neurons of the prefrontal cortex are responsible for reasoning and abstract think- ing, while neurons of the limbic system are involved in processing social and emotional information. Over the years, the main focus of scientific research has been on the characteristics of the individual components of a complex system and to understand the details of their interac- tions. We can now say that we have learnt a lot about the different types of nerve cells and the ways they communicate with each other through electrochemical signals. Analogously, we know how the individuals of a social group communicate through both spoken and body language, and the basic rules through which they learn from one another and form or match their opinions. We also understand the basic mechanisms of interactions in social animals; we know that, for example, ants produce chemicals, known as pheromones, through which they communicate, organise their work and mark the location of food. However, there is another very important, and in no way trivial, aspect of complex systems which has been explored less. This has to do with the structure of the interactions among the units of a complex system: which unit is connected to which others. For instance, if we look at the connections between the neurons in the brain and construct a similar network whose nodes are neurons and the links are the synapses which connect them, we find that such a net- work has some special mathematical properties which are fundamental for the functioning of the brain. For instance, it is always possible to move from one node to any other in a small number of steps, and, particularly if the two nodes belong to the same brain area, there are many alternative paths between them. Analogously, if we take snapshots of who is talking to whom at our hypothetical party, we immediately see that the architecture of the obtained networks, whose nodes represent individuals and links stand for interactions, plays a crucial role in both the propagation of information and the emergence of collective behaviours. Some sub-structures of a network propagate information faster than others; this means that nodes occupying strategic positions will have better access to the resources
- 19. xiv Introduction of the system. In practice, what also matters in a complex system, and it matters a lot, is the backbone of the system, or, in other words, the architecture of the network of interac- tions. It is precisely on these complex networks, i.e. on the networks of the various complex systems that populate our world, that we will be focusing in this book. Complex Networks Are All Around Us Networks permeate all aspects of our life and constitute the backbone of our modern world. To understand this, think for a moment about what you might do in a typical day. When you get up early in the morning and turn on the light in your bedroom, you are connected to the electrical power grid, a network whose nodes are either power stations or users, while links are copper cables which transport electric current. Then you meet the people of your family. They are part of your social network whose nodes are people and links stand for kinship, friendship or acquaintance. When you take a shower and cook your breakfast you are respectively using a water distribution network, whose nodes are water stations, reservoirs, pumping stations and homes, and links are pipes, and a gas distribution network. If you go to work by car you are moving in the street network of your city, whose nodes are intersections and links are streets. If you take the underground then you make use of a transportation network, whose nodes are the stations and links are route segments. When you arrive at your office you turn on your laptop, whose internal circuits form a complicated microscopic network of logic gates, and connect it to the Internet, a worldwide network of computers and routers linked by physical or logical connections. Then you check your emails, which belong to an email communication network, whose nodes are people and links indicate email exchanges among them. When you meet a colleague, you and your colleague form part of a collaboration network, in which an edge exists between two persons if they have collaborated on the same project or coauthored a paper. Your colleagues tell you that your last paper has got its first hundred citations. Have you ever thought of the fact that your papers belong to a citation network, where the nodes represent papers, and links are citations? At lunchtime you read the news on the website of your preferred newspaper: in doing this you access the World Wide Web, a huge global information network whose nodes are webpages and edges are clickable hyperlinks between pages. You will almost surely then check your Facebook account, a typical example of an online social network, then maybe have a look at the daily trending topics on Twitter, an information network whose nodes are people and links are the “following” relations. Your working day proceeds quietly, as usual. Around 4:00pm you receive a phone call from your friend John, and you immediately think about the phone call network, where two individuals are connected by a link if they have exchanged a phone call. John invites you and your family for a weekend at his cottage near the lake. Lakes are home to a variety of fishes, insects and animals which are part of a food web network, whose links indicate predation among different species. And while John tells you about the beauty of his cottage, an image of a mountain lake gradually forms in your mind, and you can see a
- 20. xv Introduction white waterfall cascading down a cliff, and a stream flowing quietly through a green valley. There is no need to say that “lake”, “waterfall”, “white”, “stream”, “cliff”, “valley” and “green” form a network of words associations, in which a link exists between two words if these words are often associated with each other in our minds. Before leaving the office, you book a flight to go to Prague for a conference. Obviously, also the air transportation system is a network, whose nodes are airports and links are airline routes. When you drive back home you feel a bit tired and you think of the various networks in our body, from the network of blood vessels which transports blood to our organs to the intricate set of relationships among genes and proteins which allow the perfect functioning of the cells of our body. Examples of these genetic networks are the transcription regula- tion networks in which the nodes are genes and links represent transcription regulation of a gene by the transcription factor produced by another gene, protein interaction networks whose nodes are protein and there is a link between two proteins if they bind together to perform complex cellular functions, and metabolic networks where nodes are chemicals, and links represent chemical reactions. During dinner you hear on the news that the total export for your country has decreased by 2.3% this year; the system of commercial relationships among countries can be seen as a network, in which links indicate import/export activities. Then you watch a movie on your sofa: you can construct an actor collaboration network where nodes represent movie actors and links are formed if two actors have appeared in the same movie. Exhausted, you go to bed and fall asleep while images of networks of all kinds still twist and dance in your mind, which is, after all, the marvellous combination of the activity of billions of neurons and trillions of synapses in your brain network. Yet another network. Why Study Complex Networks? In the late 1990s two research papers radically changed our view on complex systems, moving the attention of the scientific community to the study of the architecture of a com- plex system and creating an entire new research field known today as network science. The first paper, authored by Duncan Watts and Steven Strogatz, was published in the journal Nature in 1998 and was about small-world networks [311]. The second one, on scale-free networks, appeared one year later in Science and was authored by Albert-László Barabási and Réka Albert [19]. The two papers provided clear indications, from different angles, that: • the networks of real-world complex systems have non-trivial structures and are very different from lattices or random graphs, which were instead the standard networks commonly used in all the current models of a complex system. • some structural properties are universal, i.e. are common to networks as diverse as those of biological, social and man-made systems. • the structure of the network plays a major role in the dynamics of a complex system and characterises both the emergence and the properties of its collective behaviours.
- 21. xvi Introduction Table 1 A list of the real-world complex networks that will be studied in this book. For each network, we report the chapter of the book where the corresponding data set will be introduced and analysed. Complex networks Nodes Links Chapter Elisa’s kindergarten Children Friendships 1 Actor collaboration networks Movie actors Co-acting in a film 2 Co-authorship networks Scientists Co-authoring a paper 3 Citation networks Scientific papers Citations 6 Zachary’s karate club Club members Friendships 9 C. elegans neural network Neurons Synapses 4 Transcription regulation networks Genes Transcription regulation 8 World Wide Web Web pages Hyperlinks 5 Internet Routers Optical fibre cables 7 Urban street networks Street crossings Streets 8 Air transport network Airports Flights 10 Financial markets Stocks Time correlations 10 Both works were motivated by the empirical analysis of real-world systems. Four net- works were introduced and studied in these two papers. Namely, the neural system of a few-millimetres-long worm known as the C. elegans, a social network describing how actors collaborate in movies, and two man-made networks: the US electrical power grid and a sample of the World Wide Web. During the last decade, new technologies and increasing computing power have made new data available and stimulated the exploration of several other complex networks from the real world. A long series of papers has followed, with the analysis of new and ever larger networks, and the introduction of novel measures and models to characterise and reproduce the structure of these real-world systems. Table 1 shows only a small sample of the networks that have appeared in the literature, namely those that will be explicitly studied in this book, together with the chapter where they will be considered. Notice that the table includes different types of networks. Namely, five networks representing three different types of social interactions (namely friendships, collaborations and citations), two biological systems (respectively a neural and a gene net- work) and five man-made networks (from transportation and communication systems to a network of correlations among financial stocks). The ubiquitousness of networks in nature, technology and society has been the principal motivation behind the systematic quantitative study of their structure, their formation and their evolution. And this is also the main reason why a student of any scientific discipline should be interested in complex networks. In fact, if we want to master the interconnected world we live in, we need to understand the structure of the networks around us. We have to learn the basic principles governing the architecture of networks from different fields, and study how to model their growth. It is also important to mention the high interdisciplinarity of network science. Today, research on complex networks involves scientists with expertise in areas such as mathe- matics, physics, computer science, biology, neuroscience and social science, often working
- 22. xvii Introduction 1995 2000 2005 2010 2015 year 0 2000 4000 6000 8000 10000 # citations WS BA 1995 2000 2005 2010 2015 year 200 400 600 800 # papers t Fig. 1 Left panel: number of citations received over the years by the 1998 Watts and Strogatz (WS) article on small-world networks and by the 1999 Barabási and Albert (BA) article on scale-free networks. Right panel: number of papers on complex networks that appeared each year in the public preprint archive arXiv.org. side by side. Because of its interdisciplinary nature, the generality of the results obtained, and the wide variety of possible applications, network science is considered today a necessary ingredient in the background of any modern scientist. Finally, it is not difficult to understand that complex networks have become one of the hottest research fields in science. This is confirmed by the attention and the huge number of citations received by Watts and Strogatz, and by Barabási and Albert, in the papers mentioned above. The temporal profiles reported in the left panel of Figure 1 show the exponential increase in the number of citations of these two papers since their publication. The two papers have today about 10,000 citations each and, as already mentioned, have opened a new research field stimulating interest for complex networks in the scientific community and triggering an avalanche of scientific publications on related topics. The right panel of Figure 1 reports the number of papers published each year after 1998 on the well-known public preprint archive arXiv.org with the term “complex networks” in their title or abstract. Notice that this number has gone up by a factor of 10 in the last ten years, with almost a thousand papers on the topic published in the archive in the year 2013. The explosion of interest in complex networks is not limited to the scientific community, but has become a cultural phenomenon with the publications of various popular science books on the subject. Overview of the Book This book is mainly intended as a textbook for an introductory course on complex networks for students in physics, mathematics, engineering and computer science, and for the more mathematically oriented students in biology and social sciences. The main purpose of the book is to expose the readers to the fundamental ideas of network science, and to provide them with the basic tools necessary to start exploring the world of complex networks. We also hope that the book will be able to transmit to the reader our passion for this stimulating new interdisciplinary subject.
- 23. xviii Introduction The standard tools to study complex networks are a mixture of mathematical and com- putational methods. They require some basic knowledge of graph theory, probability, differential equations, data structures and algorithms, which will be introduced in this book from scratch and in a friendly way. Also, network theory has found many interest- ing applications in several different fields, including social sciences, biology, neuroscience and technology. In the book we have therefore included a large variety of examples to emphasise the power of network science. This book is essentially on the structure of com- plex networks, since we have decided that the detailed treatment of the different types of dynamical processes that can take place over a complex network should be left to another book, which will follow this one. The book is organised into ten chapters. The first six chapters (Chapters 1–6) form the core of the book. They introduce the main concepts of network science and the basic measures and models used to characterise and reproduce the structure of various com- plex networks. The remaining four chapters (Chapters 7–10) cover more advanced topics that could be skipped by a lecturer who wants to teach a short course based on the book. In Chapter 1 we introduce some basic definitions from graph theory, setting up the lan- guage we will need for the remainder of the book. The aim of the chapter is to show that complex network theory is deeply grounded in a much older mathematical discipline, namely graph theory. In Chapter 2 we focus on the concept of centrality, along with some of the related mea- sures originally introduced in the context of social network analysis, which are today used extensively in the identification of the key components of any complex system, not only of social networks. We will see some of the measures at work, using them to quantify the centrality of movie actors in the actor collaboration network. Chapter 3 is where we first discuss network models. In this chapter we introduce the classical random graph models proposed by Erdős and Rényi (ER) in the late 1950s, in which the edges are randomly distributed among the nodes with a uniform probability. This allows us to analytically derive some important properties such as, for instance, the number and order of graph components in a random graph, and to use ER models as term of comparison to investigate scientific collaboration networks. We will also show that the average distance between two nodes in ER random graphs increases only logarithmically with the number of nodes. In Chapter 4 we see that in real-world systems, such as the neural network of the C. ele- gans or the movie actor collaboration network, the neighbours of a randomly chosen node are directly linked to each other much more frequently than would occur in a purely ran- dom network, giving rise to the presence of many triangles. In order to quantify this, we introduce the so-called clustering coefficient. We then discuss the Watts and Strogatz (WS) small-world model to construct networks with both a small average distance between nodes and a high clustering coefficient. In Chapter 5 the focus is on how the degree k is distributed among the nodes of a network. We start by considering the graph of the World Wide Web and by showing that it is a scale-free network, i.e. it has a power–law degree distribution pk ∼ k−γ with an exponent γ ∈ [2, 3]. This is a property shared by many other networks, while neither ER random graphs nor the WS model can reproduce such a feature. Hence, we introduce the so-called
- 24. xix Introduction configuration model which generalises ER random graph models to incorporate arbitrary degree distributions. In Chapter 6 we show that real networks are not static, but grow over time with the addition of new nodes and links. We illustrate this by studying the basic mechanisms of growth in citation networks. We then consider whether it is possible to produce scale-free degree distributions by modelling the dynamical evolution of the network. For this purpose we introduce the Barabási–Albert model, in which newly arriving nodes select and link existing nodes with a probability linearly proportional to their degree. We also consider some extensions and modifications of this model. In the last four chapters we cover more advanced topics on the structure of complex networks. Chapter 7 is about networks with degree–degree correlations, i.e. networks such that the probability that an edge departing from a node of degree k arrives at a node of degree k is a function both of k and of k. Degree–degree correlations are indeed present in real- world networks, such as the Internet, and can be either positive (assortative) or negative (disassortative). In the first case, networks with small degree preferentially link to other low-degree nodes, while in the second case they link preferentially to high-degree ones. In this chapter we will learn how to take degree–degree correlations into account, and how to model correlated networks. In Chapter 8 we deal with the cycles and other small subgraphs known as motifs which occur in most networks more frequently than they would in random graphs. We consider two applications: firstly we count the number of short cycles in urban street networks of different cities from all over the world; secondly we will perform a motif analysis of the transcription network of the bacterium E. coli. Chapter 9 is about network mesoscale structures known as community structures. Com- munities are groups of nodes that are more tightly connected to each other than to other nodes. In this chapter we will discuss various methods to find meaningful divisions of the nodes of a network into communities. As a benchmark we will use a real network, the Zachary’s karate club, where communities are known a priori, and also models to construct networks with a tunable presence of communities. In Chapter 10 we deal with weighted networks, where each link carries a numerical value quantifying the intensity of the connection. We will introduce the basic measures used to characterise and classify weighted networks, and we will discuss some of the models of weighted networks that reproduce empirically observed topology–weight correlations. We will study in detail two weighted networks, namely the US air transport network and a network of financial stocks. Finally, the book’s Appendix contains a detailed description of all the main graph algo- rithms discussed in the various chapters of the book, from those to find shortest paths, components or community structures in a graph, to those to generate random graphs or scale-free networks. All the algorithms are presented in a C-like pseudocode format which allows us to understand their basic structure without the unnecessary complication of a programming language. The organisation of this textbook is another reason why it is different from all the other existing books on networks. We have in fact avoided the widely adopted separation of
- 25. xx Introduction the material in theory and applications, or the division of the book into separate chap- ters respectively dealing with empirical studies of real-world networks, network measures, models, processes and computer algorithms. Each chapter in our book discusses, at the same time, real-world networks, measures, models and algorithms while, as said before, we have left the study of processes on networks to an entire book, which will follow this one. Each chapter of this book presents a new idea or network property: it introduces a network data set, proposes a set of mathematical quantities to investigate such a network, describes a series of network models to reproduce the observed properties, and also points to the related algorithms. In this way, the presentation follows the same path of the current research in the field, and we hope that it will result in a more logical and more entertaining text. Although the main focus of this book is on the mathematical modelling of complex networks, we also wanted the reader to have direct access to both the most famous data sets of real-world networks and to the numerical algorithms to compute network proper- ties and to construct networks. For this reason, the data sets of all the real-world networks listed in Table 1 are introduced and illustrated in special DATA SET Boxes, usually one for each chapter of the book, and can be downloaded from the book’s webpage at www. complex-networks.net. On the same webpage the reader can also find an implemen- tation in the C language of the graph algorithms illustrated in the Appendix (in C-like pseudocode format). We are sure that the student will enjoy experimenting directly on real- world networks, and will benefit from the possibility of reproducing all of the numerical results presented throughout the book. The style of the book is informal and the ideas are illustrated with examples and appli- cations drawn from the recent research literature and from different disciplines. Of course, the problem with such examples is that no-one can simultaneously be an expert in social sciences, biology and computer science, so in each of these cases we will set up the relative background from scratch. We hope that it will be instructive, and also fun, to see the con- nections between different fields. Finally, all the mathematics is thoroughly explained, and we have decided never to hide the details, difficulties and sometimes also the incoherences of a science still in its infancy. Acknowledgements Writing this book has been a long process which started almost ten years ago. The book has grown from the notes of various university courses, first taught at the Physics Department of the University of Catania and at the Scuola Superiore di Catania in Italy, and more recently to the students of the Masters in “Network Science” at Queen Mary University of London. The book would not have been the same without the interactions with the students we have met at the different stages of the writing process, and their scientific curiosity. Special thanks go to Alessio Cardillo, Roberta Sinatra, Salvatore Scellato and the other students and alumni of Scuola Superiore, Salvatore Assenza, Leonardo Bellocchi, Filippo Caruso, Paolo Crucitti, Manlio De Domenico, Beniamino Guerra, Ivano Lodato, Sandro Meloni,
- 26. xxi Introduction Andrea Santoro and Federico Spada, and to the students of the Masters in “Network Science”. We acknowledge the great support of the members of the Laboratory of Complex Systems at Scuola Superiore di Catania, Giuseppe Angilella, Vincenza Barresi, Arturo Buscarino, Daniele Condorelli, Luigi Fortuna, Mattia Frasca, Jesús Gómez-Gardeñes and Giovanni Piccitto; of our colleagues in the Complex Systems and Networks research group at the School of Mathematical Sciences of Queen Mary University of London, David Arrowsmith, Oscar Bandtlow, Christian Beck, Ginestra Bianconi, Leon Danon, Lucas Lacasa, Rosemary Harris, Wolfram Just; and of the PhD students Federico Bat- tiston, Moreno Bonaventura, Massimo Cavallaro, Valerio Ciotti, Iacopo Iacovacci, Iacopo Iacopini, Daniele Petrone and Oliver Williams. We are greatly indebted to our colleagues Elsa Arcaute, Alex Arenas, Domenico Asprone, Tomaso Aste, Fabio Babiloni, Franco Bagnoli, Andrea Baronchelli, Marc Barthélemy, Mike Batty, Armando Bazzani, Stefano Boccaletti, Marián Boguñá, Ed Bullmore, Guido Caldarelli, Domenico Cantone, Gastone Castellani, Mario Chavez, Vit- toria Colizza, Regino Criado, Fabrizio De Vico Fallani, Marina Diakonova, Albert Dí az-Guilera, Tiziana Di Matteo, Ernesto Estrada, Tim Evans, Alfredo Ferro, Alessan- dro Fiasconaro, Alessandro Flammini, Santo Fortunato, Andrea Giansanti, Georg von Graevenitz, Paolo Grigolini, Peter Grindrod, Des Higham, Giulia Iori, Henrik Jensen, Renaud Lambiotte, Pietro Lió, Vittorio Loreto, Paolo de Los Rios, Fabrizio Lillo, Carmelo Maccarrone, Athen Ma, Sabato Manfredi, Massimo Marchiori, Cecilia Mascolo, Rosario Mantegna, Andrea Migliano, Raúl Mondragón, Yamir Moreno, Mirco Musolesi, Giuseppe Nicosia, Pietro Panzarasa, Nicola Perra, Alessandro Pluchino, Giuseppe Politi, Sergio Porta, Mason Porter, Giovanni Petri, Gaetano Quattrocchi, Daniele Quercia, Filippo Radic- chi, Andrea Rapisarda, Daniel Remondini, Alberto Robledo, Miguel Romance, Vittorio Rosato, Martin Rosvall, Maxi San Miguel, Corrado Santoro, M. Ángeles Serrano, Simone Severini, Emanuele Strano, Michael Szell, Bosiljka Tadić, Constantino Tsallis, Stefan Thurner, Hugo Touchette, Petra Vértes, Lucio Vinicius for the many stimulating discus- sions and for their useful comments. We thank in particular Olle Persson, Luciano Da Fontoura Costa, Vittoria Colizza, and Rosario Mantegna for having provided us with their network data sets. We acknowledge the European Commission project LASAGNE (multi-LAyer SpA- tiotemporal Generalized NEtworks), Grant 318132 (STREP), the EPSRC project GALE, Grant EP/K020633/1, and INFN FB11/TO61, which have supported and made possible our work at the various stages of this project. Finally, we thank our families for their never-ending support and encouragement.
- 28. Life is all mind, heart and relations Salvatore Latora Philosopher
- 30. 1 Graphs and Graph Theory Graphs are the mathematical objects used to represent networks, and graph theory is the branch of mathematics that deals with the study of graphs. Graph theory has a long his- tory. The notion of the graph was introduced for the first time in 1763 by Euler, to settle a famous unsolved problem of his time: the so-called Königsberg bridge problem. It is no coincidence that the first paper on graph theory arose from the need to solve a problem from the real world. Also subsequent work in graph theory by Kirchhoff and Cayley had its root in the physical world. For instance, Kirchhoff’s investigations into electric circuits led to his development of a set of basic concepts and theorems concerning trees in graphs. Nowa- days, graph theory is a well-established discipline which is commonly used in areas as diverse as computer science, sociology and biology. To give some examples, graph theory helps us to schedule airplane routing and has solved problems such as finding the max- imum flow per unit time from a source to a sink in a network of pipes, or colouring the regions of a map using the minimum number of different colours so that no neighbouring regions are coloured the same way. In this chapter we introduce the basic definitions, set- ting up the language we will need in the rest of the book. We also present the first data set of a real network in this book, namely Elisa’s kindergarten network. The two final sections are devoted to, respectively, the proof of the Euler theorem and the description of a graph as an array of numbers. 1.1 What Is a Graph? The natural framework for the exact mathematical treatment of a complex network is a branch of discrete mathematics known as graph theory [48, 47, 313, 150, 272, 144]. Dis- crete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, i.e. made up of distinct parts, not supporting or requiring the notion of continuity. Most of the objects studied in discrete mathematics are count- able sets, such as integers and finite graphs. Discrete mathematics has become popular in recent decades because of its applications to computer science. In fact, concepts and nota- tions from discrete mathematics are often useful to study or describe objects or problems in computer algorithms and programming languages. The concept of the graph is better introduced by the two following examples. 1
- 31. 2 Graphs and Graph Theory Example 1.1 (Friends at a party) Seven people have been invited to a party. Their names are Adam, Betty, Cindy, David, Elizabeth, Fred and George. Before meeting at the party, Adam knew Betty, David and Fred; Cindy knew Betty, David, Elizabeth and George; David knew Betty (and, of course, Adam and Cindy); Fred knew Betty (and, of course, Adam). The network of acquaintances can be easily represented by identifying a person by a point, and a relation as a link between two points: if two points are connected by a link, this means that they knew each other before the party. A pictorial representation of the acquaintance relationships among the seven persons is illustrated in panel (a) of the figure. Note the symmetry of the link between two persons, which reflects that if person “A” knows person “B”, then person “B” knows person “A”. Also note that the only thing which is relevant in the diagram is whether two persons are connected or not. The same acquaintance network can be represented, for example, as in panel (b). Note that in this representation the more “relevant” role of Betty and Cindy over, for example, George or Fred, is more immediate. Example 1.2 (The map of Europe) The map in the figure shows 23 of Europe’s approx- imately 50 countries. Each country is shown with a different shade of grey, so that from
- 32. 3 1.1 What Is a Graph? the image we can easily distinguish the borders between any two nations. Let us suppose now that we are interested not in the precise shape and geographical position of each coun- try, but simply in which nations have common borders. We can thus transform the map into a much simpler representation that preserves entirely that information. In order to do so we need, with a little bit of abstraction, to transform each nation into a point. We can then place the points in the plane as we want, although it can be convenient to maintain similar positions to those of the corresponding nations in the map. Finally, we connect two points with a line if there is a common boundary between the corresponding two nations. Notice that in this particular case, due to the placement of the points in the plane, it is possible to draw all the connections with no line intersections. The mathematical entity used to represent the existence or the absence of links among various objects is called the graph. A graph is defined by giving a set of elements, the graph nodes, and a set of links that join some (or all) pairs of nodes. In Example 1.1 we are using a graph to represent a network of social acquaintances. The people invited at a party are the nodes of the graph, while the existence of acquaintances between two persons defines the links in the graph. In Example 1.1 the nodes of the graph are the countries of the European Union, while a link between two countries indicates that there is a common boundary between them. A graph is defined in mathematical terms in the following way: Definition 1.1 (Undirected graph) A graph, more specifically an undirected graph, G ≡ (N, L), consists of two sets, N = ∅ and L. The elements of N ≡ {n1, n2, . . . , nN} are distinct and are called the nodes (or vertices, or points) of the graph G. The elements of L ≡ {l1, l2, . . . , lK} are distinct unordered pairs of distinct elements of N, and are called links (or edges, or lines). The number of vertices N ≡ N[G] = |N|, where the symbol | · | denotes the cardinality of a set, is usually referred as the order of G, while the number of edges K ≡ K[G] = |L| is the size of G.[1] A node is usually referred to by a label that identifies it. The label is often an integer index from 1 to N, representing the order of the node in the set N. We shall use this labelling throughout the book, unless otherwise stated. In an undirected graph, each of the links is defined by a pair of nodes, i and j, and is denoted as (i, j) or (j, i). In some cases we also denote the link as lij or lji. The link is said to be incident in nodes i and j, or to join the two nodes; the two nodes i and j are referred to as the end-nodes of link (i, j). Two nodes joined by a link are referred to as adjacent or neighbouring. As shown in Example 1.1, the usual way to picture a graph is by drawing a dot or a small circle for each node, and joining two dots by a line if the two corresponding nodes are connected by an edge. How these dots and lines are drawn in the page is in principle irrelevant, as is the length of the lines. The only thing that matters in a graph is which pairs of nodes form a link and which ones do not. However, the choice of a clear drawing can be [1] Sometimes, especially in the physical literature, the word size is associated with the number of nodes, rather than with the number of links. We prefer to consider K as the size of the graph. However, in many cases of interest, the number of links K is proportional to the number of nodes N, and therefore the concept of size of a graph can equally well be represented by the number of its nodes N or by the number of its edges K.
- 33. 4 Graphs and Graph Theory very important in making the properties of the graph easy to read. Of course, the quality and usefulness of a particular way to draw a graph depends on the type of graph and on the purpose for which the drawing is generated and, although there is no general prescription, there are various standard drawing setups and different algorithms for drawing graphs that can be used and compared. Some of them are illustrated in Box 1.1. Figure 1.1 shows four examples of small undirected graphs. Graph G1 is made of N = 5 nodes and K = 4 edges. Notice that any pair of nodes of this graph can be connected in only one way. As we shall see later in detail, such a graph is called a tree. Graphs G2 has N = K = 4. By starting from one node, say node 1, one can go to all the other nodes 2, 3, 4, and back again to 1, by visiting each node and each link just once, except of course node 1, which is visited twice, being both the starting and ending node. As we shall see, we say that the graph G2 contains a cycle. The same can be said about graph G3. Graph G3 contains an isolated node and three nodes connected by three links. We say that graphs G1 and G2 are connected, in the sense that any node can be reached, starting from any other node, by “walking” on the graph, while graph G3 is not. Notice that, in the definition of graph given above, we deliberately avoided loops, i.e. links from a node to itself, and multiple edges, i.e. pairs of nodes connected by more than one link. Graphs with either of these elements are called multigraphs [48, 47, 308]. An example of multigraph is G4 in Figure 1.1. In such a multigraph, node 1 is connected to itself by a loop, and it is connected to node 3 by two links. In this book, we will deal with graphs rather than multigraphs, unless otherwise stated. For a graph G of order N, the number of edges K is at least 0, in which case the graph is formed by N isolated nodes, and at most N(N − 1)/2, when all the nodes are pairwise adjacent. The ratio between the actual number of edges K and its maximum possible num- ber N(N − 1)/2 is known as the density of G. A graph with N nodes and no edges has zero t Fig. 1.1 Some examples of undirected graphs, namely a tree, G1; two graphs containing cycles, G2 and G3; and an undirected multigraph, G4.
- 34. 5 1.1 What Is a Graph? Box 1.1 Graph Drawing A good drawing can be very helpful to highlight the properties of a graph. In one standard setup, the so called circular layout, the nodes are placed on a circle and the edges are drawn across the circle. In another set-up, known as the spring model, the nodes and links are positioned in the plane by assuming the graph is a physical system of unit masses (the nodes) connected by springs (the links). An example is shown in the figure below, where the same graph is drawn using a circular layout (left) and a spring-based layout (right) based on the Kamada–Kawaialgorithm [173]. By nature, springs attract their endpoints when stretched and repel their endpoints when compressed. In this way, adjacent nodes on the graph are moved closer in space and, by looking for the equilibrium conditions, we get a layout where edges are short lines, and edge crossings with other nodes and edges are minimised. There are many software packages specifically focused on graph visualisation, including Pajek (http://mrvar.fdv.uni-lj.si/pajek/), Gephi (https://gephi.org/) and GraphViz (http://www.graphviz.org/). Moreover, most of the libraries for network analysis, including NetworkX (https://networkx.github.io/), iGraph (http://igraph.org/) and SNAP (Stanford Network Analysis Platform, http://snap.stanford.edu/), support differ- ent algorithms for network visualisation. density and is said to be empty, while a graph with K = N(N −1)/2 edges, denoted as KN, has density equal to 1 and is said to be complete. The complete graphs with N = 3, N = 4 and N = 5 respectively, are illustrated in Figure 1.2. In particular, K3 is called a triangle, and in the rest of this book will also be indicated by the symbol . As we shall see, we are often interested in the asymptotic properties of graphs when the order N becomes larger and larger. The maximum number of edges in a graph scales as N2. If the actual number of edges in a sequence of graphs of increasing number of nodes scales as N2, then the graphs of the sequence are called dense. It is often the case that the number of edges in a graph of a given sequence scales much more slowly than N2. In this case we say that the graphs are sparse. We will now focus on how to compare graphs with the same order and size. Two graphs G1 = (N1, L1) and G2 = (N2, L2) are the same graph if N1 = N2 and L1 = L2; that is, if both their node sets and their edge sets (i.e. the sets of unordered pairs of nodes defining L) are the same. In this case, we write G1 = G2. For example, graphs (a) and (b) in Figure 1.3
- 35. 6 Graphs and Graph Theory t Fig. 1.2 Complete graphs respectively with three, four and five nodes. t Fig. 1.3 Isomorphism of graphs. Graphs (a) and (b) are the same graph, since their edges are the same. Graphs (b) and (c) are isomorphic, since there is a bijection between the nodes that preserves the edge set. are the same. Note that the position of the nodes in the picture has no relevance, nor does the shape or length of the edges. Two graphs that are not the same can nevertheless be isomorphic. Definition 1.2 (Isomorphism) Two graphs, G1 = (N1, L1) and G2 = (N2, L2), of the same order and size, are said to be isomorphic if there exists a bijection φ : N1 → N2, such that (u, v) ∈ L1 iff (φ(u), φ(v)) ∈ L2. The bijection φ is called an isomorphism. In other words, G1 and G2 are isomorphic if and only if a one-to-one correspondence between the two vertex sets N1, N2, which preserves adjacency, can be found. In this case we write G1 G2. Isomorphism is an equivalence relation, in the sense that it is reflexive, symmetric and transitive. This means that, given any three graphs G1, G2, G3, we have G1 G1, G1 G2 ⇒ G2 G2, and finally G1 G2 and G2 G3 ⇒ G1 G3. For example, graph (c) in Figure 1.3 is not the same as graphs (a) and (b), but it is isomorphic to (a) and (b). In fact, the bijection φ(1) = 1, φ(2) = 2, φ(3) = 4, and φ(4) = 3 between the set of nodes of graph (c) and that of graph (a) satisfies the property required in Definition 1.2. It is easy to show that, once the nodes of two graphs of the same order are labelled by integers from 1 to N, a bijection φ : N1 → N2 can be always represented as a permutation of the node labels. For instance, the bijection just considered corresponds to the permutation of node 3 and node 4. In all the graphs we have seen so far, a label is attached to each node, and iden- tifies it. Such graphs are called labelled graphs. Sometimes, one is interested in the relation between nodes and their connections irrespective of the name of the nodes. In
- 36. 7 1.1 What Is a Graph? t Fig. 1.4 Two unlabelled graphs, namely the cycle C4 and the star graph S4, and one possible labelling of such two graphs. this case, no label is attached to the nodes, and the graph itself is said to be unlabelled. Figure 1.4 shows two examples of unlabelled graphs with N = 4 nodes, namely the cycle, usually indicated as C4, and the star graph with a central node and three links, S4, and two possible labellings of their nodes. Since the same unlabelled graph can be represented in several different ways, how can we state that all these representations cor- respond to the same graph? By definition, two unlabelled graphs are the same if it is possible to label them in such a way that they are the same labelled graph. In particu- lar, if two labelled graphs are isomorphic, then the corresponding unlabelled graphs are the same. It is easy to establish whether two labelled graphs with the same number of nodes and edges are the same, since it is sufficient to compare the ordered pairs that define their edges. However, it is difficult to check whether two unlabelled graphs are isomorphic, because there are N! possible ways to label the N nodes of a graph. In graph theory this is known as the isomorphism problem, and to date, there are no known algorithms to check if two generic graphs are isomorphic in polynomial time. Another definition which has to do with the permutation of the nodes of a graph, and is useful to characterise its symmetry, is that of graph automorphism. Definition 1.3 (Automorphism) Given a graph G = (N, L), an automorphism of G is a permutation φ : N → N of the vertices of G so that if (u, v) ∈ L then (φ(u), φ(v)) ∈ L. The number of different automorphisms of G is denoted as aG. In other words, an automorphism is an isomorphism of a graph on itself. Consider the first labelled graph in Figure 1.4. The simplest automorphism is the one that keeps the node labels unchanged and produces the same labelled graph, shown as the first graph in Figure 1.5. Another example of automorphism is given by φ(1) = 4, φ(2) = 1, φ(3) = 2, φ(4) = 3. Note that this automorphism can be compactly represented by the permutation (1, 2, 3, 4) → (4, 1, 2, 3). The action of such automorphism would produce the second graph shown in Figure 1.5. There are eight distinct permutations of the labels (1, 2, 3, 4) which change the first graph into an isomorphic one. The graph C4 has therefore aC4 = 8. The figure shows all possible automorphisms. Note that the permutation (1, 2, 3, 4) → (1, 3, 2, 4) is not an automorphism of the graph, because while (1, 2) ∈ L, (φ(1), φ(2)) = (1, 3) / ∈ L. Analogously, it is easy to prove that the number of different automorphisms of a triangle C3 = K3 is a = 6, and more in general, for a cycle of N nodes, CN, we have aCN = 2N.
- 37. 8 Graphs and Graph Theory t Fig. 1.5 All possible automorphisms of graph C4 in Figure 1.4. Example 1.3 Consider the star graph S4 with a central node and three links shown in Figure 1.4. There are six automorphisms, corresponding to the following transformations: identity, rotation by 120◦ counterclockwise, rotation by 120◦ clockwise and three specular reflec- tions, respectively around edge (1, 2), (1, 3), (1, 4). There are no more automorphisms, because in all permutations, node 1 has to remain fixed. Therefore, the number aG of pos- sible automorphisms is given by the number of permutations of the three remaining labels, that is, 3! = 6. Finally, we consider some basic operations to produce new graphs from old ones, for instance, by merging together two graphs or by considering only a portion of a given graph. Let us start by introducing the definition of the union of two graphs. Let G1 = (N1, L1) and G2 = (N2, L2) be two graphs. We define graph G = (N, L), where N = N1 ∪N2 and L = L1 ∪ L2, as the union of G1 and G2, and we denote it as G = G1 + G2. A concept that will be very useful in the following is that of subgraph of a given graph. Definition 1.4 (Subgraph) A subgraph of G = (N, L) is a graph G = (N, L) such that N ⊆ N and L ⊆ L. If G contains all links of G that join two nodes in N, then G is said to be the subgraph induced or generated by N, and is denoted as G = G[N]. Figure 1.6 shows some examples of subgraphs. A subgraph is said to be maximal with respect to a given property if it cannot be extended without losing that property. For example, the subgraph induced by nodes 2, 3, 4, 6 in Figure 1.6 is the maximal complete subgraph of order four of graph G. Of particular relevance for some of the definitions given in the following is the subgraph of the neighbours of a given node i, denoted as Gi. Gi is defined as the subgraph induced by Ni, the set of nodes adjacent to i, i.e. Gi = G[Ni]. In Figure 1.6, graph (c) represents the graph G6, induced by the neighbours of node 6. Let G = (N, L), and let s ∈ L. If we remove edge s from G we shall denote the new graph as G = (N, L − s), or simply G = G − s. Analogously, let L ⊆ L. We denote as
- 38. 9 1.2 Directed, Weighted and Bipartite Graphs t Fig. 1.6 A graph G with N = 6 nodes (a), and three subgraphs of G, namely an unconnected subgraph obtained by eliminating four of the edges of G (b), the subgraph generated by the set N6 = {1, 2, 3, 4, 5} (c), and a spanning tree (d) (one of the connected subgraphs which contain all the nodes of the original graph and have the smallest number of links, i.e. K = 5). G = (N, L − L), or simply G = G − L, the new graph obtained from G by removing all edges L. 1.2 Directed, Weighted and Bipartite Graphs Sometimes, the precise order of the two nodes connected by a link is important, as in the case of the following example of the shuttles running between the terminals of an airport. Example 1.4 (Airport shuttle) A large airport has six terminals, denoted by the letters A, B, C, D, E and F. The terminals are connected by a shuttle, which runs in a circular path, A → B → C → D → E → F → A, as shown in the figure. Since A and D are the main terminals, there are other shuttles that connect directly A with D, and vice versa. The network of connections among airport terminals can be properly described by a graph
- 39. 10 Graphs and Graph Theory where the N = 6 nodes represent the terminals, while the links indicate the presence of a shuttle connecting one terminal to another. Notice, however, that in this case it is neces- sary to associate a direction with each link. A directed link is usually called an arc. The graph shown in the right-hand side of the figure has indeed K = 8 arcs. Notice that there can be two arcs between the same pair of nodes. For instance, arc (A, D) is different from arc (D, A). We lose important information if we represent the system in the example as a graph accord- ing to Definition 1.1. We need therefore to extend the mathematical concept of graph, to make it better suited to describe real situations. We introduce the following definition of the directed graph. Definition 1.5 (Directed graph) A directed graph G ≡ (N, L) consists of two sets, N = ∅ and L. The elements of N ≡ {n1, n2, . . . , nN} are the nodes of the graph G. The elements of L ≡ {l1, l2, . . . , lK} are distinct ordered pairs of distinct elements of N, and are called directed links, or arcs. In a directed graph, an arc between node i and node j is denoted by the ordered pair (i, j), and we say that the link is ingoing in j and outgoing from i. Such an arc may still be denoted as lij. However, at variance with undirected graphs, this time the order of the two nodes is important. Namely, lij ≡ (i, j) stands for an arc from i to j, and lij = lji, or in other terms the arc (i, j) is different from the arc (j, i). As another example of a directed network we introduce here the first data set of this book, namely DATA SET 1. As with all the other data sets that will be provided and studied in this book, this refers to the network of a real system. In this case, the network describes friendships between children at the kindergarten of Elisa, the daughter of the first author of this book. The choice of this system as an example of a directed network is not accidental. Friendship networks of children are, in fact, among social systems, cases in which the directionality of a link can be extremely important. In the case under study, friendships have been determined by interviewing the children. As an outcome of the interview, friendship relations are directed, since it often happens that child A indicates B as his/her friend, without B saying that A is his/her friend. The basic properties of Elisa’s kindergarten network are illustrated in the DATA SET Box 1.2, and the network can be downloaded from the book’s webpage. Of course, one of the first things that catches our eye in the directed graph shown in Box 1.2 is that many of the relations are not reciprocated. This property can be quantified mathematically. A traditional measure of graph reciprocity is the ratio r between the number of arcs in the network pointing in both directions and the total number of arcs [308] (see Problem 1.2 for a mathematical expression of r, and the work by Diego Garlaschelli and Maria Loffredo for alternative measures of the reciprocity [128]). The reciprocity r takes the value r = 0 for a purely unidirectional graph, while r = 1 for a purely bidirectional one. For Elisa’s kindergarten we get a value r = 34/57 ≈ 0.6, since the number of arcs between reciprocating pairs is 34 while we have 57 arcs in total. This means that only 60 per cent of the relations are reciprocated in this network, or, more precisely, if there is an arc pointing from node i to node j, then there is a 60 per cent probability that there will also be an arc from j to i.
- 40. 11 1.2 Directed, Weighted and Bipartite Graphs Box 1.2 DATA SET 1: Elisa’s Kindergarten Network Elisa’s kindergarten network describes N = 16 children between three and five years old, and their declaredfriendshiprelations.ThenetworkgiveninthisdatasetisadirectedgraphwithK = 57arcsandis shown in the figure. The nine girls are represented as circles, while the seven boys are squares. Bidirectional relations are indicated as full-line double arrows, while purely unidirectional ones as dashed-line arrows. Notice that only a certain percentage of the relations are reciprocated. Itisinterestingtonoticethat,withtheexceptionofElvis,theyoungestboyintheclass,thereisalmostasplit between two groups, the boys and the girls. You certainly would not observe this in a network of friendship in a high school. In the kindergarten network, Matteo is the child connecting the two communities. Summing up, the most basic definition is that of undirected graph, which describes systems in which the links have no directionality. In the case, instead, in which the directionality of the connections is important, the directed graph definition is more appro- priate. Examples of an undirected graph and of a directed graph, with N = 7 nodes, and K = 8 links and K = 11 arcs respectively, are shown in Figure 1.7 (a) and (b). The directed graph in panel (b) does not contain loops, nor multiple arcs, since these elements are not allowed by the standard definition of directed graph given above. Directed graphs with either of these elements are called directed multigraphs [48, 47, 308]. Also, we often need to deal with networks displaying a large heterogeneity in the rel- evance of the connections. Typical examples are social systems where it is possible to measure the strength of the interactions between individuals, or cases such as the one discussed in the following example. Example 1.5 Suppose we have to construct a network of roads to connect N towns, so that it is possible to go from each town to any other. A natural question is: what is the
- 41. 12 Graphs and Graph Theory t Fig. 1.7 An undirected (a), a directed (b), and a weighted undirected (c) graph with N = 7 nodes. In the directed graph, adjacent nodes are connected by arrows, indicating the direction of each arc. In the weighted graph, the links with different weights are represented by lines with thickness proportional to the weight. set of connecting roads that has minimum cost? It is clear that in determining the best construction strategy one should take into account the construction cost of the hypothetical road connecting directly each pair of towns, and that the cost will be roughly proportional to the length of the road. All such systems are better described in terms of weighted graphs, i.e. graphs in which a numerical value is associated with each link. The edge values might represent the strength of social connections or the cost of a link. For instance, the systems of towns and roads in Example 1.5 can be mapped into a graph whose nodes are the towns, and the edges are roads connecting them. In this particular example, the nodes are assigned a location in space and it is natural to assume that the weight of an edge is proportional to the length of the corresponding road. We will come back to similar examples when we discuss spatial graphs in Section 8.3. Weighted graphs are usually drawn as in Figure 1.7 (c), with the links with different weights being represented by lines with thickness proportional to the weight. We will present a detailed study of weighted graphs in Chapter 10. We only observe here that a multigraph can be represented by a weighted graph with integer weights. Finally, a bipartite graph is a graph whose nodes can be divided into two disjoint sets, such that every edge connects a vertex in one set to a vertex in the other set, while there are no links connecting two nodes in the same set. Definition 1.6 (Bipartite graph) A bipartite graph, G ≡ (N, V, L), consists of three sets, N = ∅, V = ∅ and L. The elements of N ≡ {n1, n2, . . . , nN} and V ≡ {v1, v2, . . . , vV} are distinct and are called the nodes of the bipartite graph. The elements of L ≡ {l1, l2, . . . , lK} are distinct unordered pairs of elements, one from N and one from V, and are called links or edges. Many real systems are naturally bipartite. For instance, typical bipartite networks are systems of users purchasing items such as books, or watching movies. An example is shown in Figure 1.8, where we have denoted the user-set as U = {u1, u2, · · · , uN} and the object-set as O = {o1, o2, · · · , oV}. In such a case we have indeed only links between users and items, where a link indicates that the user has chosen that item. Notice that,
- 42. 13 1.3 Basic Definitions t Fig. 1.8 Illustration of a bipartite network of N = 8 users and V = 5 objects (a), as well as its user-projection (b) and object-projection (c). The link weights in (b) and (c) are set as the numbers of common objects and users, respectively. Box 1.3 Recommendation Systems Consider a system of users buying books or selecting other items, similar to the one shown Figure 1.8. A reasonableassumptionisthattheusersbuyorselectobjectstheylike.Basedonthis,itispossibletoconstruct recommendation systems, i.e. to predict the user’s opinion on those objects not yet collected, and eventually torecommendsomeofthem.Thesimplestrecommendationsystem,knownasglobalrankingmethod (GRM), sorts all the objects in descending order of degree and recommends those with the highest degrees. Such a recommendation is based on the assumption that the most-selected items are the most interesting for the average user. Despite the lack of personalisation, the GRM is widely used since it is simple to evaluate even for large networks. For example, the well-known Amazon List of Top Sellers and Yahoo Top 100 MTVs, as well as the list of most downloaded articles in many scientific journals, can all be considered as results of GRM. A more refined recommendation algorithm, known as collaborative filtering (CF), is based on similarities between users and is discussed in Example 1.13 in Section 1.6, and in Problem 1.6(c). starting from a bipartite network, we can derive at least two other graphs. The first graph is a projection of the bipartite graph on the first set of nodes: the nodes are the users and two users are linked if they have at least one object in common. We can also assign a weight to the link equal to the number of objects in common; see panel (b) in the figure. In such a way, the weight can be interpreted as a similarity between the two users. Analogously, we can construct a graph of similarities between different objects by projecting the bipartite graph on the set of objects; see panel (c) in the figure. 1.3 Basic Definitions The simplest way to characterise and eventually distinguishing the nodes of a graph is to count the number of their links, i.e. to evaluate their so-called node degree.
- 43. 14 Graphs and Graph Theory Definition 1.7 (Node degree) The degree ki of a node i is the number of edges incident in the node. If the graph is directed, the degree of the node has two components: the number of outgoing links kout i , referred to as the out-degree of the node, and the number of ingoing links kin i , referred to as the in-degree of node i. The total degree of the node is then defined as ki = kout i + kin i . In an undirected graph the list of the node degrees {k1, k2, . . . , kN} is called the degree sequence. The average degree k of a graph is defined as k = N−1 N i=1 ki, and is equal to k = 2K/N. If the graph is directed, the degree of the node has two components: the average in- and out-degrees are respectively defined as kout = N−1 N i=1 kout i and kin = N−1 N i=1 kin i , and are equal. Example 1.6 (Node degrees in Elisa’s kindergarten) Matteo and Agnese are the two nodes with the largest in-degree (kin = 7) in the kindergarten friendship network intro- duced in Box 1.2. They both have out-degrees kout = 5. Gianluca has the smallest in and out degree, kout = kin = 1. The graph average degree is kout = kin = 3.6 Another central concept in graph theory is that of the reachability of two different nodes of a graph. In fact, two nodes that are not adjacent may nevertheless be reachable from one to the other. Following is a list of the different ways we can explore a graph to visit its nodes and links. Definition 1.8 (Walks, trails, paths and geodesics) A walk W(x, y) from node x to node y is an alternating sequence of nodes and edges (or arcs) W = (x ≡ n0, e1, n1, e2, . . . , el, nl ≡ y) that begins with x and ends with y, such that ei = (ni−1, ni) for i = 1, 2, . . . , l. Usually a walk is indicated by giving only the sequence of traversed nodes: W = (x ≡ n0, n1, .., nl ≡ y). The length of the walk, l = (W), is defined as the number of edges (arcs) in the sequence. A trail is a walk in which no edge (arc) is repeated. A path is a walk in which no node is visited more than once. A shortest path (or geodesic) from node x to node y is a walk of minimal length from x to y, and in the following will be denoted as P(x, y). Basically, the definitions given above are valid both for undirected and for directed graphs, with the only difference that, in an undirected graph, if a sequence of nodes is a walk, a trail or a path, then also the inverse sequence of nodes is respectively a walk, a trail or a path, since the links have no direction. Conversely, in a directed graph there might be a directed path from x to y, but no directed path from y to x. Based on the above definitions of shortest paths, we can introduce the concept of distance in a graph. Definition 1.9 (Graph distances) In an undirected graph the distance between two nodes x and y is equal to the length of a shortest path P(x, y) connecting x and y. In a directed graph the distance from x to y is equal to the length of a shortest path P(x, y) from x to y.
- 44. 15 1.3 Basic Definitions Notice that the definition of shortest paths is of crucial importance. In fact, the very same concept of distance between two nodes in a graph is based on the length of the shortest paths between the two nodes. Example 1.7 Let us consider the graph shown in Figure 1.6(a). The sequence of nodes (5, 6, 4, 2, 4, 5) is a walk of length 5 from node 5 back to node 5. This sequence is a walk, but not a trail, since the edge (2, 4) is traversed twice. An example of a trail on the same graph is instead (5, 6, 4, 5, 1, 2, 4). This is not a path, though, since node 5 is repeated. The sequence (5, 4, 3, 2) is a path of length 3 from node 5 to node 2. However, this is not a shortest path. In fact, we can go from node 5 to node 2 in two steps in three different ways: (5, 1, 2), (5, 6, 2), (5, 4, 2). These are the three shortest paths from 5 to 2. Definition1.10(Circuitsandcycles) A circuit is a closed trail, i.e. a trail whose end vertices coincide. A cycle is a closed walk, of at least three edges (or arcs) W = (n0, n1, .., nl), l ≥ 3, with n0 = nl and ni, 0 i l, distinct from each other and from n0. An undirected cycle of length k is usually said a k-cycle and is denoted as Ck. C3 is a triangle (C3 = K3), C4 is called a quadrilater, C5 a pentagon, and so on. Example 1.8 An example of circuit on graph 1.6(a) is W = (5, 4, 6, 1, 2, 6, 5). This example is not a path on the graph, because some intermediate vertex is repeated. An example of cycle on graph 1.6(a) is (1, 2, 3, 4, 5, 6, 1). Roughly speaking a cycle is a path whose end vertices coincide. We are now ready to introduce the concept of connectedness, first for pairs of nodes, and then for graphs. This will allow us to define what is a component of a graph, and to divide a graph into components. We need here to distinguish between undirected and directed graphs, since the directed case needs more attention than the undirected one. Definition 1.11 (Connectedness and components in undirected graphs) Two nodes i and j of an undirected graph G are said to be connected if there exists a path between i and j. G is said to be connected if all pairs of nodes are connected; otherwise it is said to be unconnected or disconnected. A component of G associated with node i is the maximal connected induced subgraph containing i, i.e. it is the subgraph which is induced by all nodes which are connected to node i. Of course, the first thing we will be interested in looking at, in a graph describing a real network or produced by a model, is the number of components of the graph and their sizes. In particular, when we consider in Chapter 3 families of graphs with increasing order N, a natural question to ask will be how the order of the components grows with the order of the graph. We will therefore find it useful there to introduce the definition of the giant component, namely a component whose number of nodes is of the same order as N.
- 45. 16 Graphs and Graph Theory Box 1.4 Path-Finding Behaviours in Animals Findingtheshortestrouteisextremelyimportantalsoforanimalsmovingregularlybetweendifferentpoints. How can animals, with only limited local information, achieve this? Ants, for instance, find the shortest path betweentheirnestandtheirfoodsourcebycommunicatingwitheachotherviatheirpheromone,achemical substancethatattractsotherants.Initially,antsexploreallthepossiblepathstothefoodsource.Antstaking shorter paths will take a shorter time to arrive at the food. This causes the quantity of pheromone on the shorterpathstogrowfasterthanonthelongerones,andthereforetheprobabilitywithwhichanysingleant choosesthepathtofollowisquicklybiasedtowardstheshorterones.Thefinalresultisthat,duetothesocial cooperative behaviour of the individuals, very quickly all ants will choose the shortest path [141]. Even more striking is the fact that unicellular organisms can also exhibit similar path-finding behaviours. A well-studied case is the plasmodium of a slime mould, the Physarum polycephalum, a large amoeba-like cell. The body of the plasmodium contains a network of tubes, which enables nutrients and chemical sig- nals to circulate through the organism. When food sources are presented to a starved plasmodium that has spread over the entire surface of an agar plate, parts of the organism concentrate over the food sources and are connected by only a few tubes. It has been shown in a series of experiments that the path connecting these parts of the plasmodium is the shortest possible, even in a maze [224]. Check Ref. [296] if you want to see path-finding algorithms inspired by the remarkable process of cellular computation exhibited by the P. polycephalum. In a directed graph, the situation is more complex than in an undirected graph. In fact, as observed before, a directed path may exist through the network from vertex i to vertex j, but that does not guarantee that one exists from j to i. Consequently, we have various definitions of connectedness between two nodes, and we can define weakly and strongly connected components as below. Definition 1.12 (Connectedness and components in directed graphs) Two nodes i and j of a directed graph G are said to be strongly connected if there exists a path from i to j and a path from j to i. A directed graph G is said to be strongly connected if all pairs of nodes (i, j) are strongly connected. A strongly connected component of G associated with node i is the maximal strongly connected induced subgraph containing node i, i.e. it is the subgraph which is induced by all nodes which are strongly connected to node i. The undirected graph Gu obtained by removing all directions in the arcs of G is called the underlying undirected graph of G. A directed graph G is said to be weakly connected if the underlying undirected graph Gu is connected. A weakly connected component of G is a component of its underlying undirected graph Gu. Example 1.9 Most graphs shown in the previous figures are connected. Examples of dis- connected graphs are graph G3 in Figure 1.1 and graph (b) in Figure 1.6. Graph G3 in Figure 1.1 has two components, one given by node 1 and the other given by the subgraph induced by nodes {2, 3, 4}. Graph (b) in Figure 1.6 has also two components, one given by
- 46. 17 1.4 Trees the subgraph generated by nodes {1, 5} and the other generated by nodes {2, 3, 4, 6}. The directed graph in Example 1.4 is strongly connected, as it should be, since in an airport one wants to join any pair of terminals in both directions. We will come back to component analysis and to the study of number and size of components in real-world networks in the next chapters. 1.4 Trees Trees are a particular kind of graph that appear very commonly both in the analysis of other graphs and in various applications. Trees are important because, among all connected graphs with N nodes, are those with the smallest possible number of links. Usually a tree is defined as a connected graph containing no cycles. We then say that a tree is a connected acyclic graph. The simplest possible non-trivial tree is a graph with two nodes and two links, known as a triad, and usually indicated in this book with the symbol ∧. Triads and triangles play an important role in complex networks, and we will come back to them in Section 4.3. Together with the concept of the tree, we can also introduce that of the forest, that is, a graph whose connected components are all trees. Various examples of trees are shown in Figure 1.9. The first graph is a tree with N = 17 nodes. The second graph is one of the possible spanning trees of a graph with N = 7 nodes and K = 12 links. A spanning tree of a graph G is a tree that contains all the nodes of G, i.e. a connected subgraph which contains all the nodes of the original graph and has the smallest number of links. Finally, the third graph is a sketch of a Cayley tree, an infinite tree in which each node is connected to z neighbours, where z is called the coordination number. Namely, we plot the first three iterations to construct a Cayley tree with z = 3, starting with an origin node placed in the centre of the figure. Even if the concept of the tree graph is not familiar to you, you are bound to be familiar with many examples. t Fig. 1.9 Three examples of trees. A tree with N = 17 nodes (a). A spanning tree (solid lines) of a graph with N = 7 nodes and 12 links (solid and dashed lines) (b). Three levels of a Cayley tree with z = 3 (c).
- 47. 18 Graphs and Graph Theory Example 1.10 (Trees in the real world) Any printed material and textbook that is divided into sections and subsections is organised as a tree. Large companies are organised as trees, with a president at the top, a vice-president for each division, and so on. Mailing addresses, too, are trees. To send a mail we send it first to the correct country, then to the correct state, and similarly to the correct city, street and house number. There are abundant examples of trees on computers as well. File systems are the canonical example. Each drive is the root of an independent tree of files and directories (folders). File systems also provide a good example of the fact that any node in a tree can be identified by giving a path from the root. In nature, plants and rivers have a tree-like structure. In practice, there are several equivalent ways to characterise the concept of tree, and each one can be used as a definition. Below we introduce the three definitions most commonly found in the literature. Definition 1.13 (Trees) A tree can be alternatively defined as: (A) a connected acyclic graph; (B) a connected graph with K = N − 1 links; (C) an acyclic graph with K = N − 1 links. The three definitions can be proven to be equivalent. Here, we shall only prove that definition (A) implies definition (B), i.e. that (A) ⇒ (B), while we will defer the discussion of (B) ⇒ (C) and (C) ⇒ (A) to Problem 1.4(a). In order to show that definition (A) implies definition (B), we first need to prove the following three propositions, which also show other interesting properties of trees. Proposition 1.1 Let G = (N, L) be a tree, i.e. by using definition (A), a connected acyclic graph. Then, for any pair of vertices x, y ∈ N there exists one and only one walk that joins x and y. Proof We will provide a proof by contradiction. Let x, y ∈ N. Since G is connected, there is at least one path that joins x and y. Let us assume that there exist two paths, denoted by w1 = (x0 = x, x1, x2, . . . , xr = y) and w2 = (y0 = x, y1, y2, . . . , ys = y). Let us denote by u min(r, s) the largest index for which xi = yi. The two walks will reconnect for some indices j and j, i.e. it will be xj = yj , xi = yi , ∀i ∈ {u + 1, . . . j − 1}, i ∈ {u + 1, . . . j − 1}. As shown in Figure 1.10, it follows that there exists a cycle C = (xu, xu+1, . . . , xj = yj , yj−1, . . . , yu = xu), which contradicts the assumption that G is acyclic. t Fig. 1.10 In a tree there cannot exist two different paths that join two nodes, otherwise a cycle would form.
- 48. 19 1.5 Graph Theory and the Bridges of Königsberg Proposition 1.2 Let G = (N, L) be a graph. Suppose that for each pair of distinct nodes of the graph there exists one and only one path joining the nodes. Then G is connected and if we remove any edge ∈ L, the resulting graph G − will not be connected. Proof That G is connected follows immediately from the assumptions. Furthermore, if is an edge that joins x and y, since there is only one path joining two edges (from proposition 1.1), if we remove it there will be no path joining x and y, and therefore the resulting graph will be disconnected. Proposition 1.3 Let G be a connected graph, such that if ∈ L ⇒ G − is disconnected. Then G is connected and has K = N − 1 links. Proof We only need to prove that K = N − 1. We will do this by induction on N. For N = 1 and N = 2 one has respectively K = 0 and K = 1. Now let G be a graph with N ≥ 3, and let x, y ∈ N, (x, y) ∈ L, x = y. By assumption, G − (x, y) is not connected: it is in fact formed by two connected components, G1 and G2, having respectively N1 and N2 nodes, with N = N1 + N2. Because N1 N, N2 N, by induction one has N1 = K1 + 1 and N2 = K2 + 1. From K = K1 + K2 + 1 it follows that N = N1 + N1 = K1 + 1 + K2 + 1 = K + 1. Finally, it is clear that by the successive use of the three propositions above we have proved that definition (A) implies definition (B). 1.5 Graph Theory and the Bridges of Königsberg As an example of the powerful methods of graph theory, in this section we discuss the theorem proposed by the Swiss mathematician Leonhard Euler in 1736 as a solution to the Königsberg bridge problem. This is an important example of how the abstraction of graph theory can prove useful for solving practical problems. It is also historically significant, since Euler’s work on the Königsberg bridges is often regarded as the birth of graph theory. The problem is related to the ancient Prussian city of Königsberg (later, the city was taken over by the USSR and renamed Kaliningrad), traversed by the Pregel river. The city, with its seven bridges, as there were in Euler’s time, is graphically shown in the left-hand side of Figure 1.11. The problem to solve is whether or not it is possible to find an optimum stroll that traverses each of the bridges exactly once, and eventually returns to the starting point. A brute force approach to this problem consists in starting from a side, making an exhaustive list of possible routes, and then checking one by one all the routes. In the case that no route satisfies the requested condition, one has to start again with a different initial point and to repeat the procedure. Of course, such an approach does not provide a general solution to the problem. In fact, if we want to solve the bridges’ problem for a different city, we should repeat the enumeration for the case under study. Euler came up with an elegant
- 49. 20 Graphs and Graph Theory t Fig. 1.11 The city of Königsberg at the time of Leonhard Euler (left). The river is coloured light grey, while the seven bridges are dark grey. The associated multigraph, in which the nodes corresponds to river banks and islands, and the links represents bridges (right). way to answer the question for any given configuration of bridges. First, he introduced the idea of the graph. He recognised that the problem depends only on the set of connections between riverbanks and islands. If we collapse the whole river bank A to a point, and we do the same for river bank B and for the islands C and D, all the relevant information about the city map can, in fact, be encapsulated into a graph with four nodes (river banks and islands) and seven edges (the bridges) shown in right-hand side of Figure 1.11. The graph is actually a multigraph, but this will not affect our discussion. In graph terms, the origi- nal problem translates into the following request: “Is it possible to find a circuit (or trail) containing all the graph edges?” Such a circuit (or trail) is technically called an Eulerian circuit (Eulerian trail). Definition 1.14 (Eulerian circuits and trails) A trail in a graph G containing all the edges is said to be an Eulerian trail in G. Similarly, a circuit in G containing all the edges is said to be an Eulerian circuit in G. A graph is said to be Eulerian if it contains at least one Eulerian circuit, or semi-Eulerian if it contains at least one Eulerian trail. Example 1.11 (Difficulty of an exhaustive search) A way to perform an exhaustive search for an Eulerian trail in a graph with N nodes and K edges is to check among the walks of length l = K whether there is one containing all the edges of the graph. If there is such a walk, then it is necessarily a trail, and therefore it is an Eulerian trail. The number of walks of length l = K is thus a measure of the difficulty of an exhaustive search. This number can be calculated exactly in the case of the multigraph in Figure 1.11, although here we will only give an approximate estimate. Let us consider first the simpler case of a complete graph with N nodes, KN. The total number of walks of length l for such a graph is N(N − 1)l. In fact, we can choose the initial node in N different ways and, at each node, we can choose any of its N − 1 edges. In conclusion, to look for Eulerian trails in KN we have to check N(N − 1)N(N−1)/2 walks. This number is equal to 2916 in a complete graph with N = 4 nodes. The same argument applies to a regular graph RN,k, i.e. a graph with N nodes and k links for each node. In such a case we can choose the initial node in N different ways and, at each node, we can choose k edges. Finally, the number of walks of length l is equal to Nkl, so that if we set l = K we obtain NkK walks of length K. By using such a formula with k replaced by k = 2K/N, we can get an estimate for the number of walks of length K in a generic graph with N nodes and K links. This gives 25736 for the graph of
- 50. 21 1.5 Graph Theory and the Bridges of Königsberg Königsberg, having k = 3.5. This number is of the same order of magnitude as the exact value (see Problem 1.5). Notice that this number grows exponentially with K, so that in a city with a larger number of bridges it can become impossible to explore all the different trips. For instance, in the case of the historical part of Venice, with its 428 bridges, even by assuming a small value k = 2, we get a number of N · 2428 walks to check. Thus, an exhaustive search for an Eulerian path over the graph represented by the islands of Venice and its bridges will be far beyond the computational capabilities of modern computers. In fact, even assuming that a computer can check 1015 walks per second, it would be able to check about 1032 walks in a timespan equal to the age of the universe; this number is much smaller than N · 2428 ≈ 7N · 10128. After having shown that the problem can be rephrased in terms of a graph, Euler gave a general theorem on the conditions for a graph to be Eulerian. Theorem1.1(Eulertheorem) A connected graph is Eulerian iff each vertex has even degree. It has a Eulerian trail from vertex i to vertex j, i = j, iff i and j are the only vertices of odd degree. To be more precise, Euler himself actually proved only a necessary condition for the existence of an Eulerian circuit, i.e. he proved that if some nodes have an odd degree, then an Eulerian trail cannot exist. The proof given by Euler can be summarised as follows. Proof Suppose that there exists an Euler circuit. This means that each node i is a crossing point, therefore if we denote by pi the number of times the node i is traversed by the circuit, its degree has to be ki = 2pi, and therefore it has to be even. If we only assume the existence of an Eulerian trail, then there is no guarantee that the starting point coincides with the ending point, and therefore the degree of such two points may be odd. Euler believed that the converse was also true, i.e. that if all nodes have an even degree then there exists an Eulerian circuit, and he gave some argument about this, but he never rigorously proved the sufficient condition [105]. The proof that the condition that all nodes have even degree is sufficient for the existence of an Eulerian trail appeared more than a century later, and was due to the German mathematician Carl Hierholzer, who pub- lished the first characterisation of Eulerian graphs in 1873 [152]. The early history of graph theory, including the work of Euler and Hierholzer, is illustrated in [245]. Here we shall give a complete proof of the Euler theorem based on the concept of par- tition of a graph into cycles. Consider the set of edges L of a graph G. We say that a subset L1 ⊆ L is a cycle if there exists a cycle, Z1, that contains all and only the edges L1. We say that the set L is partitioned if there exists a certain number s of subsets of L, L1, L2, . . . , Ls, such that: Li ∩ Lj = ∅, ∀i, j ∈ [1, . . . , s], ∪s i=1Li = L Now we can state the characterisation of Eulerian graphs in the form of equivalence of the following three statements [150]:
- 51. 22 Graphs and Graph Theory t Fig. 1.12 The reduction process to prove that a graph with all nodes with even degree can be partitioned into cycles. (1) G is an Eulerian graph (i.e. it contains at least one Eulerian circuit) (2) ∀i ∈ N, ki is even (3) there exists a partition of L into cycles. Proof Proof (1) =⇒ (2). It is clear that the existence of an Eulerian circuit implies that every node i is a crossing point, i.e. every node can be considered both a “starting point” and an “ending point”, therefore its degree ki has to be even. Proof Proof (2) =⇒ (3). From property (2) it follows that in G there exists at least one cycle, Z1, otherwise G would be a tree, and it would therefore have vertices of degree one (see Section 1.4). If G Z1,[2] then (3) is proved. If G = Z1, let G2 = G1 − L1, i.e. G2 is the graph obtained from G after removing all edges of Z1. It is clear that all vertices of G2 have even degree, because the degree of each node belonging to Z1 has been decreased by 2. Let G 2 be the graph obtained from G2 by eliminating all isolated vertices of G2. Since all vertices of G 2 have even degree, this means that G2 contains at least a cycle, Z2, and the argument repeats. Proceeding with the reduction, one will at last reach a graph G Z, therefore the set of links L is partitioned in cycles L1, L2, . . . , L. The procedure is illustrated in Figure 1.12. Proof Proof of (3) =⇒ (1). We now assume that the set of edges L can be partitioned into a certain number s of cycles, L1, L2, . . . , Ls. Let us denote by Z1, Z2, . . . , Zs the cor- responding graphs. If Z1 G then (1) is proved. Otherwise, let Z2 be a cycle with a vertex i in common with Z1. The circuit that starts in i and passes through all edges of Z1 and Z2 contains all edges of Z1 and Z2 exactly once. Hence, it is an Eulerian circuit for Z1 ∪ Z2. If G Z1 ∪ Z2 the assert is proved, otherwise let Z3 be another cycle with a vertex in common with Z1 ∪ Z2, and so on. By iterating the procedure, one can construct in G an Eulerian circuit. The Euler theorem provides a general solution to the bridge problem: the request to pass over every bridge exactly once can be satisfied if and only if the vertices with odd degree are zero (starting and ending point coincide) or two (starting and ending point do not coincide). Now, if we go back to the graph of Königsberg we see that the conditions of the theorem are not verified. Actually, all the four vertices in the graph in Figure 1.11 have [2] The symbol indicates that the two graphs are isomorphic. See Section 1.1
- 52. 23 1.6 How to Represent a Graph an odd degree. Therefore Eulerian circuits and trails are not possible. In the same way, by a simple and fast inspection, we can answer the same question for the city of Venice or for any other city in the world having any number of islands and bridges. 1.6 How to Represent a Graph Drawing a graph is a certainly a good way to represent it. However, when the number of nodes and links in the graph is large, the picture we get may be useless because the graph can look like an intricate ball of wool. An alternative representation of a graph, which can also prove useful when we need to input a graph into a computer program, can be obtained by using a matrix. Matrices are tables of numbers on which we can perform certain operations. The space of matrices is a vector space, in which, in addition to the usual operations on vector spaces, one defines a matrix product. Here and in Appendices A.4 and A.5 we will recall the basic definitions and operations that we will need in the book. More information can be found in any textbook on linear algebra. There are different ways to completely describe a graph G = (N, L) with N nodes and K links by means of a matrix. One possibility is to use the so-called adjacency matrix A. Definition 1.15 (Adjacency matrix) The adjacency matrix A of a graph is a N × N square matrix whose entries aij are either ones or zeros according to the following rule: aij = 1 iff (i, j) ∈ L 0 otherwise In practice, for an undirected graph, entries aij and aji are set equal to 1 if there exists the edge (i, j), while they are zero otherwise. Thus, in this case, the adjacency matrix is symmetric. If instead the graph is directed, aij = 1 if there exists an arc from i to j. Notice that in both cases it is common convention to set aii = 0, ∀i = 1, . . . , N. Example 1.12 Consider the two graphs in the figure below. The first graph is undirected and has K = 4 links, while the second graph is directed and has K = 7 arcs. The adjacency matrices associated with the two graphs are respectively:
- 53. 24 Graphs and Graph Theory Au = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ad = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Au is symmetric and contains 2K non-zero entries. The number of ones in row i, or equiv- alently in column i, is equal to the degree of vertex i. The adjacency matrix of a directed graph is in general not symmetric. This is the case of Ad. This matrix has K elements dif- ferent from zero, and the number of ones in row i is equal to the number of outgoing links kout i , while the number of ones in column i is equal to the number of ingoing links kin i . Clearly, Definition 1.15 refers only to undirected and directed graphs without loops or multiple links, which will be our main interest in this section and in the rest of the book. While weighted graphs will be treated in detail in Chapter 10, here we want to add that it is also possible to describe a bipartite graph by means of a slightly different definition of the adjacency matrix than that given above, and which in general is not a square matrix. In fact, a bipartite graph such as that shown in Figure 1.8 can be described by an N × V adjacency matrix A, such that entry aiα, with i = 1, . . . , N and α = 1, . . . , V, is equal to 1 if node i of the first set and node α of the second set are connected, while it is 0 otherwise. Notice that we used Roman and Greek letters to avoid confusion between the two types of nodes. Using this representation of a bipartite graph in terms of an adjacency matrix, we show in the next example how to formally describe a commonly used method to recommend a set of objects to a set of users (see also Box 1.3). Example 1.13 (Recommendation systems: collaborative filtering) Consider a bipartite graph of users and objects such as that shown in Figure 1.8, in which the existence of the link between node i and node α denotes that user ui has selected object oα. A famous personalised recommendation system, known as collaborative filtering (CF), is based on the construction of a N × N user similarity matrix S = {sij}. The similarity between two users ui and uj can be expressed in terms of the adjacency matrix of the graph as: sij = V α=1 aiαajα min{kui , kuj } , (1.1) where kui = V α=1 aiα is the degree of user ui, i.e. the number of objects chosen by ui [324]. Based on the similarity matrix S, we can then construct an N × V recommen- dation matrix R = {riα}. In fact, for any user–object pair ui, oα, if ui has not yet chosen oα, i.e. if aiα = 0, we can define a recommendation score riα measuring to what extent ui may like oα, as: riα = N j=1,j=i sijajα N j=1,j=i sij . (1.2)
- 54. 25 1.6 How to Represent a Graph At the numerator, we sum the similarity between user ui and all the other users that have chosen object oα. In practice, we count the number of users that chose object oα, weight- ing each of them with the similarity with user ui. The normalisation at the denominator guarantees that riα ranges in the interval [0, 1]. Finally, in order to recommend items to a user ui, we need to compute the values of the recommendation score riα for all objects oα, α = 1, . . . , V, such that aiα = 0. Then, all the non-zero values of riα are sorted in decreasing order, and the objects in the top of the list are recommended to user ui. Let us now come back to the main issue of this section, namely how to represent an undirected or directed graph. An alternative possibility to the adjacency matrix is a N × K matrix called the incidence matrix, in which the rows represent different nodes, while the columns stand for the links. Definition 1.16 (Incidence matrix) The incidence matrix B of an undirected graph is an N × K matrix whose entry bik is equal to 1 whenever the node i is incident with the link lk, and is zero otherwise. If the graph is directed, the adopted convention is that the entry bik of B is equal to 1 if arc k points to node i, it is equal to −1 if the arc leaves node i, and is zero otherwise. Example 1.14 (Incidence matrix) The incidence matrices respectively associated with the undirected and directed graphs considered in Example 1.12 are: Bu = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Bd = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −1 1 −1 0 0 0 0 1 0 0 −1 0 1 0 0 0 0 1 −1 0 −1 0 0 0 0 1 −1 0 0 −1 1 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Notice that Bu is a 5 × 4 matrix because the first graph has N = 5 nodes and K = 4 links, while Bd is a 5×7 matrix because the second graph has N = 5 nodes and K = 7 arcs. Also, notice that there are only two non-zero entries in each column of an incidence matrix. Observe now that many elements of the adjacency and of the incidence matrix are zero. In particular, if a graph is sparse, then the adjacency matrix is sparse: for an undirected (directed) graph the number 2K (K) of non-zero elements is proportional to N, while the total number of elements of the matrix is N2. When the adjacency matrix is sparse, it is more convenient to use a different representation, in which only the non-zero elements are stored. The most commonly used representation of sparse adjacency matrices is the so-called ij-form, also known as edge list form. Definition1.17(Edgelist) The edge list of a graph, also known as ij-form of the adjacency matrix of the graph, consists of two vectors i and j of integer numbers storing the posi- tions, i.e. respectively the row and column indices of the ones of the adjacency matrix
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- 56. parted; they followed her to the ship, but not before they promised to meet again in the winter. Ragnild was very much missed by them in all the entertainments that followed, and Sigurd thought often afterwards of lovely Ragnild. One day Sigurd proposed to Thordis and Thorana a moonlight drive, as the weather was beautiful, and at that time, the beginning of September, the moon was very brilliant. It was agreed that two other friends were to go. They were pleasant men, full of life and jollity. It was a beautiful night; not a cloud was seen in the sky. The full moon, queen of the night, shone in all her glory; the stars glittered and twinkled brilliantly in the deep azure of the firmament. Waiting in front of the “skemma,” or bower, of Thordis and Thorana stood a splendid four-wheeled carriage, wagon-like in shape, drawn by two of the fleetest horses known in the country. The horses were very restive. They champed the bit, pawed the ground, and snorted incessantly. Two men held the fiery steeds firmly by the bridle, and it took all their might, and in despite of this they could hardly prevent them from getting away from them. Sigurd and his two companions were anxiously waiting for the coming of the two Vikings’ daughters. Thorana and Thordis at last made their appearance, clad in their warm, graceful evening cloaks. Their faces were radiant with expectation, for both had been looking forward to that drive by moonlight and the sail on Eagle Lake, and were anticipating great delight. Accompanying them was a middle- aged friend, a woman who was to act, as we say in our modern way, as a chaperon. She was very skilled in embroidery, and had great talent in representing on canvas all kinds of scenery, views of the sea or landscape, either weird or charming. They had hardly entered the carriage, and had had no time to be seated, when the horses, becoming apparently unmanageable, dashed forward, and, as they rounded the corner of the way leading to the high road, the vehicle seemed fairly to bend like a bow, and
- 57. was on the point of being overturned. Fortunately the great skill of the driver was equal to the emergency. Then the carriage fairly flew over the ground, an irresistible power seeming to impel the fiery steeds forward in their furious speed. The excitement was very great among all. Sigurd exclaimed that even Sleipnir, the eight-footed horse of Odin, could not go faster, neither could clouds, pushed before the tempest, fly forward more quickly. The moonlight imparted a weird appearance to the landscape, the strange shadows of the trees seemed to play all round them, and the shadows of the rocks and of the hills appeared and disappeared, one after another, in quick succession, like phantoms or ghosts. Here and there they entered a part of the road densely wooded and where the rays of the moon could not penetrate; then came a less dense part of the forest, where tall, conical-shaped pines extended their phantom-like shadows out upon the road and over themselves; then groves of aspens came in sight, with their leaves quivering and frolicking as so many merry maids. The heaven was their banqueting hall, the stars their lights, and the murmur of the wind the music. All were speechless and spell-bound at the speed of the coursers and the unearthly beauty of all that passed swiftly before them, but once in a while an exclamation of delight or of wonderment escaped from the lips of Thordis and Thorana. Sigurd, who had been silent for some time, suddenly seemed to see far off in the sky nine Valkyrias riding in the air on fiery white steeds. Skuld, the Norn personifying the future, was preceding them, and Sigurd wondered why Skuld was with them, and what her appearance forebode. She accompanied them evidently to see that the decrees of the Nornir, who had shaped the lives of each of them at their birth, should be fulfilled at the particular time. What were those decrees no human being knew. Then Sigurd said to himself, “It was well ordered that no one should know his fate beforehand.” He did not know that they had fated him to be in love with Thorana or
- 58. Thordis. Suddenly the Valkyrias and Skuld appeared to vanish from his gaze. As the carriage sped along, the horses ran faster than before; it seemed hardly possible that the axles could stand the strain put upon them. Such was the rattling, that every part of the vehicle seemed on the point of coming to pieces. All shouted that they did not mind, that the wild fun would be still greater than ever. In a word, the excitement had rendered every one perfectly reckless of danger. “Why should the daughters of Viking heroes be afraid?” exclaimed Thordis; and Thorana shouted at the top of her voice: “It is good that our mothers are not with us; my mother would have died of fright or faint, and then we should have missed all our sport.” Glimpses of Eagle Lake were finally seen through the foliage of the trees, and soon afterwards they stopped before a solitary cabin near its shores, their horses fairly covered with foam. Every one declared that never had he driven so fast, or seen such superb driving, or been so excited in his life. In a few moments two boats were seen gliding out upon the waters of Eagle Lake, which was nestled in the midst of wooded hills, while yonder was Eagle Mountain towering above all. In one boat were Sigurd and Thorana; in the other, were Thordis and her two friends. Sigurd wished that Thordis had been also in his boat. The scene was most enchanting; not a ripple was seen on the crystal-like water, which the moon had transformed into a mirror, in which the stars coquettishly looked at themselves, while images of the hills and trees were reflected along the shores. “O mother Earth,” said Sigurd to himself as he contemplated that never-to-be-forgotten night. “How beautiful thou art, when the moon rules over thee instead of the sun! The moon gives us the night, the sun the day. Some say that the nights were created for the sons of men to sleep, but if it is so, why should the nights be so beautiful to behold, when the moon shines, and the stars tremble and glitter in the blue of the sky? Do not the nightingales sing their
- 59. songs of love at night when the moon is their sun? Love was born of the night; the nights of the moon are the lover’s days, for the moon shines upon them, and kisses them with her radiant and soft light.” Thorana insisted on rowing herself. Her graceful figure bent forward and backward at each stroke of the oars, her cousin Sigurd silently admiring her all the time. Their companion enlivened the time by his bright conversation and the recital of his numerous adventures, for he had been in many distant countries, and his anecdotes were full of wit. The weird echo repeated their words in the deep silence of that night, which was only disturbed at intervals by the falling of the oars upon the water. The two boats for a while drew wide apart, and their occupants amused themselves by listening to the echo. Once Sigurd thought he saw Hugin and Munin, the ravens of Odin, flying above his head on their way to Valhalla, to tell the Ruler of Hosts all that was happening in the world. Then again, appeared to him the nine Valkyrias with Skuld, who had followed them all the way; their spears glittered in the moonlight. Skuld’s hair sent rays of light out over the night. For a while she hovered over their boat, and then threw down upon the earth a superb ball of fire, a shooting star; then with the Valkyrias she disappeared in the direction of the Well of Urd. Every day the Nornir take the holy water from the well, and, mixing it with the clay that lies round it, pour it over the ash tree, Yggdrasil, that the branches may not dry up or decay. When the two boats came close together again, Sigurd saw two shadows reflected in the water, more beautiful to him than those of stars and of all that had been reflected in the water since Odin had made the world. They were the shadows of Thorana and of Thordis. Their beautiful faces, their graceful forms, their long hair, were like an apparition from the deep. It was as if the two beautiful daughters of Ægir and Ran, Dufa “the dove,” and Ud the “loving,” had come to see the men who were in the boats.
- 60. Sigurd remained spell-bound before the sight, when, by a motion of the boat, the shadow disappeared, never to reappear on the beautiful waters of Eagle Lake, and in a short time they found themselves once more on the shore. Sigurd mentioned to no one that he had beheld Valkyrias with Skuld, and the beautiful shadows of Thorana and Thordis, but all these visions had made a deep impression on his mind, and he remained thoughtful all the way home. The following day he made a sacrifice to the goddess Var, who, as we have said, listens to the vows of love men make; but no one ever heard of that vow. But we may safely say that the drive that beautiful night and the row on Eagle Lake was never forgotten by any of those who were there, as long as they lived. A few days after the events we have just mentioned, Thorana and Thordis made ready to go to their respective homes. The last evening of their stay saw the same party together in the hall where they met first. Nothing save death could have prevented Sigurd from being present. The following morning all met on board of the ship that was to take his two lovely cousins away from the island. A host of friends came to bid them good-by, all apparently happy, for none had yet realized how they would miss each other, and the good time they had all had, and that regrets were soon to follow, and all wondered if all of them would meet again. They parted with many expressions of love and friendship, and the following day a messenger came and handed to Sigurd a message written in mystic runes. It was from his cousins, who had written it on their way home. These magic words were: “With best love, from your broken-hearted cousins!” and a flush of joy overspread his face when he read this loving message. He immediately sent a messenger to them with another message, telling of his lonely feelings. Sigurd felt utterly wretched after the departure of his two cousins, though they were to meet again. A feeling of intense loneliness
- 61. came over him; all that was bright and cheerful in the island had gone; the wind moaned; the waves, as they struck the shores, seemed to sing mournfully in his ears, “Thy three cousins are gone, the rocky cliffs will see them no more.” He even dreaded to pass the skemma where they had stayed. No maiden could cheer him, for in his eyes none were so lovely and accomplished as Thorana, Ragnild, and Thordis. He whiled away the time by writing on birch bark a saga, in which he recounted all that had happened on the island. Finally he concluded to depart, and after sailing a few days he came to a burg where a foster-brother by the name of Thorkel and he had been brought up together. But Thorkel had been dead for several years. Sigurd wanted to see his grave, and, after landing and telling his errand to the people, he went towards the mound where Thorkel and his wife lay silently side by side. They had been married but a short time ere Skuld snapped asunder the thread that measured the days of their lives. He ascended the mound and murmured to himself, “Here Thorkel and his wife lie. The thinking minds that guided and moved their actions during their lives have left them. Helpless, motionless, and without life they sleep.” Looking up, he saw a butterfly of brilliant colors, with wings of gold and rainbow tints, full of life, going merrily from one flower to another, drinking of their nectar. Whilst watching his joyous course, Sigurd exclaimed musingly: “All life is ephemeral! Man and woman, like this butterfly and the flower, are but the creatures of a day in the immensity of time and in the world which the gods have made. What a beautiful life is that of the butterfly! He lives in the air; his life is that of love and immortality. He spends his days in caressing and kissing flowers, and becomes intoxicated with their sweetness. Like love, he feeds on love. As soon as he has fulfilled his destiny, and filled brimful the cup of love and drunk it, he dies as a brilliant meteor that burst into life for an instant, like the twinkling of a star that never returns. Thus the flower is born to show her tempting beauty, her sweetness, and
- 62. intoxicating nectar to the butterfly. The flower was created for the butterfly, and the butterfly for the flower; so were man and woman created for one another, and to love each other, and, like love, their minds are immortal. Short is the life of the butterfly and of the flower, but their existence in the immensity of time is apparently not shorter than that of man. If the lives of the butterfly and the flower are ephemeral, so also is the life of man. In the immensity of time since the ‘Great Void,’ the lives of all created things appear to the gods of the same duration. Man is born, ushered into the present, and then into the future, and thenceforth belongs to the past. We are tossed,” said Sigurd, “on the sea of life, like a rudderless ship, and we sail from day to day towards the unknown called by us the future, not knowing where we are going, nor how the Nornir have shaped our lives; always hoping and hoping for something we have not been able to grasp.” In this reflective mood of mind, Sigurd left the mound, under which lay two hearts which had been bound together by love during their lives, and returned to his ship, wondering what were the number of days the Nornir had decreed at his birth he should live, and also if he would ever find a woman that he would love so much as to be impelled to ask her to become his wife. Then he sailed for Dampstadir, and there met Ivar and his two foster-brothers waiting for him.
- 64. CHAPTER XIII A VOYAGE TO THE CASPIAN The following spring, Ivar and his foster-brothers made preparations to go to the Caspian Sea, by the Volga. They had sent word to several of their young kinsmen, asking them if they would join them in their voyage. The proposal had been accepted with eagerness by them all, for most of those to whom the invitation had been sent had never gone so far south, and they longed to see the lands of which they had heard so much, or from which so many costly things came; but two or three among those invited had been there before to trade, and had made on their return great profits on their goods, and they wished to try their luck again. It was not a small undertaking to make a voyage to the Caspian, for it was tedious, and took a long time. Ivar chose three vessels of very light draught, that could sail easily on the rivers of the present Russia, leading to that sea. Special vessels were built for such voyages, and the models of these craft were beautiful, and could not even to-day be improved upon for that sort of navigation. One vessel, very much like those of Ivar, was found at Tune, in Norway, and can be seen at Christiania to-day. Provisions were collected, among which was a great deal of hard bread, very much like that used to-day in Scandinavia. Various articles necessary for barter were also collected, such as scales and weights; a great quantity of gold spiral rods of certain size and weight, which were to be cut into smaller or larger pieces if necessary, and then weighed, for the Norsemen had no coins, and
- 65. these rings or pieces were the medium of exchange. Their scale of value was according to weight. Their intercourse with Rome, however, had made them acquainted with Roman coins of gold and silver, and they knew exactly their worth, and often brought them home and kept them until they visited again the Roman province. They also had a measure called an ell, two feet in length, to measure the beautiful fabrics they intended to buy, and also a measure for wine, for they were to bring back wines with them. A man named Ulf was to go with them. He was familiar with the navigation of the Dnieper, the Don, and the Volga, and had sailed several times from the Baltic to the Black Sea. He had lived chiefly upon the River Don, where he had a large trading establishment. He was a great trader and sea-farer, whose business was to go on trading voyages to various countries. Sometimes he went by sea, at other times by land. He was an old friend of Hjorvard, who often ordered him to buy goods for him, and had been very often to Gotland. Ulf was just the man for such an expedition, and the foster- brothers and their friends congratulated themselves on his going with them. In the beginning of June, as soon as the ice allowed them to sail, they left Dampstadir, and sailed through the Gulf of Finland; thence, after a difficult navigation through lakes and rivers, and some hard rowing, they reached the great River Volga, and, descending the stream, they came to a place called Novgrad, a great mart, where a fair was held once a year in summer. Novgrad was in the great realm of Holmgard, and they found there many friends, for the people were of the same kindred. Many Vikings had married the daughters of the Holmgard people, and much intercourse took place between them and the Norsemen. Both peoples had in common the same religious belief. During the fair, many kinds of people were to be seen there with their wares. They came from the Caucasus, from the Ural Mountains, from the shores of the Caspian, from Turkestan, even from China, and many other lands. Slaves were also sold at Novgrad in the
- 66. market-places. Peace reigned at the Novgrad fair, as it did at all the fairs of the Norsemen, or at the temple or assemblies of the people. No strife or shedding of blood was allowed to take place, and no one was molested. Ivar and his friends bought nothing at Novgrad, intending to come back and do so on their return. From Novgrad they sailed down the Volga, using their oars when there was no wind. They stopped here and there at several places, and were well received everywhere. While on board, every one of the crew had to be cook, for it was then the custom of traders not to have cooks, all the messmates drawing lots to see which of them should do the cooking each day. All shipmates also had to drink together, and a tub with a lid over it stood near the mast for this purpose. When finally they reached the Caspian without any serious mishaps, there was great rejoicing on board. Ulf was the recipient of many praises for his skilful pilotage. But the most difficult part of the journey was not yet accomplished—that of crossing the Caspian to the Persian shores, and the ascent of the Oxus River remained to be done. Before undertaking the second part of the voyage, the ships were drawn ashore, scraped, repaired, and painted, for their bottoms had become foul. During those days the Vikings spent much of their time practising athletic games when they were ashore, for on no account were these exercises to be neglected. When the ships were ready, they crossed the Caspian without encountering one of those storms that make the water of that shallow sea, so full of shoals, dangerous to mariners; and after landing they wondered much at the people they saw, for they differed greatly from the Vikings. They worshipped the sun and fire; some wore large turbans like the Turks of to-day, were very industrious, and many led a nomadic life with their herds. Their women were beautiful, and the men were courteous and hospitable. The Vikings bought a good deal of beautiful velvet, which they called pell, and much rich cloth embroidered with gold and silver, brocades, and also superb linen tablecloths and napkins.
- 67. They ascended afterward the River Oxus, the ascent of which was very tedious. The current was strong and against them, but no one tried to molest them, and every one was anxious to barter with them. There they bought silk goods and velvets, and spent the beginning of the winter on the river, and the later winter months on the Caspian, whence they sailed and rowed up the Volga before the melting of the snow and ice in the north; and by the time these had melted, and swelled the stream and made the current very rapid, they were far up. Here they waited until the river should have fallen to its usual size, remaining all the time on board of their ships, spending their days in playing chess or gambling with dice, for almost all the Vikings were great gamblers. Their voyage northward was far more tedious than that southward. It was necessary to place three men at each oar on account of the current, and the end of September found them once more in the Baltic, with their ships loaded with precious merchandise. Hjorvard and the Gotlanders were delighted to see Ivar and all his companions back. Not a single death had occurred during the voyage. Ulf had had an eye to business, and a good part of the cargo belonged to him. Ivar presented his father with several casks of wine and many precious objects, and to his mother he gave costly woollen, velvets, and silk stuff. The fame of Ivar had spread far and wide all over the land. Scalds in the halls of Hersirs recounted his brilliant warlike exploits. He was very mature for his age, and gifted with great tact. People said he would be exactly like his father. He had reached the age when parents think about looking for a wife for their sons. One morning, accordingly, Hjorvard called Ivar and said to him: “Listen to what I am going to tell thee. We have been told Frey had seated himself on Hlidskjalf, the high seat from which Odin could see over all worlds. When he looked to the north he saw on an estate, or farm, a large and fine house towards which a woman was walking. When she lifted her arms to open the door, a light shone from them on the sea, and the air and all the worlds were brightened by her.
- 68. This woman, as thou knowest, was Gerd, the daughter of Gymir by his wife Orboda, and was the most beautiful of all women. Frey’s great boldness in sitting down on the holy seat was thus punished, for he went away full of sorrow, having fallen deeply in love with her. When he returned home he did not speak, nor could he sleep or drink, and no one dared to question him. Then Njord called Skirnir, the page of Frey, and told him to go to him and ask Frey with whom he was so angry that he would not speak to anyone. Skirnir obeyed, though unwillingly, and when he came to Frey he asked him why he was so sad and did not speak to anyone. Frey answered that he had seen a beautiful woman, and for her sake he was full of grief. ‘Now thou shalt go,’ he said, ‘and ask her in marriage for me, and bring her home hither, whether her father be willing or not. I will reward thee well for the deed.’ “Skirnir replied that he would go and deliver this message if Frey gave him his sword. This sword was so powerful that it fought of itself. Frey gave it to him, and then Skirnir departed and asked the woman of Gymir in marriage for Frey, and Gerd promised him that she would come after three nights, and keep her wedding with Frey. “When Skirnir had told Frey of the result of his journey, Frey sang: ‘Long is one night, long is another. How can I endure three? Often a month to me seemed shorter than one-half this forthcoming wedding night.’” Hjorvard having thus told the story of Frey, said: “Ivar, my son, when I look from Dampstadir over the sea, I see yonder, towards the west, where we often behold so many grand sunsets, a beautiful maiden, nay, three beautiful ones, walking in the green paths leading to Upsalir. These three maidens are the daughters of Yngvi, the Hersir of Svithjod, and, as thou knowest, their beauty and accomplishments are known all over our northern lands. Thou hast come to that age when it is time for thee to find a wife, and I have thought of a match for thee, kinsman, if thou wilt follow my advice, and nothing would please me more than to have thee marry one of Yngvi’s daughters. Thou mightest visit many countries and find no
- 69. maidens more accomplished than they are, and it would be of good advantage to our family and to Gotland if thou didst marry one of them, and bring our kinship still closer than it was before with the Hersir of Svithjod.” Ivar replied that he knew how much his father had his welfare at heart, but said: “Thou must not forget, father, that the daughters of Yngvi have the highest pedigree in all the Northern lands, and the realm of Gotland may not be large enough for their ambition. It may be possible that these daughters may wish to wed men having greater possessions than myself. I think it would be prudent, before thou and our kinsmen propose the match, that I obtain greater renown than I have.” “There is no difficulty, my son, about thy pedigree, for we are all of Odin’s kin, and you would be equally matched.” The conversation ended there for the present, Ivar leaving the matter of his marriage in his father’s hands, though he thought much of what Hjorvard had said, and of his earnest wish to have him happily and honorably married. He knew, too, that Yngvi and Hjorvard were great friends, and visited one another, and gave feasts to each other, and that a connection by marriage between the two families would be very advantageous and agreeable to his parents.
- 71. CHAPTER XIV HAKI’S BURNING JOURNEY TO VALHALLA On their return from the Caspian, the four foster-brothers had found the country very much disturbed; several Vikings from abroad, with a great number of ships, had been plundering here and there among the people. Peace had deserted the land, and great distress from these incursions prevailed everywhere. Among the greatest plunderers were two famous brothers of the name of Haki and Hagbard; they were great Vikings, and had a large host and a great number of powerful and swift ships. These had gray sails, and were painted of such a color that their vessels could not be seen far away. Haki and Hagbard had no lands; they lived on their ships, and never slept under a roof, nor did they ever drink at the fire-side; their men had no homes, and had left their country, preferring a life of adventure and warfare with two such famous chiefs. They attacked people ashore everywhere, and plundered them, and afterwards returned with their booty to their ships; they wintered in the rivers, and defied the power of Rome, and of all the Hersirs in the land. When their ships were old, they bought new ones or captured others. They had at last become tired of Western countries, had returned to the Norseland, and had been outlawed by all the Things, or assemblies, of the people of every realm. Haki had with him twelve champions, among whom were Starkad the old, and Ulf the valiant. All his men were berserks, who were often seized by the berserk fury. Starkad and Ulf were old men, who had been through many a bloody fight, and had served under Haki’s
- 72. father, who had never himself slept under a roof. They all had taken an oath at a great sacrifice that they would never die in a bed; neither would they ever throw themselves from a rock in order to go to Odin and Valhalla, but that they would all die by weapons in battle. Haki himself was one of the greatest of champions, and so agile as well as powerful, that he was a most dangerous enemy to deal with. One day Haki went with his host against Thorkel, a great Hersir, without warning, for he ruthlessly disregarded the laws of war, so that Thorkel had hardly time to collect his warriors. The latter had also twelve champions, among whom were the brothers Svidpad and Geigad, both far-famed in the North. A fierce battle took place, and Valhalla was destined to receive many men that day. When the battle was at its height, Svidpad and Geigad made a furious assault on Haki’s men, and many of them never saw the light again. All of Haki’s champions were badly wounded, and could fight no more, being too weak on account of loss of blood. Then he went forward and broke the shield-burg of Thorkel and slew him, as well as his standard-bearer, and also Svidpad and Geigad. He conquered the land and took possession of it, and became the ruler of the herad of Thorkel. He stayed at home during the winter, and ruled the land he had conquered, after which his champions sailed away to southern lands, on Viking expeditions, and earned much wealth for themselves. Among other great Vikings who never slept under a roof or drank by the fire-side, and who disturbed the land and had been outlawed, were the brothers Eirick and Jorund. After a great battle in which they had slain the Hersir of Gautaland, they thought themselves far greater men than before, and wished to try their strength against Haki and Hagbard, and avenge the disgrace put upon Thorkel, their kinsman; so, when they heard that Haki had allowed his champions to go away, they collected a large host. When it was known that they had come to reconquer the land for their kinsmen, the people from all the country round flocked to their standards in large numbers, and a great host marched against Haki. A mighty battle
- 73. soon took place. For a long time victory was undecided, champion fighting against champion. Finally, Haki rushed forward, and fought with such irresistible force, that he slew all near him, among them Eirick, and cut down the standard-bearers of the brothers, whereupon Jorund fled to his ships with his men. But Haki had received such severe wounds that he foresaw that his remaining days would be few. He had made ready a vessel which he prized very highly on account of its swiftness, beauty, and war power. He had it loaded with the bodies of high-born warriors that had fallen in battle, together with their weapons, and had a large pyre of tarred wood made on the ship. Then he bade his followers farewell, and told them that he was going to Odin, and ordered men to carry him, in full war dress, with chain-armor, helmet, sword, and shield, on board of his ship. Then he bade them to build a large pyre near the prow, and to lay him upon it. After they had done so, he had the rudder adjusted and the sail hoisted and set, and much gold and many weapons placed on board. Then the tarred wood was kindled. The wind blew from the shore seaward; the burning ship sailed away, and the warriors bade Haki and his men a happy journey to Valhalla. Farther and farther the funeral pyre of Haki and his men went on its way. The flames rose higher and higher towards the sky; the sail burned, and at last the mast, looking like a tower of fire, fell upon the deck. The people believed that the higher the flames rose, the greater would be the welcome in Valhalla. Then the lurid glare of the flames became less and less brilliant, and, on a sudden, the ship went down into the deep. But Haki and his warriors had sailed to Valhalla, and the people said that this great deed of Haki would live forever in the memory of man, and would be sung by the scalds to the end of time. During this time, Ivar and his foster-brothers had gathered a large host and made his vessels ready, for he intended to make war on the Viking raiders who infested the sea and brought trouble and
- 74. insecurity upon the land. As they were being launched, Hjalmar’s ship struck one man as it came down the rollers, and killed him. This accident happened once in a great while at the launching of ships— an operation that was always attended with danger, the more so if it were not carefully done. Such an accident was called “roller- reddening,” and was considered a very bad omen, therefore the intended expedition was abandoned. Ivar and his foster-brothers thought that some faithless family spirits wished them evil, and had abandoned their watch over them. The next day, when Ivar and Hjalmar were walking together, Ivar thought he saw a pet goat of his, which had been always in the habit of coming into the courtyard. No one was allowed to drive him away. Suddenly he said: “This is strange!” “What dost thou see that seems strange to thee?” asked Hjalmar. “It seems to me,” Ivar answered, “that the goat which lies in this hollow place is covered with blood.” Hjalmar, astonished, answered him that there was no goat there, nor anything else. “What is it, then?” inquired Ivar. “I am afraid,” Hjalmar returned, “that thou must be a death-fated man, and that thou hast seen the spirit that follows and protects thee, warning thee of danger; and if not thyself, some of thy kinsmen may, perhaps, be fated to die. Guard thyself well, foster- brother. I will also watch carefully over thee, so will Sigurd and Sigmund.” “That will not serve,” cried Ivar, “if death is fated to me, for no man can change his fate; but I will fall bravely.” These two successive omens made a deep impression upon Ivar; the ships were dragged ashore, and put under the sheds, and it was announced that no expeditions were to take place that year. Then Ivar made a special sacrifice to Frey, for he loved Frey more than all the other gods, and often sacrificed to him, and that day he
- 75. offered up four black oxen, and two of his most valuable horses. The following day, Hjalmar said to Ivar: “Let us find out the decrees that fate has in store for us, for I do not like the ‘roller-reddening’ that has taken place at the launching of our ships, or the vision of the bloody goat. Let us consult the oracles, as well as sacrifice to Frey. I still fear some impending misfortune is going to happen to some of us, and that some great sorrow will overtake us. Let us make ready and beware of treachery. Perhaps we may meet a witch full of evil on the way; then it is better to walk on than to lodge in her house, though the night may be stormy. Often wicked women sit near the road, who blunt both swords and sense. Let us never go out of doors without our weapons, for it is hard to know, when out on the roads, if a man may need his spear. The sons of men need eyes of foresight.” They made, therefore, another sacrifice, and dipped the sacrificial chips into the blood of the sacrificed animal, that was kept in the sacred copper bowl which stood on the altar of the temple. The sacrificing chips were thrown into the air, and the answer was that Ivar would not die, but must remain at home that year, and that a kinsman very dear to him would be killed in battle. So Ivar stayed quietly at home. The following summer Ivar made the Elidi ready and sailed for Norway; but on the voyage, while in the Cattegat, he was obliged to stay on an island on account of head winds. There they threw the sacrificial chips again to get fair winds, and, as they fell, they indicated that Odin was to receive one man out of their host before a fair wind would come. They then sailed toward the coast and cast anchor, and there they landed. Not far from their place of landing was a great sacrificing ring, in the midst of which lay a huge stone, or altar. The people were in the habit of coming there from the surrounding country to make human sacrifice and to break the backs of men given to Odin on that altar. Agnar was the name of the man whom the oracles, speaking through the sacrificial chips, had designated, and upon the altar his back was broken, and he was given to Odin, and they reddened the altar with his blood. After this
- 76. the men returned to their ships and sailed away with a fair wind. This sacrificing ring where Agnar was given to Odin is seen to this day near Blomholm in the province of Bohuslan, where a large ring composed of eleven stones is still standing, with a sacrificial boulder in the centre.
- 78. CHAPTER XV DEATH AND BURNING OF HJORVARD The warning of so many bad omens proved to be true. During a terrific sea-battle, in which many ships were engaged, between Hjorvard and Starkad, a powerful Hersir with whom he had long had a feud on account of a disputed inheritance, Hjorvard received his death-wound. During this fierce conflict, weapons buried themselves in bloody wounds, and sank deep into men’s bodies; rivers of blood gushed out on the armor; the whirlwinds of the Valkyrias, as the poetical Norsemen called battles, were abroad among men; arrows and spears played round the shields in the midst of the “tempest of Odin.” Many swords were broken, many shields were rent asunder, many suits of chain-armor were cut to pieces, and many of the host took their journey to Valhalla. Suddenly Hjorvard thought he saw during the battle a Valkyria, the mighty Skogul, leaning on her spear-shaft, and heard her say: “Now the elect of Odin are coming; a great host will enter Valhalla to-day before night.” Then looking up he thought he saw Valkyrias on horseback, in front of Skogul and Gondul, bearing themselves nobly, helmeted, with shields, with their hair floating in the air behind their backs, and with spears from which rays of light sprung. Then Hjorvard exclaimed, “Gondul and Skogul, Odin has sent to choose among chiefs who of the Ynglingar kin should to him go, and in Valhalla dwell.” It seemed to him that the Valkyrias hovered over him. He was then clad with helmet and chain-armor, and standing under his war standard; the oars had dropped, the battle was then
- 79. raging most fiercely, the spears hissed, the arrows quivered, flames of fire came from the swords. Hjorvard urged the Gotlanders and his champions to the fight; the “play of the Valkyrias” was waxing hotter and hotter. Hjorvard’s sword cut into the “cloth of Odin,” for such was the name which Norsemen gave to chain-armor, as if it were water, and reddened the ships with the blood of men. Suddenly Hjorvard beheld, as in a vision, Skuld the Norn at the head of the Valkyrias, and about to sever the thread of his own life. He was right. Odin guided a spear towards him, and Hjorvard received his death-wound. The following morning he lay on the deck of his ship amidst many dead champions. In his delirium he murmured, “Why hast thou decided the battle as thou didst, mighty Skogul? We surely deserved victory from the gods.” And Skogul seemed to answer: “We have caused thee to keep the field, and thy foes to flee. We shall now ride to Valhalla to tell Odin that Hjorvard the Wide-spreading, and his fallen host, are coming;” and in his dying ears seemed to sound the voice of Odin saying: “Hermod and Bragi, go forth to meet Hjorvard, the valiant Hersir of Gotland, for he is coming this way to the hall; he is bespattered with blood, and has a mighty host following him.” And as he dreamed of entering the portals of Valhalla he heard again the voice of Odin saying: “Welcome, Hjorvard! Thou shalt have peace with ‘the chosen,’ and cheer from the Asars; thou fighter of men, and wise ruler, who didst take care of the sacrifices and temples, thou hast more than many a chief, in many a land, reddened the sword, and carried forward the bloody blade. Twice welcome, Hjorvard! My maids, the Valkyrias, will carry wine to thee, and wait upon thee, and carry ale to those who have come with thee.” Hjorvard awoke partially, however, from his dying swoon, and lived long enough to be brought home in his ship; and before expiring he said to Sigrlin: “Wife, let my burning journey be worthy of our kinsmen; let a wide and high mound be raised over me; let the mortuary chamber be roomy; surround the mound with tents, shields, weapons of all kinds, for it is good to have them for every- day fight in Valhalla; let foreign linen, silk, and costly garments, and
- 80. riding gear go with me. Place me on the pyre in full war dress, clad with my gold helmet, my costliest chain-armor, and gird me with one of my best swords. Let many horses be killed and follow me, also my hawks, so that I may enter Valhalla as it befits a great chief and a Ynglingar; and throw gold and silver on the pyre, and throw also many weapons, so that the shining golden doors of Valhalla be not shut against me and my warriors that have fallen. Thus our journey will not be poor, for the wealth that we have earned during our life and not given away will go with us. Place by me also the sharp sword that lay between thee and me before we were wedded, while I courted thee, for thy person was holy, and that sword defended thee and guarded thy honor.” He had hardly uttered these words when he expired, and, according to holy custom, his eyes and mouth were closed and his nostrils pinched, his body and head carefully washed, and his hair combed. The people said that Odin himself had steered the ship of Hjorvard during the battle. Ivar was not in the fleet when the fight which caused his father’s death took place, nor was he at home, but two days after his father’s demise he returned to Dampstadir. He had left his ships on the other side of the island on account of contrary winds, and crossed the country on horseback. On his arrival he went immediately to the great hall, as it was his custom when he returned from an expedition, to drink with his men. He little dreamed then of the sad news that awaited him, for no one on the way had been willing to tell him of his father’s death. He had hardly seated himself on the high seat opposite to that of his father, when his eye caught sight of what he had not noticed at first on his entrance. He saw the walls covered with black and gray hangings. This had been done by his mother, for it was the custom upon occasions of this kind and importance to drape the great hall in mourning, and the hangings told of the great sorrow and loss which Gotland had sustained. By this Ivar then knew that the death of a great kinsman had taken
- 81. place, and his face at once betrayed an expression of profound anxiety. Shortly afterwards his mother came in, and seated herself by his side. Ivar looked intently at her, and after noticing the pallor of her face, said to her: “Thou must have ordered, mother, the hall to be thus draped; tell me for what purpose and for whom are those tokens of mourning?” Sigrlin answered: “My husband, the Hersir of Gotland, is no more. Hjorvard, thy father, is dead, but fell gloriously in the midst of victory.” “The tokens that forebode the death of a kinsman have then proved true,” said Ivar, with a deep sigh; “the sacrificial chips foretold this.” Then he added sorrowfully, and with a voice full of emotion: “A death-fated man cannot be saved. All is dangerous to the death-fated. A man who is not death-fated cannot receive his death-wound, he will escape in some way or other; but every one must die the day he is death-fated. The decrees which the Nornir made the day of my father’s birth had to be fulfilled.” Sigrlin was inconsolable at the death of her husband, but she did not weep, nor wring her hands, nor wail, as women often do. Very wise men came forward, who tried to console her heavy heart, but they did not succeed, for though unable to weep, her sorrow was great, and her heart broken. The high-born brides of powerful chiefs and warriors sat gold- adorned by her side, trying to soothe her sorrow; each of them related her woes, the bitterest sorrow she had suffered. The sister of Gjuki said: “No woman on earth lacks love more than I. I have suffered the loss of two husbands, of three daughters, of eight brothers, and of four sisters, and yet I live.” Still Sigrlin could not weep. Then said Herborg: “I have a harder sorrow to tell. My seven sons and my husband fell among the slain in the southern lands. The brother of Ægir, the Wind, and the nine daughters of Ran, played
- 82. with my father and mother, and with my four brothers on the deep; they were dashed against the gunwale of their ship, and they were killed. I myself had to wash, to dress, to handle, and to bury their bodies. All that I suffered in a single year, and no man gave me help. The same year I became a bond-woman. I had to dress and to tie the shoes of a Hersir’s wife every morning. She threatened me because of jealousy, and struck me with hard blows; nowhere found I a better housemaster, nor anywhere a worse housewife.” Still Sigrlin could not weep. Then Gullrond spoke thus to them: “Little comfort can you give by speaking as you have done to Sigrlin, wise though you are.” Thereupon she bade them uncover the body of Hjorvard, when she drew the sheet from it, and threw it on the ground at the feet of Sigrlin, saying to her: “Look on thy beloved husband; put thy mouth to his now silent lips, as thou wert wont when thou didst embrace him.” Sigrlin looked at her dead husband, and she saw the wound on his breast, the lips that could not speak, the ears that could not hear, the eyes that could not see, and the hands that could not caress; the cheeks were pale, and the mind and life had gone. At the sight, she sank down upon the pillow where the dead Hjorvard’s head rested. Flushed were her cheeks, and a tear fell upon her tresses, then upon her knees; and from those springs called the eyes, rivers of sorrow flowed copiously, and she was comforted. Five days after the death of Hjorvard, his funeral, or his burning journey to Valhalla, took place, for it was the law of the land that men should be laid under mound not later than the fifth day after their demise. The people believed that Odin had enacted the same laws in the northern lands as formerly prevailed among the Asar. Thus he ordered that all dead men should be burned, and that on the pyre should be placed their property, promising that with the same amount of wealth should they come to Valhalla as was burned with them; also that they should enjoy what they themselves buried in the ground, and that their ashes should be thrown into the sea or
- 83. buried in the earth; that over great men, mounds should be raised as memorials, and over men that had especially distinguished themselves for manliness, memorial stones should be erected. It had been agreed by Ivar and his kinsmen that Hjorvard’s burning journey should be on board a ship, and that the ship should not be sent to sea, but burned ashore. A fine Skuta of fifteen benches, beautifully ornamented, was chosen for the pyre, their powerful war ships never being used on such occasions. The Skuta was propped to stand up as if it were in the water; the prow looked towards the sea, as if ready to be launched for an expedition. A large quantity of tarred wood surrounded it, and in the prow of the ship the resting place of Hjorvard had been erected. When all the preparations were ready, Hjorvard’s body was carried upon the bed on which he lay; he was dressed in full war costume, clad with helmet and chain-armor, with sword by his right side and shield on his breast; spears were laid by his left hand, and at his feet lay his golden spurs. Ivar then brought forward his own saddle-horse, magnificently harnessed and equipped. Then followed a superb and profusely decorated four-wheeled carriage, with a single seat standing high in the middle, and twelve horses; the horses and falcons were slaughtered, and the carriage broken and thrown upon the pyre. Then Ivar, just as the torch was applied, bade Hjorvard his kinsman to sail, ride, or drive to Valhalla, as he liked best; and all his champions, warriors, and multitudes of people bade him a happy journey, and expressed the hope that he would welcome them there, at the proper time, when the decrees of the Nornir should be fulfilled in regard to them. So that his journey to Valhalla might be worthy of him, they threw into the pyre many costly things, weapons and quantities of gold and silver. The loose property which Hjorvard had won or got during his life, and that had remained in his possession, was also thrown into the funeral pile. All the weapons that were to follow him to Valhalla were, according to ancient customs, rendered useless. Swords and spear-heads were bent, and their edges
- 84. indented; shafts were broken, shields were rent asunder, and shield- bosses cut. Roman and Greek objects were partly destroyed, and with Roman coins were also thrown into the ship. Solemn and grand was the spectacle, and lurid the glare. Gradually the flames became less and less high, the noise of the cracking wood became fainter and fainter, and finally nothing was seen but the burning embers. Then the charred bones of Hjorvard were gathered in the midst of solemn silence. The ashes were scattered to the wind and fell into the sea. The burned bones were put in a beautiful Roman bronze vessel, and with them Roman coins of Diocletian’s time, the spear- point that had caused his death-wound, also a few draughtsmen belonging to his chess-board, and two dice. Twelve shield-bosses, with their convex side downward, were made a lid for the vase, and lay over the bones; a bent sword was placed over the cinerary urn, which was put in the mortuary chamber that had been prepared; and a large cairn, which took several days to build, was raised over Hjorvard’s remains; and a large memorial stone, with runic inscription, put on the top. Thus went to Valhalla Hjorvard, the Hersir of Gotland. “It is wise,” said Ivar to his foster-brother Hjalmar as they were mournfully conversing upon the sad ceremonies of the past few days, “that Odin has ordered that the wealth of a man, his gold and silver and his movable property, should go on his burning journey with him. This thought makes him generous during his life, and he gives away lavishly the wealth that he acquires, thus preventing his heart from being hardened towards those who are in need. So Hersirs and prominent men should not be miserly. The wealth that is thus given during one’s life is given back to them in Valhalla.” Then after a pause he added musingly: “foster-brother, I have often thought of Helgi, my first cousin, the son of Halfdan, and that if he had lived he would have been the Hersir of Gotland, instead of my father. Then I should not now be ruler over the sacrifices. How strange are the decrees of the Nornir!”
- 86. CHAPTER XVI HELGI AND THE VALKYRIAS Ivar had spoken of Helgi because he had often heard his father mention his brother, but he has not been referred to in this narrative before, for he had been dead many years. Halfdan had married Thurid, a beautiful daughter of the Hersir of Zeeland, and loved her passionately. She died about a year after their marriage, in giving birth to a son. Halfdan was so grieved at the death of his wife, that he ordered the child who was the cause of such great misfortune to him to be exposed. The infant was laid in a cradle, and a piece of pork was put in his mouth; the cradle was taken to a wood at some distance from Dampstadir, and put near the root of a tree, in such a manner that the infant should be protected against the wind and the bad weather, and thus die easily. No name had been fastened upon him, as water had not been poured upon him. A short time after the child had been exposed, an uncle of Hjorvard was passing through the forest. He heard the cries of the little one, and following the direction of the noise, he was profoundly touched at the sight, and took compassion upon the babe, and brought him up secretly on his estate, his sister taking great care of him, and both loving him tenderly. Halfdan never married again, for his love for Thurid was far too great, and in his eyes no woman could equal her. His memory and love for her never faded from his mind to his death, and the last word he uttered was her name.
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