![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/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-15-320.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/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-18-320.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/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-21-320.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/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-28-320.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/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-30-320.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/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-31-320.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/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-32-320.jpg)
![August 29th. Gillmore: Much obliged. All your fire on Sumter
materially lessened the great risk I incur.
Dahlgren.
August 29th. Gillmore: I will let you have either guns or
projectiles, as many as you wish, if you will inform me how
much you require.
Dahlgren.
September 1st. Gillmore: We have dismounted two guns on
Sumter and injured one this a. m. But two remain. We are firing
with great accuracy.
Lookout.
September 1st. Gillmore: I am glad the batteries are doing good
execution. I hope you will give me the full benefit of your fire,
as I intend to be in action to-night, if nothing prevents. I would
advise great care in handling the hand grenades, as one of my
men was killed and two wounded by a very ordinary accident.
Dahlgren.
September 2d. Gillmore: I think your fire on Sumter may be
omitted to-day. Have just returned from above and am trying to
get a little rest. I do not know what damage our fire did Sumter.
My chief of staff wounded; his leg broken.
Dahlgren.
September 2d Adams: I wish to know if Sumter fired at the
monitors last night while they were in action. Do not disturb the
admiral if he is asleep, but please get me the information, as it
will determine whether I continue firing on Sumter to-day.
Gillmore.
September 2d. Gillmore: Not to my knowledge.
[F] In his final report he said: The formidable strength of Fort
Wagner induced a modification of the plan of operations, or rather](https://image.slidesharecdn.com/24993-250615101043-f4dda60e/85/Complex-Networks-Principles-Methods-and-Applications-1st-Edition-Vito-Latora-37-320.jpg)
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- 4. 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.
- 6. 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
- 7. 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.
- 8. To Giusi, Francesca and Alessandra
- 10. 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
- 11. 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
- 12. 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
- 13. 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
- 14. 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
- 15. 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
- 16. 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
- 17. 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
- 18. 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.
- 19. 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
- 20. 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.
- 21. 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
- 22. 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
- 23. 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,
- 24. 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.
- 26. Life is all mind, heart and relations Salvatore Latora Philosopher
- 28. 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
- 29. 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
- 30. 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.
- 31. 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.
- 32. 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
- 33. Another Random Scribd Document with Unrelated Content
- 34. operation against Sumter she cannot fire a gun at you even in the daytime, if she has any to fire, which I doubt. Gillmore. August 22d. Gillmore: It is not of Sumter that I am apprehensive, but of Moultrie and adjacent forts; but most all of Sumter's have been sent to Moultrie, which makes no difference in the fire. This I am inclined to endure rather than have a monitor ashore to defend or destroy, which would change the whole course of operations. Dahlgren. August 22d. Gillmore: Wagner is firing rapidly. I fear she will dismount some of our guns. Turner. August 22d. Dahlgren: Wagner is firing very rapidly. There is great danger of dismounting our guns. What can you do to stop it? Gillmore. August 22d. Gillmore: I will send up some monitors at once. Dahlgren. August 22d. Turner: Can you not keep down Wagner's fire with mortars, 30-pounders, Parrotts and sharpshooters? Gillmore. August 22d. Gillmore: Is the fire of the ironclads effectual in silencing the sharpshooters at Fort Wagner? Dahlgren. August 22d. Dahlgren: Between the gunboats and our batteries, Wagner's fire has been considerably kept under. Gillmore. August 22d. Dahlgren: Are you going to attack to-night?
- 35. Gillmore. August 22d. Gillmore: Yes, if the weather will permit. Dahlgren. August 23d. Dahlgren: What did you ascertain as to the condition of Sumter? Gillmore. August 23d. Gillmore: It was so foggy that but little could be ascertained. We received a very heavy fire from Moultrie. The admiral is now asleep. O. C. Badger. August 23d. Badger: Did you receive any fire from Fort Sumter? Gillmore. August 23d. Gillmore: She fired two or three times only, when we first opened. Badger. August 26th. Gillmore: Would it be convenient for you to open a heavy fire on Sumter, sustaining it until nightfall? Dahlgren. August 26th. Dahlgren: I can open a pretty strong fire on Sumter, if you deem it necessary. One of my 8-inch guns is burst, and others are nearly expended. Do you think Sumter has any serviceable guns? My calcium lights can operate to-night on Sumter and the harbor, unless you wish otherwise, and we can arrange for investing Morris island. Gillmore. August 26th. Gillmore: I am going to operate on the obstructions and a portion of my men will be without cover. I do not fear heavy guns from Sumter, but wish to keep down the
- 36. fire of small guns. Your fire will help me very much. I am sorry that your guns are giving out. Dahlgren. August 26th. Dahlgren: I shall be able, I think, to light up the waters between Fort Sumter and Cummings point, so that no small boats can approach the latter without being seen by your picket boats. Gillmore. August 26th. Turner: Open all the guns in the left batteries on Sumter and keep them going through the day. Gillmore. August 26th. Gillmore: To-night I shall need all the darkness I can get. If you light up you will ruin me. What I did want was the active fire of your batteries this afternoon on Sumter. Dahlgren. August 27th. Dahlgren: Can I take from your vessel another 8- inch gun and a 100-pounder? I have burst three 8-inch guns in all. We took 68 prisoners, including 2 officers, and gained 100 yards toward Wagner yesterday. Gillmore. August 27th. Gillmore: You can take the guns with pleasure. My attempt to pass the forts last night was frustrated by the bad weather, but chiefly by the setting in of a strong flood tide. Dahlgren. August 27th. Dahlgren: Can you spare me some 200-pounder shells? My supply is very low. A constant fire on Sumter is more than my guns can stand very long. I have lost three 200- pounders. Gillmore.
- 37. August 29th. Gillmore: Much obliged. All your fire on Sumter materially lessened the great risk I incur. Dahlgren. August 29th. Gillmore: I will let you have either guns or projectiles, as many as you wish, if you will inform me how much you require. Dahlgren. September 1st. Gillmore: We have dismounted two guns on Sumter and injured one this a. m. But two remain. We are firing with great accuracy. Lookout. September 1st. Gillmore: I am glad the batteries are doing good execution. I hope you will give me the full benefit of your fire, as I intend to be in action to-night, if nothing prevents. I would advise great care in handling the hand grenades, as one of my men was killed and two wounded by a very ordinary accident. Dahlgren. September 2d. Gillmore: I think your fire on Sumter may be omitted to-day. Have just returned from above and am trying to get a little rest. I do not know what damage our fire did Sumter. My chief of staff wounded; his leg broken. Dahlgren. September 2d Adams: I wish to know if Sumter fired at the monitors last night while they were in action. Do not disturb the admiral if he is asleep, but please get me the information, as it will determine whether I continue firing on Sumter to-day. Gillmore. September 2d. Gillmore: Not to my knowledge. [F] In his final report he said: The formidable strength of Fort Wagner induced a modification of the plan of operations, or rather
- 38. a change in the order previously determined upon. The demolition of Fort Sumter was the object in view as preliminary to the entrance of the ironclads.... To save valuable time, it was determined to attempt the demolition of Sumter from ground already in our possession, so that the ironclads could enter upon the execution of their part of the programme, ... and arrangements were at once commenced, and the necessary orders given to place the breaching guns in position. Arrangements were also made to press the siege of Fort Wagner by regular approaches.
- 39. T CHAPTER XV. THE GETTYSBURG CAMPAIGN—GALLANT SERVICE OF PERRIN'S AND KERSHAW'S BRIGADES—HAMPTON'S CAVALRY AT BRANDY STATION. he spring had gone and summer had opened in Virginia, when, seeing no indications of aggressive movement on the part of the Federal army lying opposite him on the Rappahannock, General Lee determined to draw it from his Fredericksburg base and compel it to follow his movements or attack him in position. General Lee's plan involved the movement of his army by its left to Orange and Culpeper, the crossing of the Blue ridge into the Shenandoah valley, the crossing of the Potomac, and the march of his whole force directly on Harrisburg, the capital of Pennsylvania. The army of Northern Virginia was now organized in three corps, commanded by Lieutenant-Generals Longstreet, Ewell and A. P. Hill. Longstreet's division commanders were McLaws, Pickett and Hood; Ewell's, Early, Rodes and Johnson; A. P. Hill's, Anderson, Heth and Pender. Still in the division of the gallant McLaws, under Longstreet, associated with Barksdale's Mississippians and Semmes' and Wofford's Georgians, was the South Carolina brigade of Gen. J. B. Kershaw. Also in the First corps were the batteries of Capt. Hugh R. Garden (Palmetto) and Captain Bachman's German artillery, with Hood's division, and the Brooks (Rhett's) battery, Lieut. S. C. Gilbert, in Alexander's battalion of Walton's reserve artillery. Gen. Micah Jenkins' South Carolina brigade, of Pickett's division, Longstreet's corps, was detached for special duty on the Blackwater, in southeast Virginia, under Maj.-Gen. D. H. Hill. In the Third army corps (A. P. Hill's), South Carolina was represented by McGowan's brigade, Hill's light division—North Carolinians, South Carolinians and Georgians— now being commanded by Pender, and the South Carolina brigade
- 40. by Col. Abner Perrin. Maj. C. W. McCreary commanded the First regiment, Capt. W. M. Hadden the First rifles, Capt. J. L. Miller the Twelfth, Lieut.-Col. B. T. Brockman the Thirteenth, and Lieut.-Col. J. N. Brown the Fourteenth. With the Third corps also was the Pee Dee artillery, Lieut. W. E. Zimmerman. In the cavalry corps of Maj.-Gen. J. E. B. Stuart, Brig.-Gen. Wade Hampton commanded his brigade, including the First and Second South Carolina cavalry, and Capt. J. F. Hart's South Carolina battery was part of the horse artillery under Major Beckham. Thus it will be seen that there were two infantry brigades, five batteries, and two cavalry regiments of South Carolina troops in the army of General Lee on this march into Pennsylvania. Evans' and Gist's brigades were in Mississippi with General Johnston, and Manigault's brigade was with General Bragg's army at Chattanooga. Attached to those commands or serving in the West, were the batteries of Captains Ferguson, Culpeper, Waties and Macbeth. Most of the South Carolina troops of all arms were engaged in the defense of Charleston and the coast of the State, then being attacked by a powerful fleet and a Federal army. On June 7th the corps of Longstreet and Ewell, with the main body of the cavalry under Stuart, were encamped around Culpeper Court House; Hill's corps being in position at Fredericksburg in front of General Hooker. The latter, vaguely aware of a campaign at hand, sent his cavalry, under General Pleasanton, up the Rappahannock to gain information. Pleasanton crossed his cavalry, supported by infantry and artillery, at Kelly's and Beverly fords, and advanced upon Brandy Station, one column approaching that railroad station from the northeast (Beverly ford), the other from the southeast (Kelly's ford). The road from Beverly ford, before reaching the station, passes over a high ridge on which is the hamlet of Fleetwood. On the morning of June 9th, Jones' cavalry brigade was covering Beverly ford, and Robertson's, Kelly's ford. The Federal columns drove off the pickets at the two fords and marched directly to the attack. Before Robertson's brigade had assembled, General Stuart sent the First South Carolina, Col. John L. Black, down the
- 41. Kelly's Ford road to check the advance until Robertson could take position. This duty was well done by the First, until relieved by Robertson, when the regiment went into battle on the Beverly road with Hampton. As soon as the firing in front was heard, General Hampton mounted his brigade and moved from his camp rapidly through the station and over the Fleetwood ridge to support Jones on the Beverly Ford road, leaving the Second South Carolina, Col. M. C. Butler, to guard the station. Throwing his brigade immediately into action on the right of General Jones, and in support, the division, after severe fighting, drove the column of attack back. At this juncture the Federal force which moved up the Kelly's Ford road had reached the railroad and was taking possession of the Fleetwood ridge in rear of the engagement on the Beverly Ford road. General Stuart promptly ordered his brigades to concentrate upon this, the main attacking force, and the battle followed for the possession of the ridge. The brigades of Hampton, Jones and W. H. F. Lee by repeated charges, front and flank, swept the hill, captured the artillery which had been placed on its summit, and drove the enemy in full retreat for the river. His strong infantry and artillery support checked the pursuit and covered his crossing. The First South Carolina lost 3 killed and 9 wounded, among the latter the gallant Captains Robin Ap C. Jones and J. R. P. Fox. Meanwhile the Second South Carolina had been fighting, single- handed, an unequal battle on the road running from the station to Stevensburg, 5 or 6 miles south, and beyond that place on the road leading to Kelly's ford. A column of cavalry, with artillery, had advanced from Kelly's toward Stevensburg with the evident intention of moving up from that place to the support of the attack at Fleetwood, and if it had reached the field of battle in the rear of Stuart, might have turned the day in Pleasanton's favor. But, being advised of this menacing movement, General Stuart sent Colonel Butler's regiment, 220 strong, down the Stevensburg road to meet and check it. Leading the advance of Butler's regiment, Lieut.-Col. Frank Hampton met and drove back the Federal advance beyond Stevensburg. Then Butler formed his command across and to the left
- 42. of the road at Doggett's house, about 1½ miles beyond Stevensburg, and stood ready to dispute the advance of the main body of the enemy. Lieutenant-Colonel Hampton was charged with the defense of the road, with a few sharpshooters and one company, Capt. T. H. Clark's. Here he held the right for a half hour, while Butler and Major Lipscomb resisted the attack in the center and on the left, the line of defense being nearly a mile in length. Massing his squadrons, the enemy charged the right, and to break the force of the onset, Lieutenant-Colonel Hampton, with 36 men, dashed forward at the head of his column. He fell mortally wounded, and the onrushing squadrons scattered his little band. Butler retired his center and left up the Brandy Station road and took post on an eminence at Beckham's house, where his command was reinforced by a squadron from the Fourth Virginia, sent by General Stuart and led by Capt. W. D. Farley of his staff. While holding this position a shell from one of the enemy's batteries passed through Colonel Butler's horse, shattered his leg below the knee, and mortally wounded the gallant Farley. The artillery fire was sweeping the road and the hill, and the Federal squadrons were forming to charge, when the men offered to bear Farley off. Smiling, with grateful thanks, he told them to stand to their rifles, and to carry Butler out of the fire. Then, with expressions of resignation to his fate and devotion to his country, he expired on the field. Major Lipscomb took command and drew off slowly toward Brandy Station. But the battle had been won for the Confederates at Fleetwood, and Lipscomb soon had opportunity to advance and drive the Federals before him in the general retreat, until he posted his pickets at the river. In this famous cavalry battle Stuart captured 375 prisoners, 3 pieces of artillery and several colors. A few days later, being satisfied that General Lee was beyond his right flank in force, Hooker began moving his army to keep between Lee and Washington. Meanwhile Ewell marched upon Milroy at Winchester in the Valley, attacked and captured 4,000 prisoners and 28 pieces of artillery, and cleared the Valley for Lee's advance.
- 43. General Lee now ordered up A. P. Hill's corps to join in the march for the Potomac. Kershaw's brigade, with McLaws, marched to Sperryville on the 16th, thence to Ashby's gap, where Rice's battalion rejoined the command, crossed the Shenandoah at Berry's ford on the 20th, recrossed and formed line of battle to meet a threatened attack on the 21st, and then continuing, crossed the Potomac on the 26th and encamped near Williamsport. Reaching Chambersburg, Pa., on the 28th of June, they remained there until the 30th, then marching to Fayetteville. McGowan's brigade, with A. P. Hill, also occupied a position near Fayetteville on the 29th. Stuart's cavalry, moving on Longstreet's right flank, left General Hampton on the Rappahannock to watch the enemy. On the 17th, Fitzhugh Lee's brigade made a splendid fight at Aldie, but Pleasanton occupied that place with a large force, and Stuart called Hampton and his other scattered commands together at Middleburg. Here he was attacked by cavalry, infantry and artillery on the 21st. Hampton and Jones received the attack gallantly, but were compelled to retire. Here, said General Stuart in his report, one of the pieces of Captain Hart's battery of horse artillery had the axle broken by one of the enemy's shot, and the piece had to be abandoned, which is the first piece of my horse artillery that has ever fallen into the enemy's hands. Its full value was paid in the slaughter it made in the enemy's ranks, and it was well sold. The fight was renewed at Upperville, before Ashby's gap, and there, said Stuart, General Hampton's brigade participated largely and in a brilliant manner. On the night of the 24th, Stuart's brigades rendezvoused secretly near Salem Depot, and started toward Washington, encountering Hancock's corps marching north, at Gum Spring. When Hancock had passed they moved to Fairfax Station, where Hampton's advance had a brisk fight on the 27th. Stuart was now between the Federal army and Washington, and Hampton, in advance, crossed the Potomac near Dranesville, and on the 28th started northward. At Rockville a Federal army train, about 8 miles long, was captured, and the subsequent movements of the cavalry were embarrassed by the attempt to convoy the train to Lee's army.
- 44. Ewell, meanwhile, taking a more easterly route than Longstreet and Hill, on the 27th camped at Carlisle, Early's division of his corps marching to York, and menacing the Pennsylvania capital. General Hooker did not cross the Potomac until the 25th and 26th, and on the 28th General Meade was placed in command of the Federal army. On the 28th, General Lee learned from a scout that the Federal army was marching to Frederick and was in part located at the base of South mountain, and he changed his design of marching up the valley to Harrisburg and ordered Hill eastward toward Gettysburg. Heth took the lead, and the South Carolinians, with Pender, reached Cashtown, 8 miles from Gettysburg, on the last day of June. On that day both Meade and Lee were marching unconsciously to the point at which they were to fight the great and decisive battle of the year, if not of the war. It is interesting to note that the Southern general was concentrating from the north and the Northern general from the south. Ewell's corps was approaching the battlefield from Carlisle and York, and Hill's from Chambersburg. Before the close of the day Hill learned that Pettigrew's North Carolinians, of Heth's division, in advance near Gettysburg, had met a strong cavalry force, before which they withdrew without battle. Early on the morning of July 1st, General Hill pushed Heth's division forward, followed closely by Pender's. With Heth was the Pee Dee artillery, in Pegram's battalion; with Pender, the battalion of McIntosh. About 10 a. m. Heth met Buford's Federal cavalry and drove it back across Willoughby run, where the cavalry was promptly supported by the First corps of Meade's army, three divisions, under General Reynolds. General Hill deployed Heth's division on the right and left of the road, Pender's in support, and the battle became severe. Pushing his battle forward, Hill was checked at the wooded ridge known as Seminary hill, where the First corps with artillery was strongly posted. Putting his artillery in position Heth gallantly charged the heights with his four brigades, and made so strong a
- 45. battle that General Howard, with part of the Eleventh corps, reinforced the line of the First. At this juncture Ewell's two divisions came in on Hill's left, and the latter ordered Pender forward to relieve Heth. Ewell's line was at right angles to that of Hill's, and both lines now swept onward with irresistible force. Pender's advance was with Thomas' Georgians on the left of the road, and Lane, Scales and Perrin (McGowan's brigade) on the right. The combined assault of Pender and Ewell's divisions swept the hill and routed the two Federal corps, driving them through the streets, capturing 5,000 prisoners, exclusive of the wounded, several colors and 3 pieces of artillery. Reporting the advance of Pender, General Hill said: The rout of the enemy was complete, Perrin's brigade taking position after position of the enemy and driving him through the town of Gettysburg. This special mention by the corps commander of McGowan's veterans, under Perrin, was well deserved. Never was a brigade better handled in battle, and never did regiments respond more steadily to every order for advance in direct charge, or change of front under fire. The Fourteenth, under Lieut.-Col. J. N. Brown and Maj. Edward Croft, and the First, under Maj. C. W. McCreary, on the right of the brigade; and the Twelfth, under Col. J. L. Miller, and the Thirteenth, under Lieut.-Col. B. T. Brockman, on the left, stormed the stone fences on either side of the Lutheran college on Seminary hill and routed their foe from this strong position, capturing hundreds of prisoners, 2 field pieces and a number of caissons, and following the routed columns through the town of Gettysburg. The colors of the First South Carolina were the first Confederate standard raised in the town as Hill's troops were entering it. Late in the afternoon, when Perrin drew up his brigade for rest on the south of the town, a battery which had been driven before Perrin took position on Cemetery hill and fired the first shot from that memorable eminence at the South Carolina brigade. Colonel Perrin reported this fact, and stated that he had watched the battery on its retreat as it was pursued through the town, and saw it take position on the hill. But the loss of the brigade did not fall short of 500. Every one of the
- 46. color sergeants taken into the fight was killed in front of his regiment. Perrin was in position in front of Cemetery hill on the 2d, the Federal sharpshooters in his front on the Emmitsburg road. In the afternoon he was ordered by General Pender to push his skirmishers to the road. Capt. William T. Haskell, of the First regiment, commanding a select battalion of sharpshooters, was intrusted with this duty, and Major McCreary led the First regiment, now only about 100 strong, in Haskell's support. The gallant Haskell threw his sharpshooters against the Federal skirmishers, captured the road and drove his opponents up the slope and under their guns. While putting his men in favorable positions on the road, Haskell received a mortal wound and expired on the field. His fall was felt to be a serious loss to the whole brigade. South Carolina gave no better, purer, nobler man as a sacrifice to the cause of Southern independence at Gettysburg. Perrin held the skirmish line Haskell had won, and on the 3d threw forward the Fourteenth to maintain it against a strong attack. His sharpshooters from the road commanded the cannoneers on the hill, and a desperate effort was made to drive them off the road. In the fight of the Fourteenth regiment to sustain the sharpshooters, Lieutenant-Colonel Brown and Major Croft were severely wounded. The skirmish line was held until the massing of artillery and infantry on the crest made it no longer tenable. The total loss in McGowan's brigade at Gettysburg was 100 killed and 477 wounded. Including the loss on the retreat, the total was 654. Orr's Rifles, left to guard the trains, did not participate in the battle of the 1st, or the affairs of the 2d and 3d, and lost but few men. The heaviest casualties fell on the Fourteenth, two-thirds of its men being killed or wounded in the three days' engagements. Colonel Perrin mentioned particularly the conduct of the following officers: Major Croft, of the Fourteenth; Maj. I. F. Hunt, of the Thirteenth; Maj. E. F. Bookter, of the Twelfth; Capts. W. P. Shooter, T. P. Alston and A. P. Butler, of the First; Capts. James Boatwright and
- 47. E. Cowan, of the Fourteenth, and Capt. Frank Clyburn, of the Twelfth. Among the gallant dead were Lieut. A. W. Poag, of the Twelfth; Capt. W. P. Conner and Lieuts. W. C. McNinch and D. M. Leitzsey, of the Thirteenth; and Lieutenant Crooker, of the Fourteenth. Lieut. J. F. J. Caldwell, of the First, whose graphic and instructive history of the brigade has aided the writer materially, was among a host of wounded line officers. The break of day on the 2d revealed the army of General Meade in line of battle on the heights south of Gettysburg, running north and south with the Emmitsburg road in his front. General Lee thus described his position: The enemy occupied a strong position, with his right upon two commanding elevations adjacent to each other, one southeast (Culp's hill), and the other (Cemetery hill) immediately south of the town which lay at its base. His line extended thence upon the high ground along the Emmitsburg road, with a steep ridge in rear, which was also occupied. This ridge was difficult of ascent, particularly the two hills above mentioned as forming its northern extremity, and a third at the other end (Little Round Top) on which the enemy's left rested. Numerous stone and rail fences along the slope served to afford protection to his troops and impede our advance. In his front the ground was undulating and generally open for about three-quarters of a mile. Immediately south of the Federal left, as described by General Lee, was a still higher hill, known as Round Top, which commanded the whole left of the Federal position, and was not occupied early on the morning of the 2d. To attack a superior force in a position so strong presented a difficult problem for solution, and gave the Confederate general serious pause. He had Ewell's corps on his left, confronting Culp's and Cemetery hills, and facing southwest and south; and Hill's corps on the right facing east. McLaws' and Hood's divisions of Longstreet's corps camped within 4 miles of the battlefield on the night of the 1st, left camp at sunrise on the 2d, and marched to the right of Hill's corps. The Third division of Longstreet's corps
- 48. (Pickett's) was left to guard the trains at Chambersburg, and did not reach the vicinity of Gettysburg until the afternoon of the 2d. General Longstreet received his definite orders for position and attack about 11 o'clock, and by 3:30 p. m. McLaws was in position opposite the enemy's advanced position at the peach orchard, with Hood on his right facing the Round Tops. General Lee's order of attack directed that his right (Hood and McLaws), strongly supported by artillery, should envelop and drive in the Federal left; that simultaneously with this attack against the Federal left, the Confederate left should storm Culp's and Cemetery hills; and the Confederate center at the same time should so threaten the Federal center as to prevent reinforcements to either Federal wing. General Lee's plan of battle contemplated prompt movement, and concert of action along his entire line. If these conditions, essential to the success of the plan, had been given in its execution, the writer believes that the battle of Gettysburg would have been won by General Lee on July 2d by a victory as complete as Chancellorsville. They were not given and the plan failed. The actual fighting of the separate assaults was gallant and heroic, and the resistance both steady and aggressive; the Federal position along his main line being unmoved by the assaults. On the Confederate right two divisions of Longstreet's corps made the advance at 4 p. m. (Hood's and McLaws'), supported by four of the five brigades of Anderson's division from the center. Hood on the extreme right, next McLaws, and then Anderson, were fighting forward and struggling to storm the last position of the Federal army on the heights, but these divisions were fighting it out without the simultaneous battle which Lee had ordered on the left. They had carried the stone walls and numerous hills and woods, the peach orchard, the great wheat-field and rocky bluffs in their front, and were on the slopes of the Round Tops and the heights north of them, but still the battle had not opened on the left. There was not a man to reinforce Longstreet's line, and the enemy in his front was reinforced by both infantry and artillery. Hours passed (General Lee said two, General Longstreet four and Gen. Edward Johnson said it
- 49. was dark) before General Ewell's left division moved to the attack on Culp's hill, which, after some time, perhaps another hour, was followed by the attack on the north face of Cemetery hill. Edward Johnson's division made the attack on Culp's hill and Early's division on Cemetery hill. The Third division of Ewell's corps (Rodes') did not attack at all. Anderson's (of Hill's corps) was the only one of the three center divisions that attacked from the center. It is evident from these statements, which are made from a careful study of the official reports, that the prime conditions of success, concert of action and simultaneous movement, were not given the plan of the commanding general. Edward Johnson's three brigades did not begin the actual attack on Culp's hill until dusk, according to his own and General Ewell's statements. General Early, with two of his four brigades, Hays' and Hoke's, attacked Cemetery hill still later. These two brigades carried the height and actually took the enemy's batteries, but were unable without support to hold what they had gained. It is in the report of Rodes, who did not advance at all, on account of darkness, that particular mention is made of his having observed the enemy on Cemetery hill, during the afternoon, withdrawing artillery and infantry to reinforce against the attack then in progress on the Confederate right. The troops of the Federal army in position at Culp's and Cemetery hills were those beaten and routed on the 1st, and considering the success gained by the brigades of Hays and Avery, there can be no reasonable doubt that with the immediate support of Rodes, the attack being made at the earlier hour ordered, Cemetery hill would have fallen, and with its fall the Confederate left and center would have driven the Federal right in confusion and Gettysburg would have been added to the long list of General Lee's great victories. The Comte de Paris, in his review of Gettysburg, has truly said, that the way in which the fights of the 2d of July were directed does not show the same co- ordination which insured the success of the Southern arms at Gaines' Mill and Chancellorsville. But it is time that our attention was directed to the South Carolina brigade, under Kershaw, operating with McLaws, in Longstreet's
- 50. attack, and the batteries of Bachman and Garden, operating with Hood, on the extreme right of Longstreet's battle. Kershaw formed the right of McLaws' division and Barksdale his left, Semmes behind Kershaw and Wofford behind Barksdale. In front of Barksdale was the peach orchard, 500 yards distant and in front of Kershaw and on a line with the orchard a stone house, stone barn and stone fence. The peach orchard was on an eminence, and was held by infantry and a battery. Beyond the stone house was another eminence, defended by a battery, and beyond this battery a stony hill, wooded and rough. This stony hill was in front of Kershaw's center, and beyond the hill opened the great wheat-field which spread forward to the slopes of the Federal main position. Barksdale moved against the orchard and Kershaw against the stony hill and the battery in front of it. Before moving General Kershaw had detached the Fifteenth South Carolina, Colonel De Saussure, to support a battery between his right and Hood's left. Marching forward under the fire of canister from the battery in his front, and the infantry fire from the south side of the peach orchard, the Carolina brigade swept past the battery and reached the hill, Barksdale clearing the orchard and its battery on Kershaw's left. Taking possession of the rocky hill, the enemy at once advanced upon it over the wheat-field in two lines of battle. As the brigade stood on the rocky hill to receive the advance, the regiments were ranged, from right to left: The Seventh, Colonel Aiken; Third, Maj. R. C. Maffett; Second, Colonel Kennedy; Third battalion, Lieut.-Col. W. G. Rice; Eighth, Colonel Henagan. The Fifteenth, Colonel DeSaussure, was still in battle in support of artillery between Kershaw and Hood. Here, at the rocky hill, was the battle ground of the brigade. The Eighth, Third battalion and Second held their ground and beat back the attacks coming again and again against them. Moving around Kershaw's right, before Semmes could come to his support, a large force assaulted the Seventh and pushed back its right. The Third held its ground until the Seventh was crowded back at right angles, and then changed its front to support
- 51. the Seventh. A part of Semmes' brigade came up, but the enemy were so far in rear of Kershaw's right as to cut off the support. Surrounding his right, the attacking force drove back the Seventh, and the battle on Kershaw's right was with the Third and Seventh and one of Semmes' regiments at close quarters among the rocks and trees of the hill-crest and sides. Meanwhile the left was holding fast. On came Wofford toward the conflict, and on the right Semmes' other regiments and the Fifteenth South Carolina. Sweeping up to the battle everything gave way before the charge, and joining Wofford and Semmes, Kershaw's line moved forward, the advance sweeping the whole wheat-field and beyond to the foot of the mountain. Night came on, and the brigades of McLaws were put on the hill along the positions gained by the battle. General Kershaw's losses were severe and grievous. The brave and able Colonel De Saussure, of the Fifteenth, and Major McLeod, of the Eighth, gallant in fight and estimable in life, had both fallen; Colonel De Saussure killed on the field and Major McLeod mortally wounded. Among the wounded were Colonel Kennedy of the Second, Lieut.- Col. Elbert Bland of the Seventh, and Maj. D. B. Miller of the Third battalion. The writer regrets that he can find no list of the line officers killed and wounded in the brigade at Gettysburg. The brigade lost 115 killed, 483 wounded and 32 missing, making a total of 630. Bachman's and Garden's batteries with Hood's right, and Rhett's battery, under Lieutenant Gilbert, were in action during the day, but there are no reports at hand of their casualties. If the problem presented to the mind of General Lee on the morning of the 2d, as he saw his army, inferior in numbers and equipment, confronted by the army of General Meade on the heights of Gettysburg, was one which gave him the deepest concern, how much more serious was the situation on the morning of the 3d! General Longstreet's battle on the right had driven the Federal left to the crests, and the Confederate infantry and artillery of that wing were occupying the positions which the Federal forces had held on
- 52. the morning of the 2d. But now the Federal army was intrenched on those heights, with the Round Tops bristling with artillery and Cemetery hill and Culp's hill crowned by batteries, seven corps behind breastworks of stone or earth, and the slopes in front guarded by advanced lines lying behind fences or covered in the woods. There is no record of a council of war. Longstreet, second in command, continued to favor a movement around the Federal left; but General Lee disapproved, and resolutely determined to attack the Federal citadel, confident that the men who had swept Hooker's army from the heights of Chancellorsville, if properly supported, could carry victory to the heights of Gettysburg. He selected the Federal left center as the point of attack; ordered, as on the 2d, concert of action from both wings of his army, and organized his assaulting column of 15,000 men. Stuart's cavalry had come up on his left and confronted the main body of Meade's cavalry. The situation on his extreme right was more serious than the Confederate general realized. This is evident from the reports. The Round Tops were unassailable by the force at Longstreet's command, and a division of cavalry, Farnsworth's and Merritt's brigades, was in position on the right rear, confronted by a single regiment, the First South Carolina cavalry, Bachman's South Carolina battery, and three regiments of Anderson's Georgia brigade. Anderson's regiments were at right angles to Longstreet's line, and Colonel Black's cavalry was on Anderson's right flank. Black had only about 100 men in his regiment. In Longstreet's immediate front the situation was such that there was nothing to do but stand on the defensive. He was weaker in numbers on the 3d than he was on the morning of the 2d, and his enemy was stronger by reinforcements and the occupation of the greater of the two Round Tops. If, however, the assaulting column of 15,000 could break the center, the wings of General Meade's army would be so shaken that both Longstreet and Ewell could attack with good hope of success, and Lee was fixed in his purpose.
- 53. The column of attack was made up of the divisions of Pickett and Pettigrew (Heth's), to be supported by Wilcox and the brigades of Lane and Scales under Trimble. All the available artillery of Hill's and Longstreet's corps was put in position by Col. E. P. Alexander, and at 1 o'clock General Longstreet ordered the batteries to open. For two hours more than 200 cannon were in action across the plain against Federal and Confederate. At 3 the assaulting column moved out from cover and down toward the Emmitsburg road, which ran between the two armies, and at the point of attack was held by the Federal pickets. The Confederate batteries had ceased firing and could give no more support, for their ammunition was nearly exhausted, no supply near at hand, and it was essential to reserve the supply in the chests. All the reports of the advance concur in the statement that the troops moved over the field and into the fire of the enemy's batteries in beautiful order. Coming under the canister fire of the batteries on the crest, the ranks began rapidly to thin and officers to fall, but the advance was steady. General Trimble, riding with his line, then 100 yards in rear of Pettigrew, said: Notwithstanding the losses as we advanced, the men marched with the deliberation and accuracy of men on drill. I observed the same in Pettigrew's line. The enemy's batteries were on the crest. Below them 30 or 40 yards on the slope, and running almost parallel with the crest, was a stone wall, breast high. Behind this wall lay the Federal first line. Below this line, some hundred yards, concealed in the undergrowth, lay his advance line. Beyond it, at the road, ran his picket line. Meeting the pickets, they were immediately driven in, and Garnett and Kemper marched against the advance line in the undergrowth. The resistance was slight, prisoners were made, and the attack so vigorous and dashing that the Federal line was driven in rout. But the enemy's batteries opened with redoubled activity, and the fire from the stone wall was galling. A battery on Little Round Top, enfilading the front of the stone wall, and another from Cemetery
- 54. hill, plunged their shell into the ranks of Kemper and Garnett and raked the advancing line of Armistead as it moved up in support. Garnett led his brigade forward against the stone wall and got in advance, and arrived within 50 yards, where the fire was so severe that it checked his onset and he sent back to hurry up Kemper and Armistead. Both these brigades were struggling through the withering fire, and in a few moments were abreast with Garnett. At 25 yards from the wall Garnett was shot from his horse. Kemper had fallen and Armistead had been killed, but officers and men rushed for the wall and planted their standards. The fighting at this line was desperate, and hand to hand. But the conflict was too unequal to avail the gallant survivors of Garnett and Kemper and Armistead. Of the three brigades scarcely a picket line was left to grapple with the battle array of their foe. The remnant gave up the fight and left the field. If Wilcox could have reached the wall with his gallant Alabamians, the fight might have been prolonged—it might have been successful. But to reach that stone wall Wilcox must march through the fire that shot to pieces the brigades of Kemper, Garnett and Armistead. General Wilcox says that he reached the foot of the hill; that he could not see a man whom he was sent to support; that he was subjected to such an artillery fire from front and both flanks that he went back in search of a battery; that he could find none; that returning to his brigade he regarded further advance useless and ordered a retreat. On the left, Pettigrew and Trimble carried their battle to the Emmitsburg road and to the advanced line. Archer's brigade, on Garnett's immediate left, had 13 color-bearers shot one after another in gallant efforts to plant the colors of his five regiments on the stone wall. The direction of the Federal line was oblique to the general line of advance. Pettigrew's line was exposed longest to the front and flank fire, and at the Emmitsburg road he had suffered more severely than Pickett's brigades. When Pettigrew was yet 150 yards from the Emmitsburg road, says General Trimble, who was about that distance in his rear, They seemed to sink into the earth from the tempest of fire poured into them. Although wounded,
- 55. Pettigrew led his line across the road and against the first line, but his brigades were shattered too badly to make organized assault further. Archer's brigade on his right fought at the stone wall, as did Garnett's and Kemper's and Armistead's, and suffered a like repulse. Officers and men from the other brigades reached the wall and fought with desperate courage, and died beside it, but the division in its organization was torn asunder and shot to pieces by the time they reached and attacked the first line. Trimble's brigades were as helpless for successful assault as Pettigrew; and yet they moved on until within pistol shot of the main line. As General Trimble followed his line back to Seminary ridge, on horseback, under the increased fire of shell, grape and musketry, he reported his wonder that any one could escape wounds or death. And, indeed, but few did. The loss is reported for Garnett, Kemper, Armistead and Wilcox, but there is no report given of the particular loss of July 3d in Pettigrew's command, or Trimble's. The three brigades of Pickett lost their brigadiers, nearly every field officer, and nearly or quite 3,000 men. With the failure of this attack, the great contest at Gettysburg was decided. While it was in progress General Stuart, on the rear of General Lee's left, was fighting a great cavalry battle with the main body of General Meade's cavalry. Stuart had the brigades of Hampton, Fitz Lee, Chambliss, W. H. F. Lee and Jenkins. In the battle much of the fighting was at close quarters and with pistol and saber as the charging lines came together. In one of these contacts General Hampton was twice severely wounded. On the day previous, his having been the first of General Stuart's brigades to reach the vicinity of Gettysburg, he was just in time to meet a cavalry force moving from Hunterstown directly against General Lee's unprotected left. After a sharp engagement General Hampton defeated this force, and drove it beyond reach. The arrival of Stuart on the 2d was a source of infinite satisfaction to the Confederate commander; indeed, if he had not come, the three divisions of General Pleasanton would have taken complete possession of General Lee's communications, and the battle of Gettysburg would have been a still greater disaster to the Southern army.
- 56. After the defeat of the assaulting column, Meade was too cautious to risk his lines against the army that had held the heights of Fredericksburg. He stood resolutely on the defensive throughout the 4th of July. On that night General Lee began his masterly retreat to the Potomac, which he crossed in the face of his enemy on the morning of the 14th. Ewell's corps forded the river at Williamsport, Generals Longstreet and Hill crossed by pontoon at Falling Waters, and by 1 p. m. of the 14th the Gettysburg campaign was over.
- 57. T CHAPTER XVI. SOUTH CAROLINIANS AT CHICKAMAUGA—ORGANIZATION OF THE ARMIES—SOUTH CAROLINIANS ENGAGED—THEIR HEROIC SERVICE AND SACRIFICES. he armies of Generals Bragg and Rosecrans, which were to fight the battle of Chickamauga on the 19th and 20th of September, 1863, were widely separated in the early part of August, Bragg at Chattanooga and Rosecrans beyond the Cumberland mountains, with the Tennessee river rolling between them. About the middle of August, the Federal general broke up his encampments and moved his army across the mountains to the Tennessee. Crittenden's corps threatened Chattanooga through the gaps in Walden's ridge, while Thomas' corps and McCook's moved to Stevenson, Bridgeport and the vicinity. Rosecrans established his depot at Stevenson and passed his army over the river on pontoons, rafts and boats, and boldly crossed Sand mountain to Trenton. He was on the flank of General Bragg by the 8th of September, and by the 12th had crossed Lookout mountain. Bragg, having left Chattanooga on the 8th, Rosecrans sent Crittenden's corps to occupy that place and move on the railroad as far as Ringgold, while Thomas and McCook took position in McLemore's cove and down as far as Alpine. Rosecrans' corps was widely separated and his wings were by road, 50 miles or more apart! Meanwhile Bragg was on the line of Chickamauga creek, with his left at Lafayette and his headquarters at Lee Gordon's mills. General Gist's South Carolina brigade, with Ferguson's battery, was guarding his extreme left at Rome and supporting the cavalry in that quarter. Crittenden's corps at Ringgold and vicinity was at General Bragg's mercy. He was only 10 miles from Bragg's headquarters,
- 58. with the Chickamauga between himself and Thomas, and by road at least 20 miles from that general's support. McCook was fully as far from Thomas on the other flank. It was therefore a matter of life and death (says Rosecrans in his report) to effect the concentration of the army. Crittenden marched across Bragg's right, passed the Chickamauga and moved down toward Thomas, and McCook marched up from Alpine toward that general's position in McLemore's cove. Pigeon mountain range covered McCook and Thomas; but Crittenden's march was open to attack. His corps should have been beaten and driven off toward Chattanooga. General Bragg clearly saw this and endeavored to strike Crittenden at the proper moment, giving explicit orders to that effect. These orders were not executed, the opportunity passed, and Rosecrans united his corps on the west side of the Chickamauga, while Bragg confronted him on the east. The great battles of the 19th and 20th of September were now imminent. We give the organization of the two armies as they were engaged in that memorable conflict, omitting those troops which were not in the battle; as, for instance, the brigades of Hood's and McLaws' divisions, and the artillery of those commands. Longstreet had only three brigades in battle on the 19th and five on the 20th, the artillery and other commands of his corps not having arrived. Among his absent brigades was that of Gen. Micah Jenkins, composed of South Carolina regiments. BRAGG'S ARMY. RIGHT WING, LIEUTENANT-GENERAL POLK COMMANDING. Hill's corps, Lieut.-Gen. D. H. Hill: Cheatham's division, 5 brigades, 5 batteries; Cleburne's division, 3 brigades, 3 batteries; Breckinridge's division, 3 brigades, 4 batteries. Walker's corps, Maj.-Gen. W. H. T. Walker: Walker's division, 3 brigades, 2 batteries; Liddell's division, 2 brigades, 2 batteries. Total of wing, 5 divisions, 16 brigades, 16 batteries.
- 59. LEFT WING, LIEUTENANT-GENERAL LONGSTREET COMMANDING. Buckner's corps, Major-General Buckner: Stewart's division, 4 brigades, 4 batteries; Preston's division, 3 brigades, 3 batteries; Johnson's division, 2 brigades, 2 batteries. Longstreet's corps, Major-General Hood: McLaws' division, 2 brigades; Hood's division, 3 brigades; Hindman's division, 3 brigades, 3 batteries; Reserve artillery, 5 batteries. Total of wing, 6 divisions, 17 brigades, 17 batteries. Total in both wings, 11 divisions, 33 brigades, 33 batteries. Corps of cavalry, Major-General Wheeler, operating on Bragg's left: Wharton's division, 2 brigades, 1 battery; Martin's division, 2 brigades, 1 battery. Corps of cavalry, Major-General Forrest, operating on Bragg's right: Armstrong's division, 2 brigades, 2 batteries; Pegram's division, 2 brigades, 2 batteries. Total of cavalry, 4 divisions, 8 brigades, 6 batteries. ROSECRANS' ARMY. Fourteenth corps, Major-General Thomas commanding: Baird's division, 3 brigades, 3 batteries; Negley's division, 3 brigades, 3 batteries; Brannan's division, 3 brigades, 3 batteries; Reynolds' division, 3 brigades, 3 batteries. Twentieth corps, Major-General McCook commanding: Davis' division, 3 brigades, 5 batteries; Johnson's division, 3 brigades, 3 batteries; Sheridan's division, 3 brigades, 3 batteries. Twenty-first corps, Major-General Crittenden commanding: Wood's division, 3 brigades, 3 batteries; Palmer's division, 3 brigades, 4 batteries; Van Cleve's division, 3 brigades, 3 batteries.
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