![Chapter 1
From Complex to Spatial Networks
The study of spatial networks – networks embedded in space – started essentially with
quantitative geographers in the 60–70s who studied the structure and the evolution of
transportation systems. The interest for networks was revived by Watts and Strogatz
who opened the way to a statistical physics type of analysis and modeling of large
networks. This renewed interest, together with an always growing availability of data,
led to many studies of networks and their structures. Most of these studies focused
on the topological properties of networks, leaving aside their spatial properties. It is
only recently that researchers realized the importance of geometry – as opposed to
topology – for spatial networks. In this chapter, we first describe briefly the evolution
of these fields and ideas about spatial networks. Most of these objects are planar and
in the second part of this chapter, we give basic definitions and results for planar
graphs.
1.1 Early Days
The research activity on networks was intense this last decade (see [1, 2] and other
reviews) but spatial networks were already the subject of many papers and books
more than 40 years ago [3, 4]. In particular, in their great book [3], Haggett and
Chorley explored the topology and the geometry of transportation networks (road
networks, subways, and railways). In the last chapter of their book, they addressed
the problem of patterns of spatial evolution, a subject that is still at the heart of
modern studies. In another study, Kansky [5] defined many indicators to characterize
highways and roads, and in [6], Taafe, Morrill, and Gould proposed a model for the
evolution of road networks in cities, followed by many others (see [3]). Despite these
various empirical and theoretical studies, the subject of the structure of these spatial
networks was only revived later, first by geographers and then by physicists. The
important difference between now and the 70s is certainly the availability of data,
© Springer International Publishing AG 2018
M. Barthelemy, Morphogenesis of Spatial Networks,
Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-319-20565-6_1
1](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-23-320.jpg)
![2 1 From Complex to Spatial Networks
the existence of large computer capacities and most importantly, a better knowledge
of the structure of large networks.
In these early days, theoretical approaches were very simple and relied heavily
on (basic) graph theory. The topological classification of networks amounted then
essentially to distinguish between planar and nonplanar, and for planar networks,
between trees and graphs with loops. For these networks, characterizations were
basic and the indicators were mostly various combinations of the number of nodes
N, the number of edges E and for planar graphs, the number of faces F. All these
measures – the cyclomatic number, the Alpha, Beta and Gamma indices, the average
degree, the average shortest path, etc. do not take into account the spatial nature of
these networks, and therefore represent only one specific aspect of these objects. An
important goal for geographers was then to understand the evolution of these systems
and how these different network measures depend on socioeconomical indicator. For
example, Kansky [5] discussed the relation of the Beta index (given by β = E/N)
for railways with the gross energy consumption in different countries. Of course,
they also investigated some spatial aspects, such as the shape of the network, density
of roads, flow properties, etc., and we refer the interested reader to this excellent
book for more details.
1.2 Complex Networks
Independently, from studies in quantitative geography or in graph theory, physicists
started from 1998 to work intensively on networks. With the first paper on small-
world networks by Watts and Strogatz [7], the statistical physics community realized
that their tools for empirical analysis and modeling could be useful in other fields,
even far from traditional objects of study in physics. This seminal work triggered
a wealth of analysis of all possible networks available at that time. In particular, it
was realized that many complex systems are very often organized under the form
of networks and that these tools and models (together with new ones) have a strong
impact across many disciplines. An important change of paradigm occurred when
we realized [8, 9] that the usual Erdos–Renyi random network was not representative
of most networks observed in real-world, and we had to include large fluctuations of
the degree. These strong fluctuations have a crucial impact on the dynamics that take
place on these networks and many studies were devoted to this phenomenon [2].
This intense activity on networks thus led the researchers to think about the char-
acterization of large networks. All the information is a priori encoded in the adja-
cency matrix but it is usually far too large and difficult to use under this form. In
order to extract a smaller amount of information easily usable and that characterize
the network, scientists introduced a number of measures that describe the statisti-
cal features of large networks. For example, for the class of networks with strong
degree fluctuations, the degree distribution, the diameter, the clustering coefficient
and the assortativity give a reasonable coarse-grained picture of the network and
are in general enough to describe the dynamics on these networks. It appears that](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-24-320.jpg)
![1.2 Complex Networks 3
degree fluctuations are essential and govern many processes, but we note here that
other quantities (such as correlations, for example) can also play a critical role in the
dynamics on networks (see, for example, the case of epidemics [10]).
1.3 Space Matters
These various studies on complex networks largely ignored space and considered that
these networks were living in some abstract world with no metrics. In many cases
indeed, the network is introduced as a simplified way to describe interactions between
elements. Transportation and mobility networks, Internet, mobile phone networks,
power grids, social and contact networks, neural networks, are however all embedded
in space, and for these networks, space is relevant and topology alone does not
contain all the information. In other words, in order to completely characterize these
networks, we need not only the adjacency matrix but also the list of the position of the
nodes. An important consequence of space on networks is that there is naturally a cost
associated to the length of edges which in turn has dramatic effects on the topological
structure of these networks [11]. Characterizing and understanding the structure and
the evolution of these “spatial networks” became an important subject with many
consequences in various fields ranging from epidemiology, neurophysiology, to ICT,
urbanism, and transportation studies. These networks are usually very large and we
need statistical tools in order to describe the most accurately possible the salient
aspects of their organization by taking into account both the topological and spatial
aspects. In the first chapters (1–7), we will review the most important tools for their
characterization.
1.4 Definition and Representations
The representation of a network is not unique in general and we introduce here the
main definitions and representations used in the framework of spatial networks.
1.4.1 Spatial Networks
A graph G = (V, L) is usually defined as a combination of a set V of N nodes
and a set L of E links connecting these nodes. The N × N adjacency matrix A is
then simply defined by ai j = 1 if there is a link between nodes i and j and ai j = 0
otherwise. This definition can be extended to weighted networks with ai j = wi j
where the weight wi j denotes any quantity that flows on this link (i, j). Note that for
directed networks A is not symmetric and usually ai j denotes the link that goes from
j to i. Also, for practical purposes it is not necessary to store the whole matrix A as](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-25-320.jpg)
![1.4 Definition and Representations 5
Fig. 1.2 (Left) Primal and (right) dual networks for a square lattice. In this example, the lattice in
primal space has N = 8 routes. Each route has k = N/2 = 4 connections, so the total number of
connections is Ktot = k2 = 16. In the dual network, the four East–West routes (A, B, C, D) and
the four North–South routes (E, F, G, H) form the bipartite graph K4,4 with a diameter equal to 2.
Figure taken from [12]
a
b
c
d
e
f
g
h
x
y
z
a
b
c
d
e
f g
h
a
b
c
d
e
f
g
h
s
a
b
c
d
e
f
g
h
Fig. 1.3 a Direct representation of the routes (here for three different routes). b Space-of-changes
(sometimes called P space [14, 15]). A link connects two nodes if there is at least one vehicle that
stops at both nodes. c Space-of-stops. Two nodes are connected if they are consecutive stops of at
least one vehicle. d Space-of-stations. Here two stations are connected only if they are physically
connected (without any station in between) and this network reflects the real physical network.
Figure taken from [13]](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-27-320.jpg)
![6 1 From Complex to Spatial Networks
Concerning the important case of transportation networks [13], Kurant and Thiran
discuss very clearly the different representations of these systems (Fig.1.3). The
simplest representation is obtained when the nodes represent the stations and links
the physical connections. One could, however, construct other networks such as the
space-of-stops or the space-of-changes (see Fig.1.3). One also finds in the literature
on transportation systems, the notions of L and P-spaces [14, 16], where the L-space
connects nodes if they are consecutive stops in a given route. The degree in L-space
is then the number of different nodes one can reach within one segment and the
shortest path length represents the number of stops. In the P-space, two nodes are
connected if there is at least one route between them so that the degree of a node is
the number of nodes that can be reached directly. In this P-space, the shortest path
represents the number of connections needed to go from one node to another.
1.5 Planar Graphs
As we will see throughout this book, most spatial networks are well described by
planar networks. These graphs that can be represented in a two-dimensional plane
without any edge crossings (see, for example, the textbook [17]). The particular case
of random planar graphs pervade many different fields from abstract mathematics
[18, 19], to quantum gravity [20], botanics [21, 22], geography and urban studies
[11]. In particular, planar graphs are central in biology, where they can be used
to describe veination patterns of leaves or insect wings and display an interesting
architecture with many loops at different scales [21, 22]. In the study of urban
systems, planar networks are extensively used to represent, to a good approximation,
various infrastructure networks [11] such as transportation networks [3] and streets
patterns [23–41]. Understanding the structure and the evolution of these networks
is therefore interesting from a purely graph theoretical point of view, but could also
have an impact in different fields where these structures are central.
As mentioned above, a graph is planar when there is at least one plane embedding
such that no edges cross each other. However, if a certain embedding displays edge
crossing, it does not necessarily mean that the graph is nonplanar. Standard graph
theory shows that a necessary and sufficient condition for planarity is the absence
of subgraphs homeomorphic to the two graphs: K5 and K3,3 (see Fig.1.4, where the
complete graph Kn with n = 5 nodes and the complete bipartite graph Kn,m with
n = 3 and m = 3 are shown). This is the Kuratowsky theorem (see for example, the
textbook [17]) and there are efficient algorithms that can test this in O(N) time (see
for example [42]).
Basically, a planar graph is thus a graph that can be drawn in the plane in such
a way that its edges do not intersect. Not all drawings of planar graphs are without
intersection and a drawing without intersection is sometimes called a plane graph
or a planar embedding of the graph (the term planar map is also frequently used
in combinatorial studies). In real-world cases, these considerations actually do not
apply since the nodes and the edges represent in general physical objects.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-28-320.jpg)
![1.5 Planar Graphs 7
Fig. 1.4 Complete graphs K5 and K3,3. The Kuratowsky theorem states that all nonplanar graphs
have subgraphs homeomorphic to one (or both) of these graphs
We note here that it is not trivial to demonstrate that a graph is nonplanar and the
demonstration is simplified by invoking the Jordan curve theorem (see for example
[43]) which asserts that a continuous, non-self-intersecting closed-loop divides the
plane into an interior and an exterior that can be connected by a continuous path
that has to intersect the loop somewhere. In order to illustrate a non-planarity demon-
stration, we follow here [17] in the case of the complete graph K5. We assume that
K5 is planar and will reach a contradiction. We denote its vertices by v1, v2, v3, v4, v5,
and since they are all connected to each other, the loop C = v1v2v3v1 exists and is
a Jordan curve separating an inside from an exterior domain. The node v4 does not
lie on C and we assume that it is in the inside domain (there is a similar argument
in the other case where v4 is outside). The interior of C is then divided into three
different domains i for i = 1, 2, 3 delimited by the Jordan curves C1 = v1v2v4v1,
C2 = v4v2v3v4, and C3 = v4v3v1v4 (see Fig.1.5). The remaining node v5 must then lie
C
v4
v
2
v
1
v 3
Ω1
Ω2
Ω3
Fig. 1.5 Demonstration that K5 is nonplanar. Case considered in the text: v4 is inside C and divides
the interior of C in three domains](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-29-320.jpg)
![1.5 Planar Graphs 9
Chatelet
Place
Bastille
Reuilly-
Diderot
Gare de
Lyon
Bercy
Chatelet
Place
Bastille
Reuilly-
Diderot
Gare de
Lyon
Bercy
Fig. 1.7 The nonplanarity of the Paris subway. (Left) ‘Map’ representation where the relative
position of the nodes are respected. (Right) Same graph but which displays more clearly the K3,3
structure. Despite this structure, we observe that in the real graph there is only one edge crossing
For planar graphs, we thus have cr(G) = 0. In general, the crossing number is
very difficult to compute (and might be a NP-complete problem) and the interested
readers can find some discussions about this problem in the excellent graph theory
book [44] or in the more specialized survey [45].
Mathematicians worked in particular on the complete graph Kn where we have
the scaling cr(Kn) ∼ n4
. This scaling can be understood by computing the largest
number of crossings [44]: if we place the vertices on a circle, in order to produce
a crossing we have to choose 4 vertices, and if we assume that we always create a
crossing, we obtain
cr(Kn) ≤
n
4
∼ n4
(1.1)
Actually, better bounds can be found (see [44])
1
80
n4
+ O(n3
) ≤ cr(Kn) ≤
1
64
n4
+ O(n3
) (1.2)
It is interesting to see that the best drawing is actually obtained by avoiding a finite
fraction of the worst case (of order 24/64 ≈ 37%).
We also have results for the complete bipartite Kn,m (for a survey of various
results, see [45])
m(m − 1)
5
n
2
n − 1
2
≤ cr(Km,n) ≤
n
2
n − 1
2
m
2
m − 1
2
(1.3)
(where x is the lowest nearest integer of x), demonstrating a scaling of the form
cr(Km,n) ∼ m2
n2
(1.4)](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-31-320.jpg)
![10 1 From Complex to Spatial Networks
Despite these results, crossing numbers are however not well known and only
few general results are available. In particular, there is a theorem (Ajtai-Chvatal-
Newborn-Szemeredi and Leighton, see [44]), that states that for a simple graph G
with E ≥ 4N (which means an average degree k ≥ 8), the following bound holds
cr(G) ≥
1
64
E3
N2
(1.5)
1.5.2 Basic Results
Basic results for planar networks can be found in any graph theory textbook (see, for
example [17]) and we recall here briefly the most important ones.
We start with very general facts that can be demonstrated for planar graphs, and
among them Euler’s formula is probably the best known. Euler showed that a finite
connected planar graph satisfies the following formula
N − E + F = 2 (1.6)
where N is the number of nodes, E the number of edges, and F is the number
of faces. This formula can be easily proved by induction by noting that removing
an edge decreases F and E by one, leaving N − E + F invariant. We can repeat
this operation until we get a tree for which F = 1 and N = E + 1 leading to
N − E + F = E + 1 − E + 1 = 2. This argument can be repeated in the case where
the graph is made of C disconnected components and the Euler relation reads in this
case
N − E + F = C + 1 (1.7)
Moreover, for any finite connected planar graph we can obtain a bound for the
average degree k . Indeed, any face is bounded by at least three edges and every edge
separates two faces at most which implies that E ≥ 3F/2. From Euler’s formula,
we then obtain
E ≤ 3N − 6 (1.8)
In other words, planar graphs are always sparse with a bounded average degree
k ≤ 6 −
12
N
(1.9)
which is therefore always smaller than 6.
We end this part with a particular class of planar graphs that are constructed on a set
of points distributed in the plane (see the Chap.8 about tessellations). The maximal](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-32-320.jpg)
![Chapter 2
Irrelevant and Simple Measures
Many studies on complex networks were about how to characterize them and what
are the most relevant measures for understanding their structure. In particular, the
degree distribution and the existence of the second moment for an infinite network
were shown to be critical when studying dynamical processes on networks. These
behaviors are therefore strongly connected to degree fluctuations and the existence
of hubs. In the case of spatial networks, the physical constraints are usually large
and prevent the appearance of such hubs. These constraints also impact other quan-
tities that are nontrivial for complex networks but that become irrelevant for spatial
networks. We review here these measures that are essentially useless for spatial net-
works and we then discuss older, simple measures that were mostly introduced in
the context of quantitative geography.
2.1 Irrelevant Measures
Quantities that depend very much on the spatial constraints turn out in general to be
irrelevant for spatial networks. The prime example is the degree distribution which
in many complex networks was found to be a broad law and in some cases well
fitted by a power law of the form P(k) ∼ k−γ
with 1 γ 3 [1]. In this case,
degree fluctuations are very large which has a direct impact on many dynamical
processes that take place on the network, such as epidemics for example [2]. For
spatial networks, however, the degree has to satisfy steric constraints. If we consider
the road network, nodes represent the intersections and the degree of a node is the
number of streets starting from it and is therefore clearly limited as a result of space.
As a consequence, the degree distribution for most spatial networks is not broad but
displays a fast decaying tail (such as an exponential for example).
© Springer International Publishing AG 2018
M. Barthelemy, Morphogenesis of Spatial Networks,
Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-319-20565-6_2
13](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-34-320.jpg)
![14 2 Irrelevant and Simple Measures
Other irrelevant parameters include the clustering coefficient or the assortativity.
Indeedasshownbelowtheclusteringcoefficientisalwayslarge:ifanodeisconnected
to two other nodes in a spatial network, they are usually located in its neighborhood
which in turn increases the probability that they are connected to each other, leading
to a large clustering coefficient. In the following, we will discuss in more detail these
different measures.
2.1.1 Degree
We recall here that a graph with N nodes and E edges can be described by its N × N
adjacency matrix A which is defined as
Ai j =
= 1 if i and j are connected
= 0 otherwise
(2.1)
If the graph is undirected, then the matrix A is symmetric. The degree of a node is
by definition the number of its neighbors and is given by
ki =
j
Ai j (2.2)
The first simple indicator of a graph is the average degree
k =
1
N
i
ki =
2E
N
(2.3)
where here and in the following the brackets · denote the average over the nodes
of the network. In particular, the scaling of k with N indicates if the network is
sparse (which is the case when k → const. for N → ∞).
In [31, 46], measurements for street networks in different cities in the world are
reported. Based on the data from these sources, the authors of [47] plotted (Fig.2.1a)
the number of roads E (edges) versus the number of intersections N. The plot is
consistent with a linear fit with slope ≈1.44 (which is consistent with the value
k ≈ 2.5 measured in [46]). The quantity e = E/N = k/2 displays values in the
range 1.05 e 1.69, in between the values e = 1 and e = 2 that characterize
tree-like structures and 2d regular lattices, respectively. Few exact values and bounds
are available for the average degree of classical models of planar graphs. In general,
it is known that e ≤ 3, while it has been recently shown [48] that e 13/7 for planar
Erdös–Renyi graphs [48].
The distribution of degree P(k) is usually a quantity of interest and can display
large heterogeneities such as it is observed in scale-free networks (see for example
[49]). We indeed observe that for spatial networks such as airline networks or the](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-35-320.jpg)
![2.1 Irrelevant Measures 15
0 500 1000 1500 2000 2500 3000
N
0
1000
2000
3000
4000
5000
6000
E
Linear fit
Tree limit
Random planar graph
0 100 200 300 400
0
100
200
300
400
500
600
0 500 1000 1500 2000 2500 3000
N
0
20
40
60
80
l
(a)
(b)
Fig. 2.1 a Numbers of roads versus the number of nodes (i.e., intersections and centers) for data
from [31] (circles) and from [46] (squares). In the inset, we show a zoom for a small number of
nodes. b Total length versus the number of nodes. The line is a fit which predicts a growth as
√
N
(data from [31] and figures from [47])
Internet, the degrees are very heterogeneous (see [2]). However, when physical con-
straints are strong or when the cost associated with the creation of new links is large,
a cutoff appears in the degree distribution [8] and in some case the distribution can be
very peaked. This is the case for the road network for example, and more generally in
the case of planar networks for which the degree distribution P(k) is of little interest.
For example, in a study of 20 German cities, Lämmer et al. [29] showed that most
nodes have four neighbors (the full degree distribution is shown in Fig.2.2a) and that
the degree rarely exceeds 5 for various world cities [31]. These values are, however,
not very indicative: planarity imposes severe constraints on the degree of a node and
on its distribution which is generally peaked around its average value.
We note here that in real-world cases such as the road network for example,
it is natural to study the usual (or “primal”) representation where the nodes are
the intersections and the links represent the road segment between the intersection.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-36-320.jpg)
![16 2 Irrelevant and Simple Measures
Fig. 2.2 a Degree distribution of degrees for the road network of Dresden. b The frequency distri-
bution of the cells surface areas Ac obeys a power law with exponent α ≈ 1.9 (for the road network
of Dresden). Figure taken from [29]
However, in another representation, the dual graph can be of interest (see [27])
and for the road network it is constructed in the following way: the nodes are the
roads and two nodes are connected if there exists an intersection between the two
corresponding roads. One can then measure the degree of a node which represents
the number of roads which intersect a given road. Also, the shortest path length in
this network represents the number of different roads one has to take to go from one
point to another. Even if the road network has a peaked degree distribution, its dual
representation can display broad distributions [50]. Indeed, in [50], measurements
were made on the dual network for the road network in the US, England, and Denmark
and showed large fluctuations with a power-law distribution with exponent 2.0
γ 2.5.
2.1.2 Length of Segments
In Fig.2.1b, we plot the total length T of the network versus N for the cities con-
sidered in [31]. Data are well fitted by a power function of the form
T = μNβ
(2.4)
with μ ≈ 1.51 and β ≈ 0.49. In order to understand this result, one has to focus
on the street segment length distribution P(1). This quantity has been measured
for London in [34] and is shown in Fig.2.3. This figure shows that the distribution
decreases rapidly and the fit proposed by the authors of [34] suggests that
P(1) ∼
−γ
1 (2.5)](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-37-320.jpg)
![2.1 Irrelevant Measures 17
Fig. 2.3 Length distribution P(1) for the street network of London (and for the model GRPG
proposed in [34]). Figure taken from [34]
with γ 3.4 which implies that both the average and the dispersion are well defined
and finite. If we assume that this result extends to other cities, it means that we
have a typical distance 1 between nodes which is meaningful. This typical distance
between connected nodes then naturally scales as
1 ∼
1
√
ρ
(2.6)
where ρ = N/L2
is the density of vertices and L is the linear dimension of the
ambient space. This implies that the total length scales as
T ∼ E1 ∼
k
2
L
√
N (2.7)
This simple argument reproduces well the
√
N behavior observed in Fig.2.1b and
also the value (given the error bars) of the prefactor μ ≈ kL/2.
2.1.3 Clustering, Assortativity, and Average Shortest Path
Complex networks are essentially characterized by a small set of parameters which
are not all relevant for spatial networks. For example, the degree distribution which
has been the main subject of interest in complex network studies is usually peaked for
planar networks, due to the spatial constraints, and is therefore not very interesting.
In the following we will discuss the effect of spatial constraints on other important
parameters.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-38-320.jpg)
![18 2 Irrelevant and Simple Measures
2.1.3.1 Clustering Coefficient
The clustering coefficient of a node i of degree ki is defined as
C(i) =
Ei
ki (ki − 1)/2
(2.8)
where Ei is the number of edges among the neighbors of i. This quantity gives some
information about local clustering and was the object of many studies in complex
networks. For the Erdos–Renyi (ER) random graphs with finite average degree k,
the average clustering coefficient is simply given by
C = p ∼
k
N
(2.9)
where the brackets · denote the average over the network (p is the probability
to connect two nodes). In contrast, for spatial networks, closer nodes have a larger
probability to be connected, leading to a large clustering coefficient. The variation
of this clustering coefficient in space can thus bring valuable information about
the spatial structure of the network under consideration. The clustering coefficient
depends on the number of triangles or cycles of length 3 and can also be computed
by using the adjacency matrix A. Powers of the adjacency matrix give the number
of paths of variable length. For instance, the quantity 1
6
Tr(A3
) is the number C3 of
cycles of length tree and is related to the clustering coefficient. Analogously, we can
define and count cycles of various lengths (see for example [51, 52] and references
therein) and compare this number to the ones obtained on null models (lattices,
triangulations, etc.).
Finally, many studies define the clustering coefficient per degree classes which is
given by
C(k) =
1
N(k)
i/ki =k
C(i) (2.10)
The behavior of C(k) versus k thus gives an indication on how the clustering is
organized when we explore different classes of degrees. However, in order to be
useful, this quantity needs to be applied to networks with a large range of degree
variations which is usually not the case in spatial networks.
The average clustering coefficient can be calculated for the random geometric
graph (see also Chap.9) and we discuss in the following the argument presented in
[53]. If two vertices i and j are connected to a vertex k, it means that they are both
in the excluded volume of k. Now, these vertices i and j are connected only if j
is in the excluded volume of i. Putting all pieces together, the probability to have
two connected neighbors (i j) of a node k is given by the fraction of the excluded
volume of i which lies within the excluded volume of k. By averaging over all points
i in the excluded volume of k, we then obtain the average clustering coefficient.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-39-320.jpg)
![2.1 Irrelevant Measures 19
We thus have to compute the volume overlap ρd of two spheres which for spherical
symmetry reasons depends only on the distance between the two spheres. In terms
of this function, the clustering coefficient is given by
Cd =
1
Ve
Ve
ρd(r)dV (2.11)
For d = 1, we have
ρ1(r) = (2R − r)/2R = 1 − r/2R (2.12)
and we obtain
C1 = 3/4 (2.13)
For d = 2, we have to determine the area overlapping in Fig.2.4 which gives
ρ2(r) = (θ(r) − sin(θ(r)))/π (2.14)
with θ(r) = 2 arccos(r/2R) and leads to
C2 = 1 − 3
√
3/4π ≈ 0.58650 (2.15)
θ R
r/2
Fig. 2.4 The overlap between the two disks (area comprised within the bold line) gives the quantity
ρ2(r). Figure taken from [53]](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-40-320.jpg)
![20 2 Irrelevant and Simple Measures
Similarly, an expression can be derived in d dimension [53] which for large d reduces
to
Cd ∼ 3
2
πd
3
4
d+1
2
(2.16)
The average clustering coefficient thus decreases from the value 3/4 for d = 1 to
values of order 10−1
for d of order 10 and is independent from the number of nodes
which is in sharp contrast with ER graphs for which C ∼ 1/N. Random geometric
graphs are thus much more clustered than random ER graphs. The main reason—
which is in fact valid for most spatial graphs—is that long links are prohibited or rare.
This fact implies that if both i and j are connected to k, it means that there are in some
spatial neighborhood of k which increases the probability that their inter-distance is
small too, leading to a large C.
2.1.3.2 Assortativity
In general, the degrees of the two end nodes of a link are correlated and to describe
these degree correlations one needs the two-point correlation function P(k |k). This
quantity represents the probability that any edge starting at a vertex of degree k ends
at a vertex of degree k . Higher order correlation functions can be defined and we
refer the interested reader to [54] for example. The function P(k |k) is, however, not
easy to handle and one can define the assortativity [55, 56]
knn(k) =
k
P(k |k)k (2.17)
A similar quantity can be defined for each node as the average degree of the neighbor
knn(i) =
1
ki
j∈Γ (i)
kj (2.18)
where Γ (i) denotes the set of neighbors of node i. There are essentially two classes
of behaviors for the assortativity. If knn(k) is an increasing function of k, vertices
with large degrees have a larger probability to connect to similar nodes with a large
degree. In this case, we speak of an assortative network and in the opposite case of
a disassortative network. It is expected in general that social networks are mostly
assortative, while technological networks are disassortative. However, for spatial
networks spatial constraint usually implies a flat function knn(k), since it is usually
the distance that governs the existence of a link and not the degree.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-41-320.jpg)
![2.1 Irrelevant Measures 21
2.1.3.3 Average Shortest Path
Usually, there are many paths between two nodes in connected networks and if we
keep the shortest one it defines a distance on the network
(i, j) = min
paths(i→ j)
|path| (2.19)
where the length |path| of the path is defined as its number of edges. The diameter of
the graph can be defined as the maximum value of all (i, j) or can also be estimated
by the average of this distance over all pairs of nodes in order to characterize the
“size” of the network. For a d-dimensional regular lattice with N nodes, this average
shortest path scales as
∼ N1/d
(2.20)
In a small-world network (see [7] and Chap.10) constructed over a d−dimensional
lattice has a very different behavior
∼ log N (2.21)
The crossover from a large-world behavior N1/d
to a small-world one with log N
can be achieved for a density p of long links (or “shortcuts”) [57] such that
pN ∼ 1 (2.22)
The effect of space could thus in principle be detected in the behavior of (N).
It should, however, be noted that if the number of nodes is too small this can be a
tricky task. In the case of brain networks, for example, a behavior of a typical three-
dimensional network in N1/3
could easily be confused with a logarithmic behavior
if N is not large enough.
2.1.4 Empirical Illustrations
We discuss here some simple results obtained on transportation networks that illus-
trate the fact that indeed some measures that are useful for understanding complex
networks are actually irrelevant in the case of spatial networks and do not convey
interesting information.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-42-320.jpg)
![22 2 Irrelevant and Simple Measures
2.1.4.1 Power Grids and Water Distribution Networks
Power grids are one of the most important infrastructures in our society. In modern
countries, they have evolved for a rather long time (sometimes a century) and are now
complex systems with a large variety of elements and actors playing in their function-
ing. This complexity leads to the relatively unexpected result that their robustness is
actually not very well understood and large blackouts such as the huge August 2003
blackout in North America demonstrates the fragility of these systems.
The topological structure of these networks was studied in different papers such
as [8, 58, 59]. In particular, in [8, 58], the authors consider the Southern Californian
and the North American power grids. In these networks, the nodes represent the
power plants, distribution, and transmission substations, and the edges correspond to
transmission lines. These networks are typically planar (see for example the Italian
case, Fig.2.5) and we expect a peaked degree distribution, decreasing typically as an
exponential of the form P(k) ∼ exp(−k/k) with k of order 3 in Europe and 2 in
the US. The other studies on US power grids confirm that the degree distribution is
exponential (see Fig.2.6). In [58], Albert, Albert, and Nakarado also studied the load
(a quantity similar to the betweenness centrality) and found a broad distribution. The
degree being peaked, we can then expect very large fluctuations of load for the same
value of the degree, as expected in general for spatial networks. These authors also
found a large redundance in this network with, however, 15% of cut edges.
Also, as expected for these networks, the clustering coefficient is rather large and
even independent of k as shown in the case of the power grid of Western US (see
Fig.2.7).
Besides the distribution of electricity, our modern societies also rely on various
other distribution networks. The resilience of these networks to perturbations is thus
an important point in the design and operating of these systems. In [61], Yazdani
and Jeffrey study the topological properties of the Colorado Springs Utilities and the
Richmond (UK) water distribution networks (shown in Fig.2.8). Both these networks
(of size N = 1786 and N = 872, respectively) are sparse planar graphs with very
peaked degree distributions (the maximum degree is 12).
0 5 10 15
k
10
-3
10
-2
10
-1
10
0
Cumulative
distribution
UCTE
UK and Ireland
Italy
c
(a) (c)
(b)
Fig. 2.5 a Map of the Italian power grid. b Topology of the Italian power grid. c Degree distribution
for the European network (UCTE), Italy, the UK, and Ireland. In all cases, the degree distribution
is peaked and can be fitted by exponential. Figure taken from [59]](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-43-320.jpg)
![2.1 Irrelevant Measures 23
0 10 20
# of transmission lines
10
−4
10
−3
10
−2
10
−1
10
0
Cumulative
distribution Power grid
Fig. 2.6 Degree distribution of substations in Southern California (top panel) and for the North
American power grid (bottom panel). In both cases, the lines represent an exponential fit. Figure
taken from [8, 58], respectively
1 10
k
10
−2
10
−1
10
0
C(k)
Fig. 2.7 Scaling of the clustering C(k) for the power grid of the Western United States. The dashed
line has a slope −1 and the solid line corresponds to the average clustering coefficient. Figure taken
from [60]](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-44-320.jpg)
![24 2 Irrelevant and Simple Measures
Fig. 2.8 Representation of water distribution networks. Left panels (from top to bottom): Synthetic
networks(“Anytown”[62],and“EXNET”[63]).Top-rightpanel:ColoradoSpringUtilitiesnetwork.
Bottom-right panel: Richmond (UK) water distribution network. Figure taken from [61]
2.1.4.2 Subways and Buses
One of the first studies (after the Watts–Strogatz paper) on the topology of a trans-
portation network was proposed by Latora and Marchiori [64] who considered the
Boston subway network. It is a relatively small network with N = 124 stations. The
average shortest path is ∼ 16 a value which is large compared to ln 124 ≈ 5 and
closer to the two-dimensional result
√
124 ≈ 11.
In [15], Sienkiewicz and Holyst study a larger set made of public transportation
networks of buses and tramways for 22 Polish cities and in [65], von Ferber et al.
study the public transportation networks for 15 world cities. The number of nodes of
these networks varies from N = 152 to 2811 in [15] and in the range [1494, 44629]
in [65]. Interestingly enough, the authors of [15] observe a strong correlation between
the number of stations and the population which is not the case for the world cities
studied in [65] where the number of stations seems to be independent from the
population (see Sect.14.3 for a detailed discussion about the connection between](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-45-320.jpg)
![2.1 Irrelevant Measures 25
socioeconomical indicators and the properties of networks). For polish cities, the
degree has an average in the range [2.48, 3.08] and in a similar range [2.18, 3.73] for
[65]. In both cases, the degree distribution is relatively peaked (the range of variation
is usually of the order of one decade) consistently with the existence of physical
constraints [8].
Due to the relatively small range of variation of N in these various studies [15,
64, 65], the behavior of the average shortest path is not clear and could be fitted by a
logarithm or a power law as well. We can, however, note that the average shortest path
is usually large (of order 10 in [15] and in the range [6.4, 52.0] in [65]) compared to
ln N, suggesting that the behavior of might not be logarithmic with N but more
likely scales as N1/2
, a behavior typical of a two-dimensional lattice.
The average clustering coefficient C in [15] varies in the range [0.055, 0.161]
and is larger than a value of the order CE R ∼ 1/N ∼ 10−3
− 10−2
corresponding to
a random ER graph. The ratio C/CE R is explicitly considered in [65] and is usu-
ally much larger than one (in the range [41, 625]). The degree-dependent clustering
coefficient C(k) seems to present a power-law dependence, but the fit is obtained
over one decade only.
In another study [66], the authors study two urban train networks (Boston and
Vienna which are both small N = 124 and N = 76, respectively) and their results
are consistent with the previous ones.
2.1.4.3 Railways
Oneofthefirststudiesofthestructureofrailwaynetwork[67]concernsasubsetofthe
most important stations and lines of the Indian railway network and has N = 587
stations. In the P-space representation (see Chap.1), there is a link between two
stations if there is a train connecting them and in this representation, the average
shortest path is of order ≈ 5 which indicates that one needs four connections
in the worst case to go from one node to another one. In order to obtain variations
with the number of nodes, the authors considered different subgraphs with different
sizes N. The clustering coefficient varies slowly with N that is always larger than
≈0.7 which is much larger than a random graph value of order 1/N. Finally, in this
study [67], it is shown that the degree distribution is behaving as an exponential and
that the assortativity knn(k) is flat showing an absence of correlations between the
degree of a node and those of its neighbors.
In [13], Kurant and Thiran studied the railway system of Switzerland and major
trains and stations in Europe (and also the public transportation system of Warsaw,
Poland). The Swiss railway network contains N = 1613 nodes and E = 1680 edges
(Fig.2.9). All conclusions drawn here are consistent with the various cases presented
in this chapter. In particular, the average degree is k ≈ 2.1, the average shortest path
is ≈47 (consistent with the
√
N result for a two-dimensional lattice), the clustering
coefficient is much larger than its random counterpart, and the degree distribution is
peaked (exponentially decreasing).](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-46-320.jpg)
![26 2 Irrelevant and Simple Measures
Fig. 2.9 Physical map of the Swiss railway networks. Figure taken from [13]
2.1.4.4 Neural Networks
The human brain with about 1010
neurons and about 1014
connections is one of the
most complex networks that we know. The structure and functions of the brain are
the subjects of numerous studies and different recent techniques such as electroen-
cephalography, magnetoencephalography, functional RMI, etc. can be used in order
to reconstruct networks for the human brain (see Fig.2.10 and for a clear and nice
introduction see for example [68, 69]).
Brain regions that are spatially close have a larger probability of being con-
nected than remote regions as longer axons are more costly in terms of material
and energy [68]. Wiring costs depending on distance are thus certainly an impor-
tant aspect of brain networks and we can expect spatial networks to be relevant in
this rapidly evolving topic. So far, many measures seem to confirm a large value of
the clustering coefficient, and a small-world behavior with a small average shortest
path length [70, 71]. It also seems that neural networks do not optimize the total
wiring length but rather the processing paths, thanks to shortcuts [72]. This small-
world structure of neural networks could reflect a balance between local processing
andglobalintegrationwithrapidsynchronization,informationtransfer,andresilience
to damage [73].
In contrast, the nature of the degree distribution is still under debate and a recent
study on the macaque brain [74] showed that the distribution is better fitted by an
exponential rather than by a broad distribution. Besides the degree distribution, most
of the observed features were confirmed in latest studies such as [75] where Zalesky](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-47-320.jpg)
![2.1 Irrelevant Measures 27
Fig. 2.10 Structural and functional brains can be studied with graph theory by following different
methods shown step-by-step in this figure. Figure taken from [68]
et al. propose to construct the network with MRI techniques where the nodes are
distinct gray-matter regions and links represent the white-matter fiber bundles. The
spatial resolution is of course crucial here and the largest network obtained here is
of size N ≈ 4,000. These authors find large clustering coefficients with a ratio to
the corresponding random graph value of order 102
. Results for the average shortest
path length are, however, not so clear due to relatively low values of N. Indeed,
for N varying from 1,000 to 4,000, varies by a factor of order 1.7−1.8 [75]. A
small-world logarithmic behavior would predict a ratio
r =
(N = 4000)
(N = 1000)
∼
log(4000)
log(1000)
≈ 1.20 (2.23)
while a three-dimensional spatial behavior would give a ratio of order r ≈ 41/3
≈ 1.6
which is closer to the observed value. Larger sets would, however, be needed in order
to be sure about the behavior of this network concerning the average shortest path
and to distinguish a log N from a N1/3
behavior expected for a three-dimensional
lattice.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-48-320.jpg)
![28 2 Irrelevant and Simple Measures
Things are, however, more complex than it seems and even if functional connec-
tivity correlates well with anatomical connectivity at an aggregate level, a recent
study [76] shows that strong functional connections exist between regions with no
direct structural connections, demonstrating that structural and functional properties
of neural networks are entangled in a complex way and that future studies are needed
in order to understand this extremely complex system.
2.2 Simple Measures
2.2.1 Topological Indices: α and γ Indices
Different indices were defined a long time ago mainly by scientists working in quan-
titative geography since the 1960s and can be found in [3, 77, 78] (see also the more
recent paper by Xie and Levinson [32]). Most of these indices are relatively simple
but give valuable information about the structure of the network, in particular if we
are interested in planar networks. They were used to characterize the topology of
transportation networks: Garrison [79] measured some properties of the Interstate
highway system and Kansky [80] proposed up to 14 indices to characterize these
networks. The simplest index is called the gamma index and is defined by
γ =
E
Emax
(2.24)
where E is the number of edges and Emax is the maximal number of edges (for a
given number of nodes N). For nonplanar networks, Emax is given by N(N − 1)/2
for nondirected graphs and for planar graphs we saw in Chap.1 that Emax = 3N − 6
leading to
γP =
E
3N − 6
(2.25)
The gamma index is a simple measure of the density of the network but one can
define a similar quantity by counting the number of elementary cycles instead of
edges. The number of elementary cycles for a network is known as the cyclomatic
number (see for example [17]) and is equal to
Γ = E − N + 1 (2.26)
For a planar graph, this number is always less or equal to 2N −5 which leads naturally
to the definition of the alpha index (also coined “meshedness” in [46])
α =
E − N + 1
2N − 5
(2.27)](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-49-320.jpg)
![2.2 Simple Measures 29
This index lies in the interval [0, 1] and is equal to 0 for a tree and equal to 1 for a
maximal planar graph. Using the definition of the average degree k = 2E/N, the
quantity α reads in the large N limit as
α
k − 2
4
(2.28)
which shows that in fact for a large network this index α does not contain much more
information than the average degree.
2.2.2 Organic Ratio and Ringness
We note that more recently other interesting indices were proposed in order to char-
acterize specifically road networks [32, 81]. For example, in some cities, the degree
distribution is very peaked around 3−4 and the ratio
rN =
N(1) + N(3)
k=2 N(k)
(2.29)
can be defined [81] where N(k) is the number of nodes of degree k. If this ratio
is small, the number of dead ends and of “unfinished” crossing (k = 3) is small
compared to the number of regular crossings with k = 4 which signals a more
organized city. In the opposite case of large rN (i.e., close to 1), there is a dominance
of k = 1 and k = 3 nodes, which is the sign of a mode “organic” city.
The authors of [81] also define the “compactness” of a city which measures how
much a city is “filled” with roads. If we denote by A the area of a city and by T the
total length of roads, the compactness Ψ ∈ [0, 1] can be defined in terms of the hull
and city areas
Ψ = 1 −
4A
(T − 2
√
A)2
(2.30)
In the extreme case of one square city of linear size L =
√
A with only one road
encircling it, the total length is T = 4
√
A and the compactness is then Ψ = 0. At
the other extreme, if the city roads constitute a square grid of spacing a, the total
length is T = 2L2
/a and in the limit of a/L → 0 one has a very large compactness
Ψ ≈ 1 − a2
/L2
.
We end this section by mentioning the ringness. Arterial roads (including free-
ways, major highways) provide a high level of mobility and serve as the backbone of
the road system [32]. Different measures (along with many references) are discussed
and defined in this paper [32], and in particular, the ringness is defined as
φring =
ring
tot
(2.31)](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-50-320.jpg)
![30 2 Irrelevant and Simple Measures
where ring is the total length of arterials on rings, and the denominator tot is the
total length of all arterials. This quantity ranging from 0 to 1 is thus an indication of
the importance of a ring and to what extent arterials are organized as trees.
2.2.3 Cell Areas and Shape
Planar graphs naturally produce a set of nonoverlapping cells (or faces, or blocks)
and covering the embedding plane. In the case of the road network, the distribution
of the area A of these cells has been measured for the city of Dresden in Germany
(Fig.2.2b) and has the form
P(A) ∼ A−α
(2.32)
with α 1.9, which was confirmed by measures on other cities [11]. This broad law
is in sharp contrast with the simple picture of an almost regular lattice which would
predict a distribution P(A) peaked around 2
1.
It is interesting to note that if we assume that A ∼ 1/2
1 ∼ 1/ρ and that the density
ρ is distributed according to a law f (ρ) (with a finite f (0)); a simple calculation
gives
P(A) ∼
1
A2
f (1/A) (2.33)
which behaves as P(A) ∼ 1/A2
for large A. This simple argument thus suggests
that the observed value ≈2.0 of the exponent is universal and reflects the random
variation of the density. More measurements are, however, needed at this point in
order to test the validity of this hypothesis.
The authors of [29] also measured the distribution of the form or shape factor
defined as the ratio of the area of the cell to the area of the circumscribed circle:
φ =
4A
π D2
(2.34)
(for practical applications, D can be also taken as the longest distance in the cell).
They found that most cells have a form factor between 0.3 and 0.6, suggesting a
large variety of cell shapes, in contradiction with the assumption of an almost regular
lattice. These facts thus call for a model radically different from simple models of
regular or perturbed lattices. In Chaps.3 and 7, we will discuss more thoroughly this
quantity φ and its distribution.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-51-320.jpg)
![2.2 Simple Measures 31
2.2.4 Route Factor, Detour Index
When the network is embedded in a two-dimensional space, we can define at least
two distances between the pairs of nodes. There is of course the natural Euclidean
distance dE (i, j) which can also be seen as the “as crow flies” distance. There is also
the total “route” distance dR(i, j) from i to j by computing the sum of length of
segments which belong to the shortest path between i and j. The route factor (also
called the detour index or the circuity, or directness [82]) for this pair of nodes (i, j)
is then given by (see Fig.2.11 for an example)
Q(i, j) =
dR(i, j)
dE (i, j)
(2.35)
This ratio is always larger than one and the closer to one it is, the more efficient the
network. From this quantity, we can derive another one for a single node defined by
Q(i) =
1
N − 1
j
Q(i, j) (2.36)
which measures the “accessibility” for this specific node i. Indeed the smaller it is
and the easier it is to reach the node i (Accessibility is a subject in itself–see for
example [83]—and there are many other measures for this concept and we refer
the interested reader to the articles [84–86]). This quantity Q(i) is related to the
quantity so-called “straightness centrality” [87] defined as
CS
(i) =
1
N − 1
j=i
dE (i, j)
dR(i, j)
(2.37)
If one is interested in assessing the global efficiency of the network, one can compute
the average over all pairs of nodes (also used in [88])
Fig. 2.11 Example of a detour index calculation. The “as crow flies” distance between the nodes
A and B is dE (A, B) =
√
10 while the route distance over the network is dR(A, B) = 4 leading to
a detour index equal to Q(A, B) = 4/
√
10 1.265](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-52-320.jpg)
![32 2 Irrelevant and Simple Measures
Q =
1
N(N − 1)
i= j
Q(i, j) (2.38)
The average Q or the maximum Qmax , and more generally the statistics of Q(i, j),
is important and contains a lot of information about the spatial network under consid-
eration (see [89] for a discussion on this quantity for various networks). For example,
one can define the interesting quantity [89]
φ(d) =
1
Nd
i j/dE (i, j)=d
Q(i, j) (2.39)
(where Nd is the number of nodes such that dE (i, j) = d) whose shape can help
for characterizing combined spatial and topological properties (see also Chap.7 for
empirical examples).
2.2.5 Cost, Efficiency, and Robustness
The minimum number of links to connect N nodes is E = N − 1 and the corre-
sponding network is a tree. We can also look for the tree which minimizes the total
length given by the sum of the length of all links
T =
e∈E
dE (e) (2.40)
where dE (e) denotes the length of the link e. This procedure leads to the minimum
spanning tree (MST) which has a total length MST
T (see also Sect.12.2 about the
MST). Obviously, the tree is not a very efficient network (from the point of view of
transportation for example) and usually more edges are added to the network, leading
to an increase of accessibility but also of T . A natural measure of the “cost” of the
network is then given by
C =
T
MST
T
(2.41)
Adding links thus increases the cost but improves accessibility or the transport per-
formance P of the network which can be measured as the minimum distance between
all pairs of nodes, normalized to the same quantity but computed for the minimum
spanning tree
P =
MST
(2.42)](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-53-320.jpg)
![2.2 Simple Measures 33
Another measure of efficiency was also proposed in [90, 91] and is defined as
E =
1
N(N − 1)
i= j
1
(i, j)
(2.43)
where (i, j) is the shortest path distance from i to j. This quantity is zero when there
are no paths between the nodes and is equal to one for the complete graph (for which
(i, j) = 1). The combination of these different indicators and comparisons with
the MST or the maximal planar network can be constructed in order to characterize
various aspects of the networks under consideration (see for example [46]).
Finally, adding links improves the resilience of the network to attacks or dysfunc-
tions. A way to quantify this is by using the fault tolerance (FT) (see for example
[92]) measured as the probability of disconnecting part of the network with the fail-
ure of a single link. The benefit/cost ratio could then be estimated by the quantity
FT/MST
T which is a quantitative characterization of the trade-off between cost and
efficiency [92].
Buhl et al. [46] measured different indices for 300 maps corresponding mostly
to settlements located in Europe, Africa, Central America, and India. They found
that many networks depart from the grid structure with an alpha index usually low.
For various world cities, Cardillo et al. [31] found that the alpha index varies from
0.084 (Walnut Creek) to 0.348 (New York City) which reflects in fact the variation
of the average degree. Indeed for both these extreme cases, using Eq. (2.28) leads to
αNYC (3.38−2)/4 0.345 and for Walnut Creek αWC (2.33−2)/4 0.083.
This same study seems to show that triangles are less abundant than squares (except
for cities such as Brasilia or Irvine).
Measures of efficiency are relatively well correlated with the alpha index but
display broader variations demonstrating that small variations of the alpha index can
lead to large variations in the shortest path structure. Cardillo et al. [31] plotted the
relative efficiency (see Chap.1)
Erel =
E − EMST
EGT − EMST
(2.44)
versus the relative cost
Crel =
C − CMST
CGT − CMST
(2.45)
where GT refers to the greedy triangulation (the maximal planar graph). The cost
is here estimated as the total length of segments C ≡ T and the obtained result is
shown in Fig.2.12 which demonstrates two things. First, it shows—as expected—
that efficiency is increasing with the cost with an efficiency saturating at ∼0.8. In
addition, this increase is slow: typically, doubling the value of C shifts the efficiency
from ∼0.6 to ∼0.8. Second, it shows that most of the cities are located in the high-
cost–high-efficiency region. New York City, Savannah, and San Francisco have the](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-54-320.jpg)
![Churches
First Presbyterian Church
In [1785] a bill was introduced into the Legislative Assembly, at
Philadelphia, to incorporate a 'Presbyterian Congregation in
Pittsburgh, at this time under the care of the Rev. Samuel Barr,'
which, after much delay, was finally passed on the twenty-ninth of
September, 1787. The Penns gave the site for this church….
In the Spring of 1811 Reverend Francis Herron became the pastor of
the First Church, which the year before had had a membership of
sixty-five. Dr. Herron's salary was six hundred dollars per annum. For
thirty-nine years he labored ceaselessly and wisely for the church
and congregation. In 1817 the church was enlarged, and the
membership steadily increased. Killikelly's History of Pittsburgh.
Second Presbyterian Church
The Second Presbyterian Church was organized … in 1804, by those
members of the First Church to whom the methods used, regarding
the services in the First Church, were unsatisfactory. The next year
Dr. Nathaniel Snowden took charge of the congregation which
worshiped … in the Court House and other places, public and
private. Dr. John Boggs came, but remained only a short time. He
was replaced by the Rev. Mr. Hunt, in 1809. The first edifice, on
Diamond alley, near Smithfield street, was built in 1814. Killikelly's
History of Pittsburgh.
East Liberty Presbyterian Church
Mr. Jacob Negley, whose wife had been a Miss Winebiddle, and
consequently, inherited much real estate, controlled practically what](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-56-320.jpg)
![is now known as East Liberty Valley, in the early days, called
Negleystown. He was largely instrumental … in erecting a small
frame school building at what subsequently became the corner of
Penn and South Highland avenues. This was for the accommodation
of the children of the district, as well as his own. It was … a long
distance to the then established churches, and Mr. Negley very
often, for the benefit of the neighborhood, invited some minister
passing through, or one from one of the other churches, to preach in
his own house and later in the school house. In 1819 the little school
house was torn down to make way for a church building. Killikelly's
History of Pittsburgh.
Reformed Presbyterian Church
The First Reformed Presbyterian Church of Pittsburg, long
afterwards known as the 'Oak Alley Church,' was organized in 1799.
Rev. John Black, an Irishman of considerable intellectual force, who
had been graduated from the University of Glasgow, was its first
pastor…. He included, in his ministry, all societies of the same
persuasion in Western Pennsylvania. He preached here until his
death on October 25, 1849. Boucher's Century and a half of
Pittsburg.
Roman Catholic Church
The number of Catholics prior to 1800, in what is now Allegheny
county, must have been very small. They were visited occasionally
by missionaries traveling westward…. [These] priests, ministering to
a few scattered families, celebrating Mass in private houses, fill up
the long interval between the chapel of the 'Assumption of the
Blessed Virgin of the Beautiful River' in Fort Duquesne, and 'Old St.
Patrick's Church,' which was begun in 1808.
Rev. Wm. F. X. O'Brien, the first pastor, was ordained in Baltimore,
1808, and came to Pittsburg in November of the same year, and at
once devoted himself to the erection of … 'Old St. Patrick's.' It stood](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-57-320.jpg)
![at the corner of Liberty and Washington streets, at the head of
Eleventh street, in front of the new Union Station…. The structure
was of brick, plain in design and modest in size, about fifty feet in
length and thirty in width. Rt. Rev. Michael Egan dedicated the
Church in August 1811, and the dedication was the occasion of the
first visit of a Bishop to this part of the State. St. Paul's Cathedral
record.
Protestant Episcopal Church
The building of the first Trinity Church was begun about the time it
was organized and chartered, 1805. It occupied a triangular lot at
the corner of Sixth, Wood and Liberty streets. It was built in an oval
form that it might more nearly conform to the shape of the three
cornered lot and for this reason was generally known as the 'round
church.' Rev. Taylor in his latter years became known as 'Father'
Taylor. He remained with the church as its rector until 1817, when he
resigned. Boucher's Century and a half of Pittsburg.
First German United Evangelical Protestant Church
When John Penn, jr., and John Penn presented land to the
Presbyterian and Episcopal churches of Pittsburgh they, at the same
time, deeded the same amount to the already organized German
Evangelical congregation; the land given to them was bounded by
Smithfield street, Sixth avenue, Miltenberger and Strawberry alleys.
No church was built on this grant, however, until some time between
1791-94, and it was of logs. This was … replaced in 1833 by a large
brick building, which had the distinction of a cupola, in which the
first church bell in Pittsburgh was hung. Killikelly's History of
Pittsburgh.
Methodist Episcopal Church
In June, 1810, a lot was purchased for the first [Methodist] church
built in the city. It was situated on Front street, now First street,](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-58-320.jpg)
![Libraries
It was not … until the fall of 1813, that the question of a
community Library took definite shape, when in response to the
efforts 'of many leading and progressive citizens,' there was
organized 'The Pittsburgh Library Company.' On the evening of
November 27, 1813, about 40 representative people assembled in
the spacious 'bar room' of the 'Green Tree Inn,' at the northwest
corner of Fifth and Wood streets, where the First National Bank now
stands, and took the initiative in the formation of Pittsburgh's first
real public library…. Its first president was the Rev. Francis Herron,
for 40 years pastor of the First Presbyterian Church. The secretary
was Aquila M. Bolton, 'land broker and conveyancer.' The treasurer
was Col. John Spear…. Quite a sum of money was subscribed by
citizens generally for the purchase of books, while many valuable
volumes were either contributed or loaned by members. Messrs.
Baldwin, O'Hara, Wilkins and Forward being especially mentioned for
their generosity in this connection. The first head-quarters of the
library were in rooms 'on Second street, opposite Squire Robert
Graham's office,' who at that time dispensed even handed justice at
the northeast corner of Market and Second streets. Here the library
remained until the county commissioners set aside a commodious
room in the Court House for its use. A. L. Hardy, in Gazette-Times,
1913.
The triennial meeting of the shareholders [of the Pittsburgh Library
Company] was convened at their new library room, in Second street,
opposite Squire Graham's office, at six o'clock, Monday evening,
December thirtieth, 1816. The following gentlemen were then
elected by ballot to serve as a Board of Directors for the ensuing
three years, viz: George Poe, president; Aquila M. Bolton, secretary;](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-64-320.jpg)
![The Theatre
There were in 1808 two dramatic societies in Pittsburg that were
important enough to receive notice in the newspapers. The one was
composed of law students and young lawyers and the other was
composed of mechanics. The object of these societies was to study
the poets and dramatic literature and to give public performances in
the court house. William Wilkins … was a member and took a
leading part in the entertainments given by these societies. There
was no way for theatrical companies from the East to reach Pittsburg
prior to 1817, save by the state road, which was scarcely passable
for a train of pack horses, yet they came even as early as 1808 and
performed in a small room, which was secured for them when the
court room was occupied. In 1812 a third dramatic society called the
Thespian Society was organized among the young men and young
women of Pittsburg.
The society numbered among its members the brightest and best
bred young people of the city, most of whom took part in each
performance. They were given in a room on Wood street, in a
building known as Masonic Hall. Boucher's Century and a half of
Pittsburg.
The Theatre of this City has been now opened nearly a fortnight,
and the managers although they have used every exertion to please,
in the selection of their pieces, have not been enabled to pay the
contingent expenses of the House. This is a severe satire on the
taste of the place.
Tomorrow [Wednesday] evening we understand that the 'Stranger' is
to be produced—we hope under auspices more favorable to the
managers than heretofore. The part of the Stranger is to be](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-67-320.jpg)
![Police
When the borough was incorporated into a city [March 1816], the
act incorporating it authorized the authorities to establish a police
force, but there was none established for some years afterwards.
The act limited the city taxation to five mills on a dollar, and the
corporation could scarcely have paid a police force, even if one had
been required. The city authorities did, however, pass an ordinance
on August 24, 1816, establishing a night watchman, but soon found
they had no money with which to pay him. They accordingly
repealed the ordinance and for some years the city slept in darkness
without the benefit of police protection. Boucher's Century and a
half of Pittsburg.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-73-320.jpg)
![but the prowling wolf, or the restless and cruel savage held their
haunts. Mercury, April 20, 1816.
At the beginning of the century the site of Allegheny City was a
wilderness. In 1812 a few settlers had made inroads upon the forest,
and had builded their cabins. Notice is called to the fact in the
minutes of the Presbytery of Erie, in April of that year, in the
following words: 'An indigent and needy neighborhood, situated on
the Allegheny, opposite Pittsburgh, having applied for supplies,' the
matter was laid before the Presbytery.
Joseph Stockton seems to have been the first stated minister,
preaching a part of his time there until 1819. Centenary memorial
of Presbyterianism in western Pennsylvania.
The facility for getting to and from Pittsburg [from Allegheny] was
quite a different matter from what it is to-day. The only highway (if it
may be called such) leading west from Federal Street to the Bottoms
at that early day, was the erratic Bank Lane, which owing to the
natural unevenness of the ground upon which it was located, and
total neglect of the authorities of Ross township to put it in a
condition for travel, … was for many years only accessible for foot-
passengers. Parke's Recollections of seventy years.
Lawrenceville was laid out in 1815 by Wm. B. Foster, and had begun
with the building of the United States arsenal.](https://image.slidesharecdn.com/3420369-250618165533-b1a5d2fd/85/Morphogenesis-Of-Spatial-Networks-1st-Edition-Marc-Barthelemy-Auth-81-320.jpg)
Morphogenesis Of Spatial Networks 1st Edition Marc Barthelemy Auth
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- 5. Series Editor: Alessandro Sarti Lecture Notes in Morphogenesis Marc Barthelemy Morphogenesis of Spatial Networks
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- 9. Marc Barthelemy Institut de Physique Théorique Commissariat à l'Energie Atomique Gif-sur-Yvette France and Ecole des Hautes Études en Sciences Sociales Centre d’Analyse et de Mathématique Sociales Paris France ISSN 2195-1934 ISSN 2195-1942 (electronic) Lecture Notes in Morphogenesis ISBN 978-3-319-20564-9 ISBN 978-3-319-20565-6 (eBook) https://doi.org/10.1007/978-3-319-20565-6 Library of Congress Control Number: 2017955663 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 10. Long years I devoted to learning the order and arrangement of the spots on the tiger’s skin. During the course of each blind day I was granted an instant of light, and thus was I able to fix in my mind the black shapes that mottled the yellow skin. Some made circles; others formed transverse stripes on the inside of its legs; others, ringlike, occurred over and over again—perhaps they were the same sound, or the same word. The Writing of the God, 1949 Translated from “La escritura del dios” Jorge Luis Borges v
- 11. Preface Watts and Strogatz, with the publication in 1998 of their seminal paper on small-world networks, opened the golden era of complex networks studies and showed in particular how statistical physics could contribute to the understanding of these objects. The first studies that followed considered the characterization of large graphs, their degree distribution, clustering coefficient, or their average shortest path. New models of random graphs, beyond the well-known Erdos–Renyi archetype, were then proposed in order to understand some of the empirically observed features. However, many complex networks encountered in the real-world are embedded in space: nodes and links correspond to physical objects and there is usually a cost associated with the formation of edges. This aspect turns out to be crucial as it determines many features of the structure of these networks that we can call “spatial”. It is difficult to consider that spatial networks actually form a subclass of complex networks, but rather constitute their own family specified by a set of properties that differ from the “usual” complex networks. In particular, one of the most salient properties in complex network is a broad degree distribution with the existence of hubs. This feature has a dramatic impact on dynamical processes occurring on these networks and is at the heart of studies on scale-free networks. In contrast, the physical constraints in spatial networks prohibit in general the formation of hubs and their most interesting properties lie in their spatial organi- zation and in the relation between space and topology. Spatial networks—even if this was not the standard name at that time—were the subject of numerous studies in the 70s in regional science followed by quantitative geographers who were interested in characterizing the structure of transportation networks, from roads to subways and railways, and produced a number of important results about these networks and their evolution. The recent revival of the interest in this subject, combined with an always larger amount of data, allowed to make some progress in our understanding of these objects. The recent advances obtained in the understanding of spatial networks have generated an increased attention toward the potential implication of new theoretical models in agreement with data. Questions such as the structure and resilience of infrastructures and the vii
- 12. impact of space on the formation of biological networks are fundamental questions that we hope to solve in a near future. Most of these spatial networks are—to a good approximation—planar graphs for which edge crossing is not allowed. Planar networks were for a long time the subject of numerous studies in graph theory, but we are still lacking models and tools for their characterization. In this book, we will discuss different aspects of spatial networks, focusing essentially on the characterization of their structure and on their modeling. Each chapter is as much as possible self-contained and for the sake of clarity and readability, we tried to be as modular as possible in order to allow the reader interested in just one specific model or tool to focus essentially on the corre- sponding chapter. The first chapter introduces the subject with some definitions and basic results about planar graphs together with less trivial results about the crossing number of a graph. We will insist on the distinction between topological non-planarity and non-planarity of the physical embedding. As discussed above, many measures that were extensively used for complex networks are in fact irrelevant for spatial net- works, due to constraints that make the degree bounded, and the clustering and assortativity trivial. We review both the irrelevant and the simplest measures in Chap. 2, and also a discussion on the more advanced tool that is community detection. In Chaps. 3–7, we discuss various tools and measures for spatial networks. An important object in spatial networks, and in particular in planar graphs, is the face (or cell, block depending on the context). We discuss in Chap. 3 the statistics of the area and shape of these faces and the possibility of a mapping of a planar graph to a tree. We discuss here both an approximate mapping introduced for weighted graphs and an exact bijection obtained in mathematics for (rooted) planar graphs. In Chap. 4, we discuss the important quantity which is the betweenness centrality. It was introduced in the 70s for quantifying the importance of a node in a network and this particular “centrality” seems to be very interesting for characterizing the organi- zation of spatial networks. We first expose general properties of the betweenness centrality such as the scaling of the maximum value or the effect of adding or removing edges. We then present empirical results about the spatial patterns of the betweenness centrality in various networks and theoretical aspects as well, such as the centrality of loops in random graphs. In Chap. 5, we also consider other path-related quantities that were used in spatial networks. The simplicity compares shortest paths and simplest paths—the paths with the smallest number of turns— and the entropy quantifies the complexity of paths in these networks. In Chap. 6, we address a subject whose importance might grow in the future and which concerns spatial networks with attributes. In these systems, nodes have a certain attribute (a real number such as the population of a city for example) and we have to characterize the interplay between the value of the attribute and the spatial location of a node. We discuss for these objects a measure of spatial “dominance” that was developed by Okabe and his collaborators. We end this chapter with a viii Preface
- 13. discussion on community detection whose results depend strongly on the existence of correlations between space and attribute, and on the choice of a null model. In Chap. 7, we address the important problem of time-evolving spatial networks and their characterization. We focus in this more empirically oriented part on the evolution of the street network and the growth of subways. The large number of parameters and possible measures is, maybe surprisingly, not very helpful and we will see how to identify the most relevant tools for the characterization of the evolution of these systems. This is a very timely subject and we can expect many development and progress about this problem in the coming years. In Chaps. 8–14, we discuss modeling aspects of spatial networks. We start in Chap. 8 with a description of tessellations which are good “null” models for planar graphs and which also allow to characterize the statistics of a distribution of points. We will naturally discuss the Voronoi tessellation and its properties (in particular in the case of a Poisson distribution of points), but also other models such as cracks and STIT tessellations. In Chap. 9, we discuss the random geometric graph, probably the simplest model of spatial network and some of its variants such as the soft random geometric graph, the Bluetooth graph, and the k-nearest neighbor model. We also discuss a dynamical version of the random geometric graph where agents are mobile in a plane and create a network of connections. In Chap. 10, we present generalizations of the Erdos–Renyi random graph to the spatial case. In particular, we will discuss the Waxman model that is considered as a simple model for the structure of the Internet. We will also present spatial gener- alizations of the Watts–Strogatz model and its properties. In particular, after having discussed some models, we will focus on the navigability on these networks as it has important practical applications. In Chap. 11, we discuss a particular class of spatial networks that are made of branches radiating from a node and a loop (or ring) connecting these different branches. We will see the conditions under which the loop can have a larger betweenness centrality than the origin and we will also discuss the impact of congestion at the center on the overall pattern of shortest paths. In Chap. 12, we present optimal networks and their properties, and discuss the most important illustrations of this class of graphs such as the minimum spanning tree that minimizes the total length of the network. We will discuss the statistical properties of this tree and we will present a more general class of optimal trees that minimize a combination of length and betweenness centrality, allowing to inter- polate between the minimum spanning tree and the star graph (that minimizes the average shortest path). We end this chapter with a discussion of the conditions for the appearance of loops or a hub-and-spoke structure in this optimization framework. In Chaps. 13 and 14, we present models of network growth where a new node is added at each time step and connects to the existing network according to certain rules. In Chap. 13, we first consider spatial variants of preferential attachment where the new node will preferentially connect to well-connected nodes, up to a distance-dependent factor. We will also consider the “potential” approach where the Preface ix
- 14. addition of a new node is governed by a potential that gives the probability to choose a specific location and depends in general on the state of the network at this time. We describe in this chapter the general philosophy of this approach and detail the example of the growth of road networks. In Chap. 14, we consider the case of “local” optimization where each node (added sequentially) optimizes a given function. The minimization is therefore local and the resulting network at large time does not in general minimize a simple quantity. An important example in this class of greedy models is the cost–benefit model which we will discuss thoroughly here. This framework will allow us to understand some of the properties of transportation networks such as subways or railways and how they are affected by the substrate where their evolution take place. We end this book with a (subjective) discussion in Chap. 15 about what seems to be interesting and important research directions in the study of spatial networks. As can be seen in this short outline of the book, several disciplines are con- cerned. Scientists from statistical physics, random geometry, probability, and computer sciences produced a wealth of interesting results and this book cannot cover all new studies about spatial networks. Owing to personal biases, space limitations, and lack of knowledge, important topics might have been omitted, and I apologize in advance for omissions or errors and to those colleagues who feel that their work is not well represented here. Incomplete and imperfect as it is, I hope, however, that this book will be helpful to scientists interested in the formation and evolution of spatial networks, a fascinating subject at the crossroad of so many disciplines. Paris, France Marc Barthelemy June 2017 x Preface
- 15. Acknowledgements My path in the network world started with my visit to Gene Stanley’s group in Boston where I worked in particular with Luis Amaral and Shlomo Havlin. I thank Gene for the freedom that he left me at that time and Luis and Shlomo for having introduced me to the analysis of empirical data. Back to Paris, I continued my exploration of networks with Alain Barrat and Alessandro Vespignani with whom we focused on the spread of epidemics and the impact of mobility on this process. I thank them both warmly for all the things I learned with them, from technical methods to the way of doing science. These studies on epidemic spread naturally led me to analyze transport networks at different spatial scales, and most impor- tantly to understand the effect of space on the topology of these structures. These systems are indeed embedded in space and since the beginning of network studies, this aspect was mostly ignored. These different reasons, together with my fasci- nation for maps (a fascination shared with many !), pushed me to look further about what we can now call spatial networks. In particular, I started to work on the most common example—road networks—and thanks to many discussions with Alessandro Flammini, we proposed a model for the formation and evolution of these systems. After some time, I joined the Institut de Physique Théorique in Saclay and I could continue in this interdisciplinary direction, thanks to Henri Orland who was the director at that time and thanks to his successors Michel Bauer and now Francois David who provided such a great interdisciplinary environment for these fundamental studies. In particular, I could meet colleagues at the IPhT with a strong mathematical background and from whom I could learn so much. In particular, I thank Jean-Marc Luck and Kirone Mallick for many discussions on many subjects in statistical physics, and Jeremie Bouttier, Emmanuel Guitter and Philippe Di Francesco—their knowledge in combinatorics and planar maps helped me to understand small parts of this important topic in mathematical physics. A constant interaction with another point of views and the need to explain yourself clearer are fundamental aspects of scientific research and I thank all my collaborators, colleagues, together with my postdocs and Ph.D. students with whom xi
- 16. I worked on different subjects related to networks. In particular, I thank A. Bourges, G. Carra, J. Depersin, R. Gallotti, B. Lion, T. Louail, R. Louf, R. Morris, E. Strano, and V. Volpati for their continuous input. Another crucial aspect in this field is interdisciplinarity. This brought me to meet many scientists from whom I learned a lot about completely different aspects going from applied mathematics, probability, and combinatorics, to economics, geography, and history. For all these discussions and interactions, I warmly thank E. Arcaute, A. Arenas, M. Batty, H. Berestycki, A. Blanchet, M. Boguna, P. Bordin, J.-P. Bouchaud, A. Bretagnolle, M. Breuillé, O. Cantu, G. Carra, A. Chessa, V. Colizza, J. Coon, Y. Crozet, M. De Nadai, S. Derrible, C. P. Dettmann, S. Dobson, A. Flammini, M. Fosgerau, E. Frias, R. Gallotti, G. Ghoshal, J. Gleeson, M. Gonzalez, M. Gribaudi, J. Le Gallo, R. Le Goix, R. Herranz, E. Katifori, M. Kivela, P. Krapivsky, R. Lambiotte, V. Latora, F. Le Nechet, M. Lenormand, C. Mascolo, Y. Moreno, I. Mulalic, J.-P. Nadal, V. Nicosia, A. Noulas, M. O’Kelly, J. Perret, S. Porta, M. A. Porter, D. Pumain, D. Quercia, J. J. Ramasco, C. Roth, M. San Miguel, F. Santambroggio, M. A. Serrano, S. Shai, and A. Vignes. I also thank the Springer staff for its excellent support and reactivity. In par- ticular, I thank Alessandro Sarti who serves as an editor for this series and the publishing editor Jan-Philip Schmidt, for many discussions and help about this project and for their constant support. For everything, I thank my loving family, Esther, Rebecca, and Catherine. xii Acknowledgements
- 17. Contents 1 From Complex to Spatial Networks . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Early Days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Space Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Definition and Representations. . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4.1 Spatial Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4.2 Representations of Networks . . . . . . . . . . . . . . . . . . . . 4 1.5 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.1 Planarity and Crossing Number . . . . . . . . . . . . . . . . . . 8 1.5.2 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Irrelevant and Simple Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Irrelevant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Length of Segments . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.3 Clustering, Assortativity, and Average Shortest Path. . . 17 2.1.4 Empirical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Simple Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Topological Indices: a and c Indices . . . . . . . . . . . . . . 28 2.2.2 Organic Ratio and Ringness . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Cell Areas and Shape . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.4 Route Factor, Detour Index . . . . . . . . . . . . . . . . . . . . . 31 2.2.5 Cost, Efficiency, and Robustness . . . . . . . . . . . . . . . . . 32 3 Statistics of Faces and Typology of Planar Graphs . . . . . . . . . . . . . 35 3.1 Area and Shape of Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.1 Characterizing Blocks . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.2 A Typology of Planar Graphs . . . . . . . . . . . . . . . . . . . 39 3.2 Approximate Mapping of a Planar Graph to a Tree. . . . . . . . . . 42 3.3 An Exact Bijection Between a Planar Graph and a Tree . . . . . . 48 xiii
- 18. 4 Betweenness Centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Definition of the BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 General Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2.1 Numerical Calculation: Brandes’ Algorithm . . . . . . . . . 52 4.2.2 The Average BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2.3 Edge Versus Node BC . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.4 Adding Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.5 Scaling of the Maximum BC . . . . . . . . . . . . . . . . . . . 57 4.3 The Spatial Distribution of Betweenness Centrality . . . . . . . . . . 59 4.3.1 Regular Lattice and Scale-Free Networks . . . . . . . . . . . 59 4.3.2 Giant Percolation Cluster . . . . . . . . . . . . . . . . . . . . . . 60 4.3.3 Real-World Planar Graphs . . . . . . . . . . . . . . . . . . . . . 61 4.3.4 Summary: Stylized Facts . . . . . . . . . . . . . . . . . . . . . . 66 4.4 The BC of a Loop Versus the Center: A Toy Model . . . . . . . . 67 4.4.1 Approximate Formulas . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4.2 A Transition to a Central Loop . . . . . . . . . . . . . . . . . . 69 4.5 The BC in a Disk: The Continuous Limit. . . . . . . . . . . . . . . . . 71 5 Simplicity and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.1 Simplest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.2 The Simplicity Index and the Simplicity Profile . . . . . . 77 5.1.3 A Null Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1.4 Measures on Real-World Networks . . . . . . . . . . . . . . . 81 5.2 Information Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.1 Entropy and Simplest Paths . . . . . . . . . . . . . . . . . . . . 84 5.2.2 Navigating in the City . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.3 Quantifying the Complexity . . . . . . . . . . . . . . . . . . . . 87 6 Spatial Dominance and Community Detection . . . . . . . . . . . . . . . . 93 6.1 Spatial Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Community Detection in Spatial Networks . . . . . . . . . . . . . . . . 97 6.2.1 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2.2 A Null Model for Spatial Networks with Attributes . . . 100 6.2.3 Synthetic Spatial Network Benchmarks . . . . . . . . . . . . 105 6.2.4 Modifying the Modularity . . . . . . . . . . . . . . . . . . . . . . 105 7 Measuring the Time Evolution of Spatial Networks . . . . . . . . . . . . 111 7.1 Road Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.1.1 Organic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.1.2 Effect of Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.1.3 Simplicity Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.2 Subways. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.2.2 Network Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 xiv Contents
- 19. 7.2.3 Standard Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2.4 Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.5 Temporal Statistics: Bursts . . . . . . . . . . . . . . . . . . . . . 141 7.2.6 Core and Branches: Measures and Model . . . . . . . . . . 143 7.2.7 Spatial Organization of the Core and Branches . . . . . . 152 8 Tessellations of the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.1 The Voronoi Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.1.1 The Delaunay Graph . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.1.2 Average Properties of the Poisson-Voronoi Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.1.3 Cell Area Probability Distribution . . . . . . . . . . . . . . . . 162 8.1.4 Probability Distribution of the Number of Sides and the Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.1.5 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . 166 8.2 Effect of the Density of Points . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.3 Crack and STIT Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.4 Planar Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.5 A Null Model for Spatial Multilayer Networks . . . . . . . . . . . . . 173 9 Random Geometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.1 The Hard Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.1.1 Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.1.2 The Clustering Coefficient . . . . . . . . . . . . . . . . . . . . . 179 9.1.3 Calculation of the Giant Component . . . . . . . . . . . . . . 181 9.2 Soft Random Geometric Graphs . . . . . . . . . . . . . . . . . . . . . . . 183 9.2.1 The Full Connectivity Probability . . . . . . . . . . . . . . . . 183 9.3 Bluetooth and Gabriel Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.3.1 Bluetooth Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.3.2 Gabriel Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.4 The k nearest Neighbor Model . . . . . . . . . . . . . . . . . . . . . . . . 188 9.4.1 Definition and Connectivity Properties . . . . . . . . . . . . . 188 9.4.2 A Scale-Free Network on a Lattice . . . . . . . . . . . . . . . 189 9.5 A Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.5.2 Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 9.5.3 Percolation Properties . . . . . . . . . . . . . . . . . . . . . . . . . 192 9.5.4 Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.6 Other Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 9.6.1 Random Geometric Graphs in Hyperbolic Space . . . . . 194 9.6.2 Apollonian Networks . . . . . . . . . . . . . . . . . . . . . . . . . 195 10 Spatial Generalizations of Random Graphs . . . . . . . . . . . . . . . . . . 197 10.1 Spatial Version of Erdos–Renyi Graphs . . . . . . . . . . . . . . . . . . 197 10.1.1 The Erdos–Renyi Graph . . . . . . . . . . . . . . . . . . . . . . . 197 Contents xv
- 20. 10.1.2 Random Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . 198 10.2 The Hidden Variable Model for Spatial Networks. . . . . . . . . . . 200 10.2.1 Spatial Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.2.2 Effect of Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.2.3 The Waxman Model . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.3 Spatial Small Worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.3.1 The Watts–Strogatz Model . . . . . . . . . . . . . . . . . . . . . 206 10.3.2 Spatial Generalizations in Dimension d . . . . . . . . . . . . 207 10.3.3 Percolation in Small Worlds . . . . . . . . . . . . . . . . . . . . 210 10.3.4 Navigability in the Kleinberg Model . . . . . . . . . . . . . . 213 10.3.5 Searching in Spatial Scale-Free Networks . . . . . . . . . . 218 11 Loops and Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 11.1 Reducing the Complexity of a Spatial Network . . . . . . . . . . . . 221 11.2 A Loop and Branches Toy Model . . . . . . . . . . . . . . . . . . . . . . 224 11.2.1 Exact and Approximate Formulas . . . . . . . . . . . . . . . . 225 11.2.2 Threshold Value of w and Optimal ‘ . . . . . . . . . . . . . . 229 11.3 Analyzing the Impact of Congestion Cost . . . . . . . . . . . . . . . . 233 11.3.1 An Exactly Solvable Hub-and-Spoke Model . . . . . . . . 235 11.3.2 Congestion and Centralized Organization . . . . . . . . . . . 238 12 Optimal Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 12.1 Optimization, Complexity, and Efficiency . . . . . . . . . . . . . . . . . 241 12.1.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 12.1.2 Efficiency of Transport Network . . . . . . . . . . . . . . . . . 242 12.2 Minimum Spanning Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 12.2.1 Minimum Spanning Tree on a Complete Graph . . . . . . 245 12.2.2 Euclidean Minimum Spanning Tree . . . . . . . . . . . . . . . 247 12.3 Optimal Trees: Generalization . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.4 Beyond Optimal Trees: Noise and Loops . . . . . . . . . . . . . . . . . 258 12.5 Hub-and-Spoke Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 13 Models of Network Growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 13.1 Preferential Attachment and Space . . . . . . . . . . . . . . . . . . . . . . 265 13.1.1 Preferential Attachment and Distance Selection . . . . . . 267 13.2 Attraction Potential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 13.2.1 The Connection Rule . . . . . . . . . . . . . . . . . . . . . . . . . 275 13.2.2 Uniform Distribution of Nodes . . . . . . . . . . . . . . . . . . 276 13.2.3 Exponential Distribution of Centers . . . . . . . . . . . . . . . 277 13.2.4 Effect of Centrality and Density . . . . . . . . . . . . . . . . . 279 13.2.5 The Appearance of Core Districts . . . . . . . . . . . . . . . . 285 14 Greedy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 14.1 A Model for Distribution Networks . . . . . . . . . . . . . . . . . . . . . 288 14.2 Cost-Benefit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 xvi Contents
- 21. 14.2.1 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . 291 14.2.2 Crossover Between the Star Graph and the MST . . . . . 292 14.2.3 Spatial Hierarchy and Scaling . . . . . . . . . . . . . . . . . . . 295 14.2.4 Understanding the Scaling with a Toy Model. . . . . . . . 299 14.2.5 Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 14.2.6 The Model and Real-World Railways . . . . . . . . . . . . . 304 14.3 Cost-Benefit Analysis: General Scaling Theory . . . . . . . . . . . . . 304 14.3.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . 305 14.3.2 Subways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 14.3.3 Railways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 15 Discussion and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 315 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Contents xvii
- 22. Acronyms BA Barabasi–Albert network BC Betweenness Centrality CBA Cost–Benefit Analysis DT Delaunay triangulation dEMST Dynamical Euclidean minimum spanning tree EMST Euclidean minimum spanning tree ER Erdos–Renyi graph GDP Gross domestic product GT Greedy triangulation MST Minimum spanning tree OTT Optimal traffic tree SPT Shortest path tree STIT Stability under iteration WS Watts–Strogatz graph xix
- 23. Chapter 1 From Complex to Spatial Networks The study of spatial networks – networks embedded in space – started essentially with quantitative geographers in the 60–70s who studied the structure and the evolution of transportation systems. The interest for networks was revived by Watts and Strogatz who opened the way to a statistical physics type of analysis and modeling of large networks. This renewed interest, together with an always growing availability of data, led to many studies of networks and their structures. Most of these studies focused on the topological properties of networks, leaving aside their spatial properties. It is only recently that researchers realized the importance of geometry – as opposed to topology – for spatial networks. In this chapter, we first describe briefly the evolution of these fields and ideas about spatial networks. Most of these objects are planar and in the second part of this chapter, we give basic definitions and results for planar graphs. 1.1 Early Days The research activity on networks was intense this last decade (see [1, 2] and other reviews) but spatial networks were already the subject of many papers and books more than 40 years ago [3, 4]. In particular, in their great book [3], Haggett and Chorley explored the topology and the geometry of transportation networks (road networks, subways, and railways). In the last chapter of their book, they addressed the problem of patterns of spatial evolution, a subject that is still at the heart of modern studies. In another study, Kansky [5] defined many indicators to characterize highways and roads, and in [6], Taafe, Morrill, and Gould proposed a model for the evolution of road networks in cities, followed by many others (see [3]). Despite these various empirical and theoretical studies, the subject of the structure of these spatial networks was only revived later, first by geographers and then by physicists. The important difference between now and the 70s is certainly the availability of data, © Springer International Publishing AG 2018 M. Barthelemy, Morphogenesis of Spatial Networks, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-319-20565-6_1 1
- 24. 2 1 From Complex to Spatial Networks the existence of large computer capacities and most importantly, a better knowledge of the structure of large networks. In these early days, theoretical approaches were very simple and relied heavily on (basic) graph theory. The topological classification of networks amounted then essentially to distinguish between planar and nonplanar, and for planar networks, between trees and graphs with loops. For these networks, characterizations were basic and the indicators were mostly various combinations of the number of nodes N, the number of edges E and for planar graphs, the number of faces F. All these measures – the cyclomatic number, the Alpha, Beta and Gamma indices, the average degree, the average shortest path, etc. do not take into account the spatial nature of these networks, and therefore represent only one specific aspect of these objects. An important goal for geographers was then to understand the evolution of these systems and how these different network measures depend on socioeconomical indicator. For example, Kansky [5] discussed the relation of the Beta index (given by β = E/N) for railways with the gross energy consumption in different countries. Of course, they also investigated some spatial aspects, such as the shape of the network, density of roads, flow properties, etc., and we refer the interested reader to this excellent book for more details. 1.2 Complex Networks Independently, from studies in quantitative geography or in graph theory, physicists started from 1998 to work intensively on networks. With the first paper on small- world networks by Watts and Strogatz [7], the statistical physics community realized that their tools for empirical analysis and modeling could be useful in other fields, even far from traditional objects of study in physics. This seminal work triggered a wealth of analysis of all possible networks available at that time. In particular, it was realized that many complex systems are very often organized under the form of networks and that these tools and models (together with new ones) have a strong impact across many disciplines. An important change of paradigm occurred when we realized [8, 9] that the usual Erdos–Renyi random network was not representative of most networks observed in real-world, and we had to include large fluctuations of the degree. These strong fluctuations have a crucial impact on the dynamics that take place on these networks and many studies were devoted to this phenomenon [2]. This intense activity on networks thus led the researchers to think about the char- acterization of large networks. All the information is a priori encoded in the adja- cency matrix but it is usually far too large and difficult to use under this form. In order to extract a smaller amount of information easily usable and that characterize the network, scientists introduced a number of measures that describe the statisti- cal features of large networks. For example, for the class of networks with strong degree fluctuations, the degree distribution, the diameter, the clustering coefficient and the assortativity give a reasonable coarse-grained picture of the network and are in general enough to describe the dynamics on these networks. It appears that
- 25. 1.2 Complex Networks 3 degree fluctuations are essential and govern many processes, but we note here that other quantities (such as correlations, for example) can also play a critical role in the dynamics on networks (see, for example, the case of epidemics [10]). 1.3 Space Matters These various studies on complex networks largely ignored space and considered that these networks were living in some abstract world with no metrics. In many cases indeed, the network is introduced as a simplified way to describe interactions between elements. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are however all embedded in space, and for these networks, space is relevant and topology alone does not contain all the information. In other words, in order to completely characterize these networks, we need not only the adjacency matrix but also the list of the position of the nodes. An important consequence of space on networks is that there is naturally a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks [11]. Characterizing and understanding the structure and the evolution of these “spatial networks” became an important subject with many consequences in various fields ranging from epidemiology, neurophysiology, to ICT, urbanism, and transportation studies. These networks are usually very large and we need statistical tools in order to describe the most accurately possible the salient aspects of their organization by taking into account both the topological and spatial aspects. In the first chapters (1–7), we will review the most important tools for their characterization. 1.4 Definition and Representations The representation of a network is not unique in general and we introduce here the main definitions and representations used in the framework of spatial networks. 1.4.1 Spatial Networks A graph G = (V, L) is usually defined as a combination of a set V of N nodes and a set L of E links connecting these nodes. The N × N adjacency matrix A is then simply defined by ai j = 1 if there is a link between nodes i and j and ai j = 0 otherwise. This definition can be extended to weighted networks with ai j = wi j where the weight wi j denotes any quantity that flows on this link (i, j). Note that for directed networks A is not symmetric and usually ai j denotes the link that goes from j to i. Also, for practical purposes it is not necessary to store the whole matrix A as
- 26. 4 1 From Complex to Spatial Networks Fig. 1.1 These two networks have the same adjacency matrix and are topologically equivalent. However, as shown here they can have a very different spatial representation and this information is encoded in the list of nodes’ positions networks are often sparse and many elements are zero. The convenient way to store the network is then to introduce the adjacency list which contains all the neighbors of a given node. While the full matrix necessitates to store N2 elements, the adjacency list requires only to store a number at most equal to N × (G) where (G) is the largest degree in the graph and is in general much smaller than N. This standard representation of a graph is, however, not enough to describe a spatial network. The same graph can indeed be embedded in a plane in an infinite number of ways (see Fig.1.1 for a simple example) and if we are interested in spatial features of the graph we need to specify this embedding. The minimum information needed (in addition to the adjacency matrix) for describing this aspect is the list of position of nodes: we denote X = {xi } this list. We will consider in most of the book that the quantity xi for node i is a two-dimensional vector but for three-dimensional networks (such as the neural network, for example, xi is a 3d vector). Once we have G = (V, L)and X,everythingisknowninprincipleaboutthisspatialnetworkandthe purpose of simple characterizations is to extract useful, coarse-grained information from these large datasets. 1.4.2 Representations of Networks Spatial networks can be represented directly by their embedding which is specified by the graph and the position of nodes, and which forms a ‘map’. However, in some cases (in particular, for transportation systems, or the road network) it is useful to define other types of graphs. A specific example is the so-called dual network, where we first identify “lines” in the network (which are straight lines in the road network case). These lines (see Fig.1.2 for a simple example) will be chosen as the nodes of the dual network and we connect two lines if they intersect. Note that the dual here is not the same as the dual graph in general as in the Voronoi-Delaunay construction, for example (see Chap.8).
- 27. 1.4 Definition and Representations 5 Fig. 1.2 (Left) Primal and (right) dual networks for a square lattice. In this example, the lattice in primal space has N = 8 routes. Each route has k = N/2 = 4 connections, so the total number of connections is Ktot = k2 = 16. In the dual network, the four East–West routes (A, B, C, D) and the four North–South routes (E, F, G, H) form the bipartite graph K4,4 with a diameter equal to 2. Figure taken from [12] a b c d e f g h x y z a b c d e f g h a b c d e f g h s a b c d e f g h Fig. 1.3 a Direct representation of the routes (here for three different routes). b Space-of-changes (sometimes called P space [14, 15]). A link connects two nodes if there is at least one vehicle that stops at both nodes. c Space-of-stops. Two nodes are connected if they are consecutive stops of at least one vehicle. d Space-of-stations. Here two stations are connected only if they are physically connected (without any station in between) and this network reflects the real physical network. Figure taken from [13]
- 28. 6 1 From Complex to Spatial Networks Concerning the important case of transportation networks [13], Kurant and Thiran discuss very clearly the different representations of these systems (Fig.1.3). The simplest representation is obtained when the nodes represent the stations and links the physical connections. One could, however, construct other networks such as the space-of-stops or the space-of-changes (see Fig.1.3). One also finds in the literature on transportation systems, the notions of L and P-spaces [14, 16], where the L-space connects nodes if they are consecutive stops in a given route. The degree in L-space is then the number of different nodes one can reach within one segment and the shortest path length represents the number of stops. In the P-space, two nodes are connected if there is at least one route between them so that the degree of a node is the number of nodes that can be reached directly. In this P-space, the shortest path represents the number of connections needed to go from one node to another. 1.5 Planar Graphs As we will see throughout this book, most spatial networks are well described by planar networks. These graphs that can be represented in a two-dimensional plane without any edge crossings (see, for example, the textbook [17]). The particular case of random planar graphs pervade many different fields from abstract mathematics [18, 19], to quantum gravity [20], botanics [21, 22], geography and urban studies [11]. In particular, planar graphs are central in biology, where they can be used to describe veination patterns of leaves or insect wings and display an interesting architecture with many loops at different scales [21, 22]. In the study of urban systems, planar networks are extensively used to represent, to a good approximation, various infrastructure networks [11] such as transportation networks [3] and streets patterns [23–41]. Understanding the structure and the evolution of these networks is therefore interesting from a purely graph theoretical point of view, but could also have an impact in different fields where these structures are central. As mentioned above, a graph is planar when there is at least one plane embedding such that no edges cross each other. However, if a certain embedding displays edge crossing, it does not necessarily mean that the graph is nonplanar. Standard graph theory shows that a necessary and sufficient condition for planarity is the absence of subgraphs homeomorphic to the two graphs: K5 and K3,3 (see Fig.1.4, where the complete graph Kn with n = 5 nodes and the complete bipartite graph Kn,m with n = 3 and m = 3 are shown). This is the Kuratowsky theorem (see for example, the textbook [17]) and there are efficient algorithms that can test this in O(N) time (see for example [42]). Basically, a planar graph is thus a graph that can be drawn in the plane in such a way that its edges do not intersect. Not all drawings of planar graphs are without intersection and a drawing without intersection is sometimes called a plane graph or a planar embedding of the graph (the term planar map is also frequently used in combinatorial studies). In real-world cases, these considerations actually do not apply since the nodes and the edges represent in general physical objects.
- 29. 1.5 Planar Graphs 7 Fig. 1.4 Complete graphs K5 and K3,3. The Kuratowsky theorem states that all nonplanar graphs have subgraphs homeomorphic to one (or both) of these graphs We note here that it is not trivial to demonstrate that a graph is nonplanar and the demonstration is simplified by invoking the Jordan curve theorem (see for example [43]) which asserts that a continuous, non-self-intersecting closed-loop divides the plane into an interior and an exterior that can be connected by a continuous path that has to intersect the loop somewhere. In order to illustrate a non-planarity demon- stration, we follow here [17] in the case of the complete graph K5. We assume that K5 is planar and will reach a contradiction. We denote its vertices by v1, v2, v3, v4, v5, and since they are all connected to each other, the loop C = v1v2v3v1 exists and is a Jordan curve separating an inside from an exterior domain. The node v4 does not lie on C and we assume that it is in the inside domain (there is a similar argument in the other case where v4 is outside). The interior of C is then divided into three different domains i for i = 1, 2, 3 delimited by the Jordan curves C1 = v1v2v4v1, C2 = v4v2v3v4, and C3 = v4v3v1v4 (see Fig.1.5). The remaining node v5 must then lie C v4 v 2 v 1 v 3 Ω1 Ω2 Ω3 Fig. 1.5 Demonstration that K5 is nonplanar. Case considered in the text: v4 is inside C and divides the interior of C in three domains
- 30. 8 1 From Complex to Spatial Networks Planar graph Representation: 2d embedding Non planar Planar Fig. 1.6 A graph can be planar and have either nonplanar or planar 2d representations in either 1, 2, 3, or in the exterior of C. If v5 ∈ ext(C) then since v4 ∈ int(C) the Jordan theorem implies that the edge v4v5 must cross C. If v5 ∈ 1 (the two other cases are similar), we note that v3 is exterior to this domain and according to the Jordan curve theorem the edge v3v5 must cross the curve C1. We thus find that K5 cannot be planar. 1.5.1 Planarity and Crossing Number We have however to carefully distinguish the planarity of the graph – a topological notion – and the planarity of the embedding. When a graph is nonplanar, it means that it is impossible to find a two-dimensional representation without edge crossings. In contrast, even if a graph is planar, we can, of course, have embeddings that are not planar (see Fig.1.6). The planarity is thus a topological concept and edge-crossing is a geometrical feature. It is therefore not obvious to relate the non-planarity of a graph and the local edge-crossing of the spatial network (which can be seen as an embedding of the graph). We can illustrate this on the case of the Paris subway. The planarity testing algorithm gives the subgraph shown in Fig.1.7. We first represent the map, where we respect the relative spatial locations of the nodes. This is in contrast with the usual graph representation shown in Fig.1.7 (right), where we recognize the complete bipartite graph K3,3. This example shows clearly the difference between planarity in the topological sense with the presence of subgraphs that are either K3,3 or K5, and the existence of edge-crossings in the real spatial network. Here, we observe that we have only one crossing between the two lines Chatelet-Gare de Lyon and Bastille-Place d’Italie. In this respect, even if the graph is nonplanar, the number of planarity violations in the spatial network is limited to one such event. This notion of edge crossing has been formalized in graph theory with the crossing number cr(G) of a graph G, defined as the lowest number of edge crossings of a plane drawing of G. This number is of practical importance: for example, a circuit laid out on a chip corresponds to drawing a graph in 2d and wire crossings can cause potential problems and their number should be minimized.
- 31. 1.5 Planar Graphs 9 Chatelet Place Bastille Reuilly- Diderot Gare de Lyon Bercy Chatelet Place Bastille Reuilly- Diderot Gare de Lyon Bercy Fig. 1.7 The nonplanarity of the Paris subway. (Left) ‘Map’ representation where the relative position of the nodes are respected. (Right) Same graph but which displays more clearly the K3,3 structure. Despite this structure, we observe that in the real graph there is only one edge crossing For planar graphs, we thus have cr(G) = 0. In general, the crossing number is very difficult to compute (and might be a NP-complete problem) and the interested readers can find some discussions about this problem in the excellent graph theory book [44] or in the more specialized survey [45]. Mathematicians worked in particular on the complete graph Kn where we have the scaling cr(Kn) ∼ n4 . This scaling can be understood by computing the largest number of crossings [44]: if we place the vertices on a circle, in order to produce a crossing we have to choose 4 vertices, and if we assume that we always create a crossing, we obtain cr(Kn) ≤ n 4 ∼ n4 (1.1) Actually, better bounds can be found (see [44]) 1 80 n4 + O(n3 ) ≤ cr(Kn) ≤ 1 64 n4 + O(n3 ) (1.2) It is interesting to see that the best drawing is actually obtained by avoiding a finite fraction of the worst case (of order 24/64 ≈ 37%). We also have results for the complete bipartite Kn,m (for a survey of various results, see [45]) m(m − 1) 5 n 2 n − 1 2 ≤ cr(Km,n) ≤ n 2 n − 1 2 m 2 m − 1 2 (1.3) (where x is the lowest nearest integer of x), demonstrating a scaling of the form cr(Km,n) ∼ m2 n2 (1.4)
- 32. 10 1 From Complex to Spatial Networks Despite these results, crossing numbers are however not well known and only few general results are available. In particular, there is a theorem (Ajtai-Chvatal- Newborn-Szemeredi and Leighton, see [44]), that states that for a simple graph G with E ≥ 4N (which means an average degree k ≥ 8), the following bound holds cr(G) ≥ 1 64 E3 N2 (1.5) 1.5.2 Basic Results Basic results for planar networks can be found in any graph theory textbook (see, for example [17]) and we recall here briefly the most important ones. We start with very general facts that can be demonstrated for planar graphs, and among them Euler’s formula is probably the best known. Euler showed that a finite connected planar graph satisfies the following formula N − E + F = 2 (1.6) where N is the number of nodes, E the number of edges, and F is the number of faces. This formula can be easily proved by induction by noting that removing an edge decreases F and E by one, leaving N − E + F invariant. We can repeat this operation until we get a tree for which F = 1 and N = E + 1 leading to N − E + F = E + 1 − E + 1 = 2. This argument can be repeated in the case where the graph is made of C disconnected components and the Euler relation reads in this case N − E + F = C + 1 (1.7) Moreover, for any finite connected planar graph we can obtain a bound for the average degree k . Indeed, any face is bounded by at least three edges and every edge separates two faces at most which implies that E ≥ 3F/2. From Euler’s formula, we then obtain E ≤ 3N − 6 (1.8) In other words, planar graphs are always sparse with a bounded average degree k ≤ 6 − 12 N (1.9) which is therefore always smaller than 6. We end this part with a particular class of planar graphs that are constructed on a set of points distributed in the plane (see the Chap.8 about tessellations). The maximal
- 33. 1.5 Planar Graphs 11 Fig. 1.8 Example of a triangulation constructed on a small set of point. If a face is not a triangle, we can always divide it into smaller triangles and preserve the planarity (we represented such an additional division by the dashed line in this figure) planar graph is obtained on a set of points if we cannot add another edge without violating the planarity. Such a planar graph is necessary a triangulation, where all faces are triangles (indeed if a face is not a triangle we can always “break” it into smaller triangles while preserving planarity – see the example of the dashed edge in Fig.1.8). Such a planar network is useful in practical applications in order to assess for example the efficiency of a real-world planar network and provides an interesting null model. For such a triangulation, we have the equality 3F = 2E and using Euler’s relation, we obtain that the number of edges and faces are maximal and are equal to the bounds E = 3N − 6 and F = 2N − 4, respectively. Obviously, an important aspect of spatial, planar networks is the shape of faces that will contribute to the whole visual pattern. In the next chapter, we discuss the distribution of the area and the shape of faces and in Chap.3 we will discuss how these measures can be used for constructing a typology of planar graphs.
- 34. Chapter 2 Irrelevant and Simple Measures Many studies on complex networks were about how to characterize them and what are the most relevant measures for understanding their structure. In particular, the degree distribution and the existence of the second moment for an infinite network were shown to be critical when studying dynamical processes on networks. These behaviors are therefore strongly connected to degree fluctuations and the existence of hubs. In the case of spatial networks, the physical constraints are usually large and prevent the appearance of such hubs. These constraints also impact other quan- tities that are nontrivial for complex networks but that become irrelevant for spatial networks. We review here these measures that are essentially useless for spatial net- works and we then discuss older, simple measures that were mostly introduced in the context of quantitative geography. 2.1 Irrelevant Measures Quantities that depend very much on the spatial constraints turn out in general to be irrelevant for spatial networks. The prime example is the degree distribution which in many complex networks was found to be a broad law and in some cases well fitted by a power law of the form P(k) ∼ k−γ with 1 γ 3 [1]. In this case, degree fluctuations are very large which has a direct impact on many dynamical processes that take place on the network, such as epidemics for example [2]. For spatial networks, however, the degree has to satisfy steric constraints. If we consider the road network, nodes represent the intersections and the degree of a node is the number of streets starting from it and is therefore clearly limited as a result of space. As a consequence, the degree distribution for most spatial networks is not broad but displays a fast decaying tail (such as an exponential for example). © Springer International Publishing AG 2018 M. Barthelemy, Morphogenesis of Spatial Networks, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-319-20565-6_2 13
- 35. 14 2 Irrelevant and Simple Measures Other irrelevant parameters include the clustering coefficient or the assortativity. Indeedasshownbelowtheclusteringcoefficientisalwayslarge:ifanodeisconnected to two other nodes in a spatial network, they are usually located in its neighborhood which in turn increases the probability that they are connected to each other, leading to a large clustering coefficient. In the following, we will discuss in more detail these different measures. 2.1.1 Degree We recall here that a graph with N nodes and E edges can be described by its N × N adjacency matrix A which is defined as Ai j = = 1 if i and j are connected = 0 otherwise (2.1) If the graph is undirected, then the matrix A is symmetric. The degree of a node is by definition the number of its neighbors and is given by ki = j Ai j (2.2) The first simple indicator of a graph is the average degree k = 1 N i ki = 2E N (2.3) where here and in the following the brackets · denote the average over the nodes of the network. In particular, the scaling of k with N indicates if the network is sparse (which is the case when k → const. for N → ∞). In [31, 46], measurements for street networks in different cities in the world are reported. Based on the data from these sources, the authors of [47] plotted (Fig.2.1a) the number of roads E (edges) versus the number of intersections N. The plot is consistent with a linear fit with slope ≈1.44 (which is consistent with the value k ≈ 2.5 measured in [46]). The quantity e = E/N = k/2 displays values in the range 1.05 e 1.69, in between the values e = 1 and e = 2 that characterize tree-like structures and 2d regular lattices, respectively. Few exact values and bounds are available for the average degree of classical models of planar graphs. In general, it is known that e ≤ 3, while it has been recently shown [48] that e 13/7 for planar Erdös–Renyi graphs [48]. The distribution of degree P(k) is usually a quantity of interest and can display large heterogeneities such as it is observed in scale-free networks (see for example [49]). We indeed observe that for spatial networks such as airline networks or the
- 36. 2.1 Irrelevant Measures 15 0 500 1000 1500 2000 2500 3000 N 0 1000 2000 3000 4000 5000 6000 E Linear fit Tree limit Random planar graph 0 100 200 300 400 0 100 200 300 400 500 600 0 500 1000 1500 2000 2500 3000 N 0 20 40 60 80 l (a) (b) Fig. 2.1 a Numbers of roads versus the number of nodes (i.e., intersections and centers) for data from [31] (circles) and from [46] (squares). In the inset, we show a zoom for a small number of nodes. b Total length versus the number of nodes. The line is a fit which predicts a growth as √ N (data from [31] and figures from [47]) Internet, the degrees are very heterogeneous (see [2]). However, when physical con- straints are strong or when the cost associated with the creation of new links is large, a cutoff appears in the degree distribution [8] and in some case the distribution can be very peaked. This is the case for the road network for example, and more generally in the case of planar networks for which the degree distribution P(k) is of little interest. For example, in a study of 20 German cities, Lämmer et al. [29] showed that most nodes have four neighbors (the full degree distribution is shown in Fig.2.2a) and that the degree rarely exceeds 5 for various world cities [31]. These values are, however, not very indicative: planarity imposes severe constraints on the degree of a node and on its distribution which is generally peaked around its average value. We note here that in real-world cases such as the road network for example, it is natural to study the usual (or “primal”) representation where the nodes are the intersections and the links represent the road segment between the intersection.
- 37. 16 2 Irrelevant and Simple Measures Fig. 2.2 a Degree distribution of degrees for the road network of Dresden. b The frequency distri- bution of the cells surface areas Ac obeys a power law with exponent α ≈ 1.9 (for the road network of Dresden). Figure taken from [29] However, in another representation, the dual graph can be of interest (see [27]) and for the road network it is constructed in the following way: the nodes are the roads and two nodes are connected if there exists an intersection between the two corresponding roads. One can then measure the degree of a node which represents the number of roads which intersect a given road. Also, the shortest path length in this network represents the number of different roads one has to take to go from one point to another. Even if the road network has a peaked degree distribution, its dual representation can display broad distributions [50]. Indeed, in [50], measurements were made on the dual network for the road network in the US, England, and Denmark and showed large fluctuations with a power-law distribution with exponent 2.0 γ 2.5. 2.1.2 Length of Segments In Fig.2.1b, we plot the total length T of the network versus N for the cities con- sidered in [31]. Data are well fitted by a power function of the form T = μNβ (2.4) with μ ≈ 1.51 and β ≈ 0.49. In order to understand this result, one has to focus on the street segment length distribution P(1). This quantity has been measured for London in [34] and is shown in Fig.2.3. This figure shows that the distribution decreases rapidly and the fit proposed by the authors of [34] suggests that P(1) ∼ −γ 1 (2.5)
- 38. 2.1 Irrelevant Measures 17 Fig. 2.3 Length distribution P(1) for the street network of London (and for the model GRPG proposed in [34]). Figure taken from [34] with γ 3.4 which implies that both the average and the dispersion are well defined and finite. If we assume that this result extends to other cities, it means that we have a typical distance 1 between nodes which is meaningful. This typical distance between connected nodes then naturally scales as 1 ∼ 1 √ ρ (2.6) where ρ = N/L2 is the density of vertices and L is the linear dimension of the ambient space. This implies that the total length scales as T ∼ E1 ∼ k 2 L √ N (2.7) This simple argument reproduces well the √ N behavior observed in Fig.2.1b and also the value (given the error bars) of the prefactor μ ≈ kL/2. 2.1.3 Clustering, Assortativity, and Average Shortest Path Complex networks are essentially characterized by a small set of parameters which are not all relevant for spatial networks. For example, the degree distribution which has been the main subject of interest in complex network studies is usually peaked for planar networks, due to the spatial constraints, and is therefore not very interesting. In the following we will discuss the effect of spatial constraints on other important parameters.
- 39. 18 2 Irrelevant and Simple Measures 2.1.3.1 Clustering Coefficient The clustering coefficient of a node i of degree ki is defined as C(i) = Ei ki (ki − 1)/2 (2.8) where Ei is the number of edges among the neighbors of i. This quantity gives some information about local clustering and was the object of many studies in complex networks. For the Erdos–Renyi (ER) random graphs with finite average degree k, the average clustering coefficient is simply given by C = p ∼ k N (2.9) where the brackets · denote the average over the network (p is the probability to connect two nodes). In contrast, for spatial networks, closer nodes have a larger probability to be connected, leading to a large clustering coefficient. The variation of this clustering coefficient in space can thus bring valuable information about the spatial structure of the network under consideration. The clustering coefficient depends on the number of triangles or cycles of length 3 and can also be computed by using the adjacency matrix A. Powers of the adjacency matrix give the number of paths of variable length. For instance, the quantity 1 6 Tr(A3 ) is the number C3 of cycles of length tree and is related to the clustering coefficient. Analogously, we can define and count cycles of various lengths (see for example [51, 52] and references therein) and compare this number to the ones obtained on null models (lattices, triangulations, etc.). Finally, many studies define the clustering coefficient per degree classes which is given by C(k) = 1 N(k) i/ki =k C(i) (2.10) The behavior of C(k) versus k thus gives an indication on how the clustering is organized when we explore different classes of degrees. However, in order to be useful, this quantity needs to be applied to networks with a large range of degree variations which is usually not the case in spatial networks. The average clustering coefficient can be calculated for the random geometric graph (see also Chap.9) and we discuss in the following the argument presented in [53]. If two vertices i and j are connected to a vertex k, it means that they are both in the excluded volume of k. Now, these vertices i and j are connected only if j is in the excluded volume of i. Putting all pieces together, the probability to have two connected neighbors (i j) of a node k is given by the fraction of the excluded volume of i which lies within the excluded volume of k. By averaging over all points i in the excluded volume of k, we then obtain the average clustering coefficient.
- 40. 2.1 Irrelevant Measures 19 We thus have to compute the volume overlap ρd of two spheres which for spherical symmetry reasons depends only on the distance between the two spheres. In terms of this function, the clustering coefficient is given by Cd = 1 Ve Ve ρd(r)dV (2.11) For d = 1, we have ρ1(r) = (2R − r)/2R = 1 − r/2R (2.12) and we obtain C1 = 3/4 (2.13) For d = 2, we have to determine the area overlapping in Fig.2.4 which gives ρ2(r) = (θ(r) − sin(θ(r)))/π (2.14) with θ(r) = 2 arccos(r/2R) and leads to C2 = 1 − 3 √ 3/4π ≈ 0.58650 (2.15) θ R r/2 Fig. 2.4 The overlap between the two disks (area comprised within the bold line) gives the quantity ρ2(r). Figure taken from [53]
- 41. 20 2 Irrelevant and Simple Measures Similarly, an expression can be derived in d dimension [53] which for large d reduces to Cd ∼ 3 2 πd 3 4 d+1 2 (2.16) The average clustering coefficient thus decreases from the value 3/4 for d = 1 to values of order 10−1 for d of order 10 and is independent from the number of nodes which is in sharp contrast with ER graphs for which C ∼ 1/N. Random geometric graphs are thus much more clustered than random ER graphs. The main reason— which is in fact valid for most spatial graphs—is that long links are prohibited or rare. This fact implies that if both i and j are connected to k, it means that there are in some spatial neighborhood of k which increases the probability that their inter-distance is small too, leading to a large C. 2.1.3.2 Assortativity In general, the degrees of the two end nodes of a link are correlated and to describe these degree correlations one needs the two-point correlation function P(k |k). This quantity represents the probability that any edge starting at a vertex of degree k ends at a vertex of degree k . Higher order correlation functions can be defined and we refer the interested reader to [54] for example. The function P(k |k) is, however, not easy to handle and one can define the assortativity [55, 56] knn(k) = k P(k |k)k (2.17) A similar quantity can be defined for each node as the average degree of the neighbor knn(i) = 1 ki j∈Γ (i) kj (2.18) where Γ (i) denotes the set of neighbors of node i. There are essentially two classes of behaviors for the assortativity. If knn(k) is an increasing function of k, vertices with large degrees have a larger probability to connect to similar nodes with a large degree. In this case, we speak of an assortative network and in the opposite case of a disassortative network. It is expected in general that social networks are mostly assortative, while technological networks are disassortative. However, for spatial networks spatial constraint usually implies a flat function knn(k), since it is usually the distance that governs the existence of a link and not the degree.
- 42. 2.1 Irrelevant Measures 21 2.1.3.3 Average Shortest Path Usually, there are many paths between two nodes in connected networks and if we keep the shortest one it defines a distance on the network (i, j) = min paths(i→ j) |path| (2.19) where the length |path| of the path is defined as its number of edges. The diameter of the graph can be defined as the maximum value of all (i, j) or can also be estimated by the average of this distance over all pairs of nodes in order to characterize the “size” of the network. For a d-dimensional regular lattice with N nodes, this average shortest path scales as ∼ N1/d (2.20) In a small-world network (see [7] and Chap.10) constructed over a d−dimensional lattice has a very different behavior ∼ log N (2.21) The crossover from a large-world behavior N1/d to a small-world one with log N can be achieved for a density p of long links (or “shortcuts”) [57] such that pN ∼ 1 (2.22) The effect of space could thus in principle be detected in the behavior of (N). It should, however, be noted that if the number of nodes is too small this can be a tricky task. In the case of brain networks, for example, a behavior of a typical three- dimensional network in N1/3 could easily be confused with a logarithmic behavior if N is not large enough. 2.1.4 Empirical Illustrations We discuss here some simple results obtained on transportation networks that illus- trate the fact that indeed some measures that are useful for understanding complex networks are actually irrelevant in the case of spatial networks and do not convey interesting information.
- 43. 22 2 Irrelevant and Simple Measures 2.1.4.1 Power Grids and Water Distribution Networks Power grids are one of the most important infrastructures in our society. In modern countries, they have evolved for a rather long time (sometimes a century) and are now complex systems with a large variety of elements and actors playing in their function- ing. This complexity leads to the relatively unexpected result that their robustness is actually not very well understood and large blackouts such as the huge August 2003 blackout in North America demonstrates the fragility of these systems. The topological structure of these networks was studied in different papers such as [8, 58, 59]. In particular, in [8, 58], the authors consider the Southern Californian and the North American power grids. In these networks, the nodes represent the power plants, distribution, and transmission substations, and the edges correspond to transmission lines. These networks are typically planar (see for example the Italian case, Fig.2.5) and we expect a peaked degree distribution, decreasing typically as an exponential of the form P(k) ∼ exp(−k/k) with k of order 3 in Europe and 2 in the US. The other studies on US power grids confirm that the degree distribution is exponential (see Fig.2.6). In [58], Albert, Albert, and Nakarado also studied the load (a quantity similar to the betweenness centrality) and found a broad distribution. The degree being peaked, we can then expect very large fluctuations of load for the same value of the degree, as expected in general for spatial networks. These authors also found a large redundance in this network with, however, 15% of cut edges. Also, as expected for these networks, the clustering coefficient is rather large and even independent of k as shown in the case of the power grid of Western US (see Fig.2.7). Besides the distribution of electricity, our modern societies also rely on various other distribution networks. The resilience of these networks to perturbations is thus an important point in the design and operating of these systems. In [61], Yazdani and Jeffrey study the topological properties of the Colorado Springs Utilities and the Richmond (UK) water distribution networks (shown in Fig.2.8). Both these networks (of size N = 1786 and N = 872, respectively) are sparse planar graphs with very peaked degree distributions (the maximum degree is 12). 0 5 10 15 k 10 -3 10 -2 10 -1 10 0 Cumulative distribution UCTE UK and Ireland Italy c (a) (c) (b) Fig. 2.5 a Map of the Italian power grid. b Topology of the Italian power grid. c Degree distribution for the European network (UCTE), Italy, the UK, and Ireland. In all cases, the degree distribution is peaked and can be fitted by exponential. Figure taken from [59]
- 44. 2.1 Irrelevant Measures 23 0 10 20 # of transmission lines 10 −4 10 −3 10 −2 10 −1 10 0 Cumulative distribution Power grid Fig. 2.6 Degree distribution of substations in Southern California (top panel) and for the North American power grid (bottom panel). In both cases, the lines represent an exponential fit. Figure taken from [8, 58], respectively 1 10 k 10 −2 10 −1 10 0 C(k) Fig. 2.7 Scaling of the clustering C(k) for the power grid of the Western United States. The dashed line has a slope −1 and the solid line corresponds to the average clustering coefficient. Figure taken from [60]
- 45. 24 2 Irrelevant and Simple Measures Fig. 2.8 Representation of water distribution networks. Left panels (from top to bottom): Synthetic networks(“Anytown”[62],and“EXNET”[63]).Top-rightpanel:ColoradoSpringUtilitiesnetwork. Bottom-right panel: Richmond (UK) water distribution network. Figure taken from [61] 2.1.4.2 Subways and Buses One of the first studies (after the Watts–Strogatz paper) on the topology of a trans- portation network was proposed by Latora and Marchiori [64] who considered the Boston subway network. It is a relatively small network with N = 124 stations. The average shortest path is ∼ 16 a value which is large compared to ln 124 ≈ 5 and closer to the two-dimensional result √ 124 ≈ 11. In [15], Sienkiewicz and Holyst study a larger set made of public transportation networks of buses and tramways for 22 Polish cities and in [65], von Ferber et al. study the public transportation networks for 15 world cities. The number of nodes of these networks varies from N = 152 to 2811 in [15] and in the range [1494, 44629] in [65]. Interestingly enough, the authors of [15] observe a strong correlation between the number of stations and the population which is not the case for the world cities studied in [65] where the number of stations seems to be independent from the population (see Sect.14.3 for a detailed discussion about the connection between
- 46. 2.1 Irrelevant Measures 25 socioeconomical indicators and the properties of networks). For polish cities, the degree has an average in the range [2.48, 3.08] and in a similar range [2.18, 3.73] for [65]. In both cases, the degree distribution is relatively peaked (the range of variation is usually of the order of one decade) consistently with the existence of physical constraints [8]. Due to the relatively small range of variation of N in these various studies [15, 64, 65], the behavior of the average shortest path is not clear and could be fitted by a logarithm or a power law as well. We can, however, note that the average shortest path is usually large (of order 10 in [15] and in the range [6.4, 52.0] in [65]) compared to ln N, suggesting that the behavior of might not be logarithmic with N but more likely scales as N1/2 , a behavior typical of a two-dimensional lattice. The average clustering coefficient C in [15] varies in the range [0.055, 0.161] and is larger than a value of the order CE R ∼ 1/N ∼ 10−3 − 10−2 corresponding to a random ER graph. The ratio C/CE R is explicitly considered in [65] and is usu- ally much larger than one (in the range [41, 625]). The degree-dependent clustering coefficient C(k) seems to present a power-law dependence, but the fit is obtained over one decade only. In another study [66], the authors study two urban train networks (Boston and Vienna which are both small N = 124 and N = 76, respectively) and their results are consistent with the previous ones. 2.1.4.3 Railways Oneofthefirststudiesofthestructureofrailwaynetwork[67]concernsasubsetofthe most important stations and lines of the Indian railway network and has N = 587 stations. In the P-space representation (see Chap.1), there is a link between two stations if there is a train connecting them and in this representation, the average shortest path is of order ≈ 5 which indicates that one needs four connections in the worst case to go from one node to another one. In order to obtain variations with the number of nodes, the authors considered different subgraphs with different sizes N. The clustering coefficient varies slowly with N that is always larger than ≈0.7 which is much larger than a random graph value of order 1/N. Finally, in this study [67], it is shown that the degree distribution is behaving as an exponential and that the assortativity knn(k) is flat showing an absence of correlations between the degree of a node and those of its neighbors. In [13], Kurant and Thiran studied the railway system of Switzerland and major trains and stations in Europe (and also the public transportation system of Warsaw, Poland). The Swiss railway network contains N = 1613 nodes and E = 1680 edges (Fig.2.9). All conclusions drawn here are consistent with the various cases presented in this chapter. In particular, the average degree is k ≈ 2.1, the average shortest path is ≈47 (consistent with the √ N result for a two-dimensional lattice), the clustering coefficient is much larger than its random counterpart, and the degree distribution is peaked (exponentially decreasing).
- 47. 26 2 Irrelevant and Simple Measures Fig. 2.9 Physical map of the Swiss railway networks. Figure taken from [13] 2.1.4.4 Neural Networks The human brain with about 1010 neurons and about 1014 connections is one of the most complex networks that we know. The structure and functions of the brain are the subjects of numerous studies and different recent techniques such as electroen- cephalography, magnetoencephalography, functional RMI, etc. can be used in order to reconstruct networks for the human brain (see Fig.2.10 and for a clear and nice introduction see for example [68, 69]). Brain regions that are spatially close have a larger probability of being con- nected than remote regions as longer axons are more costly in terms of material and energy [68]. Wiring costs depending on distance are thus certainly an impor- tant aspect of brain networks and we can expect spatial networks to be relevant in this rapidly evolving topic. So far, many measures seem to confirm a large value of the clustering coefficient, and a small-world behavior with a small average shortest path length [70, 71]. It also seems that neural networks do not optimize the total wiring length but rather the processing paths, thanks to shortcuts [72]. This small- world structure of neural networks could reflect a balance between local processing andglobalintegrationwithrapidsynchronization,informationtransfer,andresilience to damage [73]. In contrast, the nature of the degree distribution is still under debate and a recent study on the macaque brain [74] showed that the distribution is better fitted by an exponential rather than by a broad distribution. Besides the degree distribution, most of the observed features were confirmed in latest studies such as [75] where Zalesky
- 48. 2.1 Irrelevant Measures 27 Fig. 2.10 Structural and functional brains can be studied with graph theory by following different methods shown step-by-step in this figure. Figure taken from [68] et al. propose to construct the network with MRI techniques where the nodes are distinct gray-matter regions and links represent the white-matter fiber bundles. The spatial resolution is of course crucial here and the largest network obtained here is of size N ≈ 4,000. These authors find large clustering coefficients with a ratio to the corresponding random graph value of order 102 . Results for the average shortest path length are, however, not so clear due to relatively low values of N. Indeed, for N varying from 1,000 to 4,000, varies by a factor of order 1.7−1.8 [75]. A small-world logarithmic behavior would predict a ratio r = (N = 4000) (N = 1000) ∼ log(4000) log(1000) ≈ 1.20 (2.23) while a three-dimensional spatial behavior would give a ratio of order r ≈ 41/3 ≈ 1.6 which is closer to the observed value. Larger sets would, however, be needed in order to be sure about the behavior of this network concerning the average shortest path and to distinguish a log N from a N1/3 behavior expected for a three-dimensional lattice.
- 49. 28 2 Irrelevant and Simple Measures Things are, however, more complex than it seems and even if functional connec- tivity correlates well with anatomical connectivity at an aggregate level, a recent study [76] shows that strong functional connections exist between regions with no direct structural connections, demonstrating that structural and functional properties of neural networks are entangled in a complex way and that future studies are needed in order to understand this extremely complex system. 2.2 Simple Measures 2.2.1 Topological Indices: α and γ Indices Different indices were defined a long time ago mainly by scientists working in quan- titative geography since the 1960s and can be found in [3, 77, 78] (see also the more recent paper by Xie and Levinson [32]). Most of these indices are relatively simple but give valuable information about the structure of the network, in particular if we are interested in planar networks. They were used to characterize the topology of transportation networks: Garrison [79] measured some properties of the Interstate highway system and Kansky [80] proposed up to 14 indices to characterize these networks. The simplest index is called the gamma index and is defined by γ = E Emax (2.24) where E is the number of edges and Emax is the maximal number of edges (for a given number of nodes N). For nonplanar networks, Emax is given by N(N − 1)/2 for nondirected graphs and for planar graphs we saw in Chap.1 that Emax = 3N − 6 leading to γP = E 3N − 6 (2.25) The gamma index is a simple measure of the density of the network but one can define a similar quantity by counting the number of elementary cycles instead of edges. The number of elementary cycles for a network is known as the cyclomatic number (see for example [17]) and is equal to Γ = E − N + 1 (2.26) For a planar graph, this number is always less or equal to 2N −5 which leads naturally to the definition of the alpha index (also coined “meshedness” in [46]) α = E − N + 1 2N − 5 (2.27)
- 50. 2.2 Simple Measures 29 This index lies in the interval [0, 1] and is equal to 0 for a tree and equal to 1 for a maximal planar graph. Using the definition of the average degree k = 2E/N, the quantity α reads in the large N limit as α k − 2 4 (2.28) which shows that in fact for a large network this index α does not contain much more information than the average degree. 2.2.2 Organic Ratio and Ringness We note that more recently other interesting indices were proposed in order to char- acterize specifically road networks [32, 81]. For example, in some cities, the degree distribution is very peaked around 3−4 and the ratio rN = N(1) + N(3) k=2 N(k) (2.29) can be defined [81] where N(k) is the number of nodes of degree k. If this ratio is small, the number of dead ends and of “unfinished” crossing (k = 3) is small compared to the number of regular crossings with k = 4 which signals a more organized city. In the opposite case of large rN (i.e., close to 1), there is a dominance of k = 1 and k = 3 nodes, which is the sign of a mode “organic” city. The authors of [81] also define the “compactness” of a city which measures how much a city is “filled” with roads. If we denote by A the area of a city and by T the total length of roads, the compactness Ψ ∈ [0, 1] can be defined in terms of the hull and city areas Ψ = 1 − 4A (T − 2 √ A)2 (2.30) In the extreme case of one square city of linear size L = √ A with only one road encircling it, the total length is T = 4 √ A and the compactness is then Ψ = 0. At the other extreme, if the city roads constitute a square grid of spacing a, the total length is T = 2L2 /a and in the limit of a/L → 0 one has a very large compactness Ψ ≈ 1 − a2 /L2 . We end this section by mentioning the ringness. Arterial roads (including free- ways, major highways) provide a high level of mobility and serve as the backbone of the road system [32]. Different measures (along with many references) are discussed and defined in this paper [32], and in particular, the ringness is defined as φring = ring tot (2.31)
- 51. 30 2 Irrelevant and Simple Measures where ring is the total length of arterials on rings, and the denominator tot is the total length of all arterials. This quantity ranging from 0 to 1 is thus an indication of the importance of a ring and to what extent arterials are organized as trees. 2.2.3 Cell Areas and Shape Planar graphs naturally produce a set of nonoverlapping cells (or faces, or blocks) and covering the embedding plane. In the case of the road network, the distribution of the area A of these cells has been measured for the city of Dresden in Germany (Fig.2.2b) and has the form P(A) ∼ A−α (2.32) with α 1.9, which was confirmed by measures on other cities [11]. This broad law is in sharp contrast with the simple picture of an almost regular lattice which would predict a distribution P(A) peaked around 2 1. It is interesting to note that if we assume that A ∼ 1/2 1 ∼ 1/ρ and that the density ρ is distributed according to a law f (ρ) (with a finite f (0)); a simple calculation gives P(A) ∼ 1 A2 f (1/A) (2.33) which behaves as P(A) ∼ 1/A2 for large A. This simple argument thus suggests that the observed value ≈2.0 of the exponent is universal and reflects the random variation of the density. More measurements are, however, needed at this point in order to test the validity of this hypothesis. The authors of [29] also measured the distribution of the form or shape factor defined as the ratio of the area of the cell to the area of the circumscribed circle: φ = 4A π D2 (2.34) (for practical applications, D can be also taken as the longest distance in the cell). They found that most cells have a form factor between 0.3 and 0.6, suggesting a large variety of cell shapes, in contradiction with the assumption of an almost regular lattice. These facts thus call for a model radically different from simple models of regular or perturbed lattices. In Chaps.3 and 7, we will discuss more thoroughly this quantity φ and its distribution.
- 52. 2.2 Simple Measures 31 2.2.4 Route Factor, Detour Index When the network is embedded in a two-dimensional space, we can define at least two distances between the pairs of nodes. There is of course the natural Euclidean distance dE (i, j) which can also be seen as the “as crow flies” distance. There is also the total “route” distance dR(i, j) from i to j by computing the sum of length of segments which belong to the shortest path between i and j. The route factor (also called the detour index or the circuity, or directness [82]) for this pair of nodes (i, j) is then given by (see Fig.2.11 for an example) Q(i, j) = dR(i, j) dE (i, j) (2.35) This ratio is always larger than one and the closer to one it is, the more efficient the network. From this quantity, we can derive another one for a single node defined by Q(i) = 1 N − 1 j Q(i, j) (2.36) which measures the “accessibility” for this specific node i. Indeed the smaller it is and the easier it is to reach the node i (Accessibility is a subject in itself–see for example [83]—and there are many other measures for this concept and we refer the interested reader to the articles [84–86]). This quantity Q(i) is related to the quantity so-called “straightness centrality” [87] defined as CS (i) = 1 N − 1 j=i dE (i, j) dR(i, j) (2.37) If one is interested in assessing the global efficiency of the network, one can compute the average over all pairs of nodes (also used in [88]) Fig. 2.11 Example of a detour index calculation. The “as crow flies” distance between the nodes A and B is dE (A, B) = √ 10 while the route distance over the network is dR(A, B) = 4 leading to a detour index equal to Q(A, B) = 4/ √ 10 1.265
- 53. 32 2 Irrelevant and Simple Measures Q = 1 N(N − 1) i= j Q(i, j) (2.38) The average Q or the maximum Qmax , and more generally the statistics of Q(i, j), is important and contains a lot of information about the spatial network under consid- eration (see [89] for a discussion on this quantity for various networks). For example, one can define the interesting quantity [89] φ(d) = 1 Nd i j/dE (i, j)=d Q(i, j) (2.39) (where Nd is the number of nodes such that dE (i, j) = d) whose shape can help for characterizing combined spatial and topological properties (see also Chap.7 for empirical examples). 2.2.5 Cost, Efficiency, and Robustness The minimum number of links to connect N nodes is E = N − 1 and the corre- sponding network is a tree. We can also look for the tree which minimizes the total length given by the sum of the length of all links T = e∈E dE (e) (2.40) where dE (e) denotes the length of the link e. This procedure leads to the minimum spanning tree (MST) which has a total length MST T (see also Sect.12.2 about the MST). Obviously, the tree is not a very efficient network (from the point of view of transportation for example) and usually more edges are added to the network, leading to an increase of accessibility but also of T . A natural measure of the “cost” of the network is then given by C = T MST T (2.41) Adding links thus increases the cost but improves accessibility or the transport per- formance P of the network which can be measured as the minimum distance between all pairs of nodes, normalized to the same quantity but computed for the minimum spanning tree P = MST (2.42)
- 54. 2.2 Simple Measures 33 Another measure of efficiency was also proposed in [90, 91] and is defined as E = 1 N(N − 1) i= j 1 (i, j) (2.43) where (i, j) is the shortest path distance from i to j. This quantity is zero when there are no paths between the nodes and is equal to one for the complete graph (for which (i, j) = 1). The combination of these different indicators and comparisons with the MST or the maximal planar network can be constructed in order to characterize various aspects of the networks under consideration (see for example [46]). Finally, adding links improves the resilience of the network to attacks or dysfunc- tions. A way to quantify this is by using the fault tolerance (FT) (see for example [92]) measured as the probability of disconnecting part of the network with the fail- ure of a single link. The benefit/cost ratio could then be estimated by the quantity FT/MST T which is a quantitative characterization of the trade-off between cost and efficiency [92]. Buhl et al. [46] measured different indices for 300 maps corresponding mostly to settlements located in Europe, Africa, Central America, and India. They found that many networks depart from the grid structure with an alpha index usually low. For various world cities, Cardillo et al. [31] found that the alpha index varies from 0.084 (Walnut Creek) to 0.348 (New York City) which reflects in fact the variation of the average degree. Indeed for both these extreme cases, using Eq. (2.28) leads to αNYC (3.38−2)/4 0.345 and for Walnut Creek αWC (2.33−2)/4 0.083. This same study seems to show that triangles are less abundant than squares (except for cities such as Brasilia or Irvine). Measures of efficiency are relatively well correlated with the alpha index but display broader variations demonstrating that small variations of the alpha index can lead to large variations in the shortest path structure. Cardillo et al. [31] plotted the relative efficiency (see Chap.1) Erel = E − EMST EGT − EMST (2.44) versus the relative cost Crel = C − CMST CGT − CMST (2.45) where GT refers to the greedy triangulation (the maximal planar graph). The cost is here estimated as the total length of segments C ≡ T and the obtained result is shown in Fig.2.12 which demonstrates two things. First, it shows—as expected— that efficiency is increasing with the cost with an efficiency saturating at ∼0.8. In addition, this increase is slow: typically, doubling the value of C shifts the efficiency from ∼0.6 to ∼0.8. Second, it shows that most of the cities are located in the high- cost–high-efficiency region. New York City, Savannah, and San Francisco have the
- 55. Exploring the Variety of Random Documents with Different Content
- 56. Churches First Presbyterian Church In [1785] a bill was introduced into the Legislative Assembly, at Philadelphia, to incorporate a 'Presbyterian Congregation in Pittsburgh, at this time under the care of the Rev. Samuel Barr,' which, after much delay, was finally passed on the twenty-ninth of September, 1787. The Penns gave the site for this church…. In the Spring of 1811 Reverend Francis Herron became the pastor of the First Church, which the year before had had a membership of sixty-five. Dr. Herron's salary was six hundred dollars per annum. For thirty-nine years he labored ceaselessly and wisely for the church and congregation. In 1817 the church was enlarged, and the membership steadily increased. Killikelly's History of Pittsburgh. Second Presbyterian Church The Second Presbyterian Church was organized … in 1804, by those members of the First Church to whom the methods used, regarding the services in the First Church, were unsatisfactory. The next year Dr. Nathaniel Snowden took charge of the congregation which worshiped … in the Court House and other places, public and private. Dr. John Boggs came, but remained only a short time. He was replaced by the Rev. Mr. Hunt, in 1809. The first edifice, on Diamond alley, near Smithfield street, was built in 1814. Killikelly's History of Pittsburgh. East Liberty Presbyterian Church Mr. Jacob Negley, whose wife had been a Miss Winebiddle, and consequently, inherited much real estate, controlled practically what
- 57. is now known as East Liberty Valley, in the early days, called Negleystown. He was largely instrumental … in erecting a small frame school building at what subsequently became the corner of Penn and South Highland avenues. This was for the accommodation of the children of the district, as well as his own. It was … a long distance to the then established churches, and Mr. Negley very often, for the benefit of the neighborhood, invited some minister passing through, or one from one of the other churches, to preach in his own house and later in the school house. In 1819 the little school house was torn down to make way for a church building. Killikelly's History of Pittsburgh. Reformed Presbyterian Church The First Reformed Presbyterian Church of Pittsburg, long afterwards known as the 'Oak Alley Church,' was organized in 1799. Rev. John Black, an Irishman of considerable intellectual force, who had been graduated from the University of Glasgow, was its first pastor…. He included, in his ministry, all societies of the same persuasion in Western Pennsylvania. He preached here until his death on October 25, 1849. Boucher's Century and a half of Pittsburg. Roman Catholic Church The number of Catholics prior to 1800, in what is now Allegheny county, must have been very small. They were visited occasionally by missionaries traveling westward…. [These] priests, ministering to a few scattered families, celebrating Mass in private houses, fill up the long interval between the chapel of the 'Assumption of the Blessed Virgin of the Beautiful River' in Fort Duquesne, and 'Old St. Patrick's Church,' which was begun in 1808. Rev. Wm. F. X. O'Brien, the first pastor, was ordained in Baltimore, 1808, and came to Pittsburg in November of the same year, and at once devoted himself to the erection of … 'Old St. Patrick's.' It stood
- 58. at the corner of Liberty and Washington streets, at the head of Eleventh street, in front of the new Union Station…. The structure was of brick, plain in design and modest in size, about fifty feet in length and thirty in width. Rt. Rev. Michael Egan dedicated the Church in August 1811, and the dedication was the occasion of the first visit of a Bishop to this part of the State. St. Paul's Cathedral record. Protestant Episcopal Church The building of the first Trinity Church was begun about the time it was organized and chartered, 1805. It occupied a triangular lot at the corner of Sixth, Wood and Liberty streets. It was built in an oval form that it might more nearly conform to the shape of the three cornered lot and for this reason was generally known as the 'round church.' Rev. Taylor in his latter years became known as 'Father' Taylor. He remained with the church as its rector until 1817, when he resigned. Boucher's Century and a half of Pittsburg. First German United Evangelical Protestant Church When John Penn, jr., and John Penn presented land to the Presbyterian and Episcopal churches of Pittsburgh they, at the same time, deeded the same amount to the already organized German Evangelical congregation; the land given to them was bounded by Smithfield street, Sixth avenue, Miltenberger and Strawberry alleys. No church was built on this grant, however, until some time between 1791-94, and it was of logs. This was … replaced in 1833 by a large brick building, which had the distinction of a cupola, in which the first church bell in Pittsburgh was hung. Killikelly's History of Pittsburgh. Methodist Episcopal Church In June, 1810, a lot was purchased for the first [Methodist] church built in the city. It was situated on Front street, now First street,
- 59. nearly opposite … the present Monongahela House. The erection of a church was commenced at once, for on August 26th of that year Bishop Asbury preached on the foundation of it. His journal says: 'Preached on the foundation of the new chapel to about five hundred souls. I spoke again at 5 o'clock to about twice as many. The society here is lively and increasing in numbers.' The building was a plain brick structure, 30 × 40 feet. We do not know certainly when it was completed, but probably in the autumn of 1810. In this church the society continued to worship in peace and prosperity for eight years. But near the close of this period it had become too small, and a new and larger one became a necessity. Consequently, in May, 1817, three lots were purchased on the corner of Smithfield and Seventh streets, and the erection of a larger church commenced. It was completed the following year. Warner's History of Allegheny county. Baptist Church The first church of this denomination in Pittsburg was organized in April, 1812, when the city had about five thousand people. It was an independent organization and included about six families with perhaps not more than twelve people in all who had come from New England. The chief organizer and pastor was Rev. Edward Jones, also from New England. The society was too poor then to build a church, but worshiped in private houses and in rented halls. Boucher's Century and a half of Pittsburg.
- 60. Schools Robert Steele, who afterward became a Presbyterian preacher, opened a school in Pittsburg in January, 1803, at his house on Second street…. His rates were four dollars per quarter. In 1803, a teacher named Carr opened a school for both boys and girls. The next year he advertised that his school was moved to larger quarters over Dubac's store, where he probably taught till 1808, when he opened a boarding school for boys. In 1818 he removed his school to Third street where Mrs. Carr 'instructed young ladies in a separate room in the usual branches, and in all kinds of needle work.' William Jones began a school in 1804, and charged but two dollars per quarter for tuition. In February, 1808, Samuel Kingston opened a school in a stone house on Second street…. A teacher named Graham opened a school on Second street, using the room formerly occupied by Mr. Kingston, in which he proposed to give his pupils an English and classical education on moderate terms. The advertisement stated that Mrs. Graham would at the same time open a school for 'young ladies' in an adjoining room, and that she would instruct them in all branches of an English education and in needle work. In 1811 Thomas Hunt opened a school 'for the instruction of females exclusively.' The hours he advertised were from 8 to 12 a.m., and from 2 to 5 p.m…. In the same year this advertisement appears: 'Messrs. Chute and Noyes' evening school commences the first of October next. They also propose on Sabbath morning, the 22 instant, to open a Sunday morning school to commence at the hour of eight a.m., and continue until ten. They propose to divide the males and females into separate departments. The design of the school is to instruct those who wish to attend, the Catechism and hear them read the Holy Scriptures. No pecuniary compensation is desired, a consciousness of doing good will be an ample reward.' In 1812 John Brevost opened a French school, and with his wife and daughter opened a boarding school in connection with it in 1814.
- 61. Their terms were, 'for reading, writing, arithmetic, English grammar, history and geography, with the use of maps, globes, etc., $8.00 quarterly. Playing on the piano, $10.00 quarterly; vocal music, $5.00 quarterly. Drawing and painting of flowers, $6.00 quarterly. French language, $5.00 quarterly. Boarding $37.00, payable in advance. Dancing, books, materials, drawing, sewing, bed and bedding to be paid for separately or furnished by parents.' Mrs. Gazzam had opened a seminary for young ladies by this time, and advertised its removal to Fifth street. Her pupils were instructed in the elementary studies of an English education, and in needle work at four dollars per quarter. She taught them to cut, make and repair their clothes. The pupils were permitted to visit their homes once each week, but no young men were allowed to visit them unless attended by a servant. She boarded them for $125 per year. The two sisters, Miss Anna and Arabella Watts, instructed young ladies solely in needle work. In almost all schools needle work was a requisite part of the education of young women. In fact it was considered the all important part of a woman's training and not infrequently other branches were taught if required, or if thought necessary. Boucher's Century and a half of Pittsburg. CITY ACADEMY The subscriber, respectfully informs his fellow citizens, and others, that he has happily secured the co-operation of Mr. Edward Jones— hopes their most sanguine expectations, relative to his seminary, will be fully justified. All the most important branches of education, taught as in the best academies, on either side the Atlantick.—Mathematics in general, as in the city of Edinburgh.—During four years, the subscriber taught the only Mathematical school in the capital of New-Hampshire. A class of young gentlemen will shortly commence the study of Navigation, Gunnery, Bookkeeping, Geography and English grammar. George Forrester. Mercury, May 18, 1816.
- 62. THE LANCASTER SCHOOL. Will continue at the room where it is now kept in Market street. In addition to the common branches of reading, orthography, etc., the teacher gives lessons in English grammar, geography and Book- keeping. Penmanship is taught on a most approved system at all hours. To those who are acquainted with this mode of instructing children, its superior excellence need not be pointed out, and such as have never seen a school on this plan in actual operation, and are not intimately conversant with its theory, are invited (if they have the curiosity) to visit the institution in Market street; where, although the number of pupils is small, yet the school will afford a sufficient illustration of the Lancaster system to convince the most incredulous that 500 or even 1000 pupils by the aid of this wonderful invention, may be taught with prodigious facility by a single teacher. Commonwealth, April 3, 1816. UNIVERSITY OF PITTSBURGH The first charter to an institution of learning west of the mountains granted by the legislature of Pennsylvania, February 28, 1787, created the Pittsburg Academy. The school was in existence earlier than this…. The principals of the academy from the very beginning were men of high attainments, some of them attaining great distinction. George Welch, the first principal, took office April 13, 1789. Rev. Robert Steele, pastor of the First Presbyterian Church, Rev. John Taylor, Mr. Hopkins and James Mountain successively were at the head of the academy. From 1807 to 1810, Rev. Robert Patterson, of excellent fame, successfully carried on the work. He was succeeded in the latter year by Rev. Joseph Stockton, author of the 'Western Calculator' and 'Western Spelling Book,' who continued in office until
- 63. the re-incorporation of the academy as the Western University of Pennsylvania, in 1819. Boucher's Century and a half of Pittsburg.
- 64. Libraries It was not … until the fall of 1813, that the question of a community Library took definite shape, when in response to the efforts 'of many leading and progressive citizens,' there was organized 'The Pittsburgh Library Company.' On the evening of November 27, 1813, about 40 representative people assembled in the spacious 'bar room' of the 'Green Tree Inn,' at the northwest corner of Fifth and Wood streets, where the First National Bank now stands, and took the initiative in the formation of Pittsburgh's first real public library…. Its first president was the Rev. Francis Herron, for 40 years pastor of the First Presbyterian Church. The secretary was Aquila M. Bolton, 'land broker and conveyancer.' The treasurer was Col. John Spear…. Quite a sum of money was subscribed by citizens generally for the purchase of books, while many valuable volumes were either contributed or loaned by members. Messrs. Baldwin, O'Hara, Wilkins and Forward being especially mentioned for their generosity in this connection. The first head-quarters of the library were in rooms 'on Second street, opposite Squire Robert Graham's office,' who at that time dispensed even handed justice at the northeast corner of Market and Second streets. Here the library remained until the county commissioners set aside a commodious room in the Court House for its use. A. L. Hardy, in Gazette-Times, 1913. The triennial meeting of the shareholders [of the Pittsburgh Library Company] was convened at their new library room, in Second street, opposite Squire Graham's office, at six o'clock, Monday evening, December thirtieth, 1816. The following gentlemen were then elected by ballot to serve as a Board of Directors for the ensuing three years, viz: George Poe, president; Aquila M. Bolton, secretary;
- 65. Lewis Bollman, treasurer; James Lea, Benjamin Bakewell, Robert Patterson, Walter Forward, Alexander Johnson, jr., William Eichbaum, jr., Benjamin Page, Alexander McClurg, J. P. Skelton, Ephraim Pentland, Charles Avery, J. R. Lambdin, directors. Killikelly's History of Pittsburgh. It has been published, that the Library of this city contains two thousand volumes. Through the politeness of J. Armstrong, the librarian, I gained admittance, and having examined the catalogue, am enabled to state that the whole collection is only about five hundred volumes. The books, however, are well chosen, and of the best editions. How the error originated is of no consequence except to him who made it. Thomas's Travels through the western country in 1816.
- 66. The New Books of 1816 Austen. Emma. Byron. Childe Harold (Canto III). The dream. Hebrew melodies. Parisina. Prisoner of Chillon. Siege of Corinth. Coleridge. Christabel. Crabbe. Dictionary of English synonymes. D'Israeli. Character of James I. Goethe. Italianische reise. Hunt. A story of Rimini. Moore. Elegy on Sheridan. Irish melodies. Peacock. Headlong Hall. Scott. Antiquary. Black dwarf. Guy Mannering. Lord of the Isles. Old Mortality. Shelley. Alastor. Southey. Carmen triumphale. Wordsworth. Poems. White doe of Rylstone.
- 67. The Theatre There were in 1808 two dramatic societies in Pittsburg that were important enough to receive notice in the newspapers. The one was composed of law students and young lawyers and the other was composed of mechanics. The object of these societies was to study the poets and dramatic literature and to give public performances in the court house. William Wilkins … was a member and took a leading part in the entertainments given by these societies. There was no way for theatrical companies from the East to reach Pittsburg prior to 1817, save by the state road, which was scarcely passable for a train of pack horses, yet they came even as early as 1808 and performed in a small room, which was secured for them when the court room was occupied. In 1812 a third dramatic society called the Thespian Society was organized among the young men and young women of Pittsburg. The society numbered among its members the brightest and best bred young people of the city, most of whom took part in each performance. They were given in a room on Wood street, in a building known as Masonic Hall. Boucher's Century and a half of Pittsburg. The Theatre of this City has been now opened nearly a fortnight, and the managers although they have used every exertion to please, in the selection of their pieces, have not been enabled to pay the contingent expenses of the House. This is a severe satire on the taste of the place. Tomorrow [Wednesday] evening we understand that the 'Stranger' is to be produced—we hope under auspices more favorable to the managers than heretofore. The part of the Stranger is to be
- 68. performed by a Young Gentleman of the City, who has never before graced the Boards.—If report speaks correctly of his talents, he bids fair to excel any person who has yet appeared upon the stage on this side the Mountains. It is hoped that this novelty, together with the correct and manly acting of Mr. Savage, a stranger here, and the chastened elegance which Mrs. Savage is said to exhibit, will attract to the Theatre, for this one evening at least, the friends to this rational amusement. Commonwealth, Nov. 12, 1816. On Friday evening, June 7, will be presented, Shakespear's celebrated comedy, in 3 acts called Catharine Petruchio after which, a much admired comic opera called The Highland Reel. For particulars, see bills. And, that every person should have the opportunity of seeing the most splendid spectacle ever exhibited in Pittsburgh, on Saturday evening, June 8, will be presented, the grand romantic drama, called Timour the Tartar; or, the Princess of Mingrelia. Which will positively be the last time, of its being performed, as the scenery will be appropriated to other purposes. With other Entertainments. For particulars, see box bills. A few days after the performance of Hamlet, Mr. Entwistle, the manager, had for his benefit, that irresistibly amusing burlesque,
- 69. 'Hamlet Travestie.' His line of acting is a broad-farce caricature of that of Liston. He personated the modern Danish prince. The audience were solemn, serious, and dull. The affecting entrance of the deranged Ophelia, who, instead of rosemary, rue, c. had an ample supply of turnips and carrots, did not move a muscle of their intelligent faces—the ladies, indeed, excepted, who evinced by the frequent use of their pocket handkerchiefs, that their sympathies were engaged on the side of the love-sick maiden. Some who had seen the original Hamlet for the first time a few evenings before, gave vent to their criticisms when the curtain fell. They thought Mr. Entwistle did not look sufficiently grave; and that, as it was his benefit, he acted very dishonourably in shaving (cheating) them out of two acts; for that they guessed when Mr. Hutton played that'ere king's mad son, he gave them five acts for their dallar. Mr. —— assured me that on the following morning, a respectable lawyer of Pittsburgh met him, and said, 'I was at the play last night, Sir, and do not think that Mr. Entwistle acted Hamlet quite so well as Mr. Hutton.' Fearon's Sketches of America, 1818. Thespian Society The Public are respectfully informed that on this evening, Jan. 14th, will be presented the much admired Drama, called the Man of Fortitude. The proceeds to be appropriated to the benefit of the Sunday Male Charitable School. Recitation, Alonzo the brave or the fair Imogen. Song, I have loved thee, dearly loved thee.—Mrs. Menier. ——, America, Commerce and Freedom.
- 70. After which the much admired Farce, called, The Review, Or, the Wag of Windsor. Doors to be opened at half past 5 o'clock, and the curtain to rise at half past six. Box, one dollar; Pit, Fifty cents. A citizen of Pittsburgh, and a lover of the useful and rational amusement of the Theatre, begs leave to observe to his fellow citizens, that on Monday evening next Mr. Alexander will stand forward for public recompense, for his exertions in his profession…. It must be readily acknowledged that no young gentleman of more transcendent talent ever graced the dramatic floor of Pittsburgh; it is, therefore, but just that he who has so often made us smile, should from us receive a something to make him smile in turn. Commonwealth, Nov. 4, 1815. The Theatre in this city is now opened by the Thespian Society, for the double purpose of gratifying the public taste by a moral and rational amusement and adding to the funds of the Male Charitable Sunday School. The Man of Fortitude and the Farce of the Review have been selected for representation this evening. Since society has been released from the chains of superstition, the propriety of Theatrick amusements has not been doubted by any man of liberal feelings and enlightened understanding…. The stage conveys a moral in colours more vivid than the awful and elevated station of the preacher permits him to use—it is his coadjutor in good, and goes with him hand in hand exposing vice to ridicule and honouring virtue. Gazette, Jan. 14, 1817.
- 71. The Morals Efficiency Society of 1816 The Moral Society of Pittsburgh announce to the public their formation. The object of their association is the suppression of vice and immorality, as far as their influence shall extend, and they shall be authorized by the laws of the commonwealth, and the ordinances of this city…. We hereby give this public information of our intention to aid the civil officers in the execution of the laws of this commonwealth, and the ordinances of the city, against all vice and crime cognizable by said laws and ordinances. Such as profane swearing, gambling, horse racing, irregular tippling houses and drunkenness, profanation of the Lord's day by unnecessary work of any kind, such as driving of waggons, carts, carriages of pleasure and amusement, or other conveyances not included under the exception of the laws of the commonwealth in case of necessity and mercy. Commonwealth, Nov. 26, 1816.
- 72. Fourth of July, 1816 A numerous and respectable concourse of citizens met at Hog Island, nearly opposite the village of Middletown, on the Ohio river, to celebrate the birth day of American independence. Colonel James Martin, was nominated president, and Captain Robert Vance, vice- president.—The utmost harmony and unanimity prevailed; and it was a pleasing sight to see citizens of opposite political sentiments, bury their former animosity, and with great cordiality join in celebrating the American anniversary. After performing the manual exercise, the company partook of an elegant dinner, prepared for the occasion, and the cloth being removed … patriotic toasts were drank with great hilarity, accompanied by the discharge of musketry, and appropriate music…. The citizens retired at a late hour in the utmost harmony. Mercury, July 20, 1816.
- 73. Police When the borough was incorporated into a city [March 1816], the act incorporating it authorized the authorities to establish a police force, but there was none established for some years afterwards. The act limited the city taxation to five mills on a dollar, and the corporation could scarcely have paid a police force, even if one had been required. The city authorities did, however, pass an ordinance on August 24, 1816, establishing a night watchman, but soon found they had no money with which to pay him. They accordingly repealed the ordinance and for some years the city slept in darkness without the benefit of police protection. Boucher's Century and a half of Pittsburg.
- 74. Eagle Fire Company In 1811 the second epoch in the company's history may be said to have started, the younger element having gradually crept in and assumed control of affairs, and the older men had to some extent lost interest and perhaps gained rheumatism in the fire service. The company was now re-organized on a more active and vigorous basis. The first engineer to take charge under the new regime was William Eichbaum, who continued to act in that capacity until 1832, when he was elected First Chief Engineer of the Fire Department on its organization…. In the company organization the most important duty devolved upon the Bucket Committee. Every citizen was required to keep two or three heavy leather buckets with his name painted on them, and in case of fire these were all brought on the ground. Two lines of men and women were formed to the water supply, to pass the full buckets to and the empty ones from the engine…. When the fire was extinguished all the buckets were left on the ground till next day. Then, as many of the inscriptions were obliterated, there was some stealing of buckets and consequent fights. Certain folks … picked out the best buckets, just as in modern times some people get the best hats, or umbrellas, at the conclusion of a party. The Bucket Committee, to put a stop to this, decided to deliver all buckets to their respective owners. Dawson's Our firemen.
- 75. Water-Supply The water supply was gained, up to 1802, from wells and springs which flowed from out the hillsides, these being sufficient for a small town. An ordinance passed August 9, of that year, called for the making of four wells, not less than forty-seven feet in depth. Three of these were to be located on Market street, and were to be walled with stone…. Wells, with the springs at Grant's Hill, furnished the supply of water for public use until 1826. Boucher's Century and a half of Pittsburg.
- 76. Banks As early as the year 1815, there were only three banks in Pittsburgh; viz., the Bank of Pennsylvania, located on the north side of Second Avenue, between Chancery Lane and Ferry Street; Bank of Pittsburg, south-west corner of Market and Third Streets; Farmers and Mechanics' Bank, north side of Third, between Wood and Market Streets,—the aggregate capital amounting to less than two million dollars, which was considered abundantly adequate to the business of that period. Parke's Recollections of seventy years. The Bank of Pittsburgh is situated on the s.w. corner of Market and Third streets. President, William Wilkins, Directors, George Anchutz, Jun. Thos. Cromwell Nicholas Cunningham John Darragh William Hays Wm. McCandless James Morrison John M. Snowden Craig Ritchie (Cannonsbr'g) George Allison James Brown (baker) T. P. Skelton Cashier Alexander Johnston, Jun. Open daily from 9 o'clock a.m. till 3 p.m., except Sunday, Fourth of July, Christmas and Fast days. Discount day, Wednesday. Capital $600,000. Shares $50 each. Dividends, first Mondays in May and November. Pittsburgh directory, 1815.
- 77. The Office of Discount and Deposit of the Bank of Pennsylvania is situated on the north side of Second between Market and Ferry streets. President, James O'Hara. Directors, Joseph Barker Ebenezer Denny Anthony Beelen Boyle Irwin Thomas Baird George Wallace David Evans Pittsburgh directory, 1815. THE FARMERS' AND MECHANICS' BANK OF PITTSBURGH. Cashier George Poe, Jun. Open daily from 9 o'clock a.m. till 3 p.m., except Sunday, Fourth of July, Christmas and Fast days. Discount day, Thursday. Is situated on the north side of Third, between Market and Wood streets. President, John Scull Directors, William Eichbaum, Jun. William Leckey John Ligget Jacob Negley Pittsburgh directory, 1815.
- 78. Post-Office POST-OFFICE ESTABLISHMENT. Arrival and Departure of the MAILS, At the Post-Office—Pittsburgh The Eastern Mail arrives on Sunday, Wednesday, and Friday evenings, and closes on Tuesday, Thursday, and Saturday, at 8 o'clock A.M. The Western Mail arrives on Tuesday, Thursday, and Saturday, and closes on Sunday at sunset, and Wednesday and Friday at 1 o'clock P.M. The Beaver Mail arrives on Monday evening, and closes the same day at sun-set. The Erie Mail arrives on Monday evening and closes the same day at sun-set. The Steubenville Mail arrives on Sunday and Wednesday evenings, and closes same days at sun-set. The Huntingdon Mail, via Ebensburgh and Indiana, arrives on Tuesday, and closes same day at half past twelve P.M…. As there are several places of the same name in the United States, it is necessary that the directions should be particular, the states should be distinguished, and, where it might otherwise be doubtful, the counties….
- 79. Those who send letters may either pay the postage in advance, or leave it to be paid by their correspondents. Rates of Postage For Single Letters Cents For any distance not exceeding 40 miles 12 Over 40 and not exceeding 90 do 15 Over 90 and not exceeding 150 do 18¾ Over 150 and not exceeding 300 do 25½ Over 500 37½ Pittsburgh directory, 1815.
- 80. The Suburbs Birmingham is a small village across the Monongahela, about one mile south of Pittsburgh. It has works for green glass, furnaces for casting hollow ware, c. from pigs, and a saw mill, which is moved by a steam engine. The coal for all these, is used fresh from the mine, without mixture, coaking or desulphuration. Many of the balls for Perry's fleet, were cast in this foundery. But instead of forming such ministers of havoc, the metal is now moulded for softer hands, and flat or smoothing irons are produced in abundance. These are ground on a stone which revolves by a band from the steam engine. Thomas's Travels through the western country in 1816. At a respectable meeting of the inhabitants of Birmingham and its vicinity, convened at the school-house, on Friday evening the 28th of March, 1816, in order to take into consideration the expediency of erecting a Market-House, in said town; Nathaniel Bedford, was called to the Chair, and George Patterson, appointed secretary. The chairman having stated the object of the meeting, the following resolutions were proposed and unanimously adopted, viz.— Resolved, That a Market-House be built on the plan exhibited by Mr. Benjamin Yoe. Resolved, That the site of the structure be the centre of the square. Thus, another thriving and Manufacturing Town, is added to the many which have been established in the western section of Pennsylvania; and social order, with its concomitants, the arts and sciences, illuminate those wild and dreary shades, where lately none
- 81. but the prowling wolf, or the restless and cruel savage held their haunts. Mercury, April 20, 1816. At the beginning of the century the site of Allegheny City was a wilderness. In 1812 a few settlers had made inroads upon the forest, and had builded their cabins. Notice is called to the fact in the minutes of the Presbytery of Erie, in April of that year, in the following words: 'An indigent and needy neighborhood, situated on the Allegheny, opposite Pittsburgh, having applied for supplies,' the matter was laid before the Presbytery. Joseph Stockton seems to have been the first stated minister, preaching a part of his time there until 1819. Centenary memorial of Presbyterianism in western Pennsylvania. The facility for getting to and from Pittsburg [from Allegheny] was quite a different matter from what it is to-day. The only highway (if it may be called such) leading west from Federal Street to the Bottoms at that early day, was the erratic Bank Lane, which owing to the natural unevenness of the ground upon which it was located, and total neglect of the authorities of Ross township to put it in a condition for travel, … was for many years only accessible for foot- passengers. Parke's Recollections of seventy years. Lawrenceville was laid out in 1815 by Wm. B. Foster, and had begun with the building of the United States arsenal.
- 82. Courts The Supreme Court holds a term in Pittsburgh, on the 1st Monday in September annually, to continue two weeks if necessary, for the Western District, composed of the counties of Somerset, Westmoreland, Fayette, Greene, Washington, Allegheny, Beaver, Butler, Mercer, Crawford, Erie, Warren, Venango, Armstrong, Cambria, Indiana and Jefferson. Pittsburgh directory, 1815. Mr. Lacock submitted an important resolution for instructing the committee on the Judiciary to enquire into the expediency of dividing the state of Pennsylvania into two Judicial Districts, and establishing a district court of the U. States at the city of Pittsburgh, which was agreed to. Commonwealth, Jan. 6, 1817.
- 83. County Elections Henry Baldwin is elected to congress for the district composed of the counties of Allegheny and Butler, by a majority of about 800 votes. John Gilmore, William Woods, Samuel Douglass and Andrew Christy are elected to the assembly. Lazarus Stewart is elected Sheriff of Allegheny county, by a majority of 181 votes. Joseph Davis is elected commissioner by a majority of 249 votes, and Charles Johnson, Auditor by a majority of 28 votes. Gazette, Oct. 15, 1816.
- 84. The State Legislature The bill for erecting the two Bridges at Pittsburgh has passed both houses. The sites are fixed at St. Clair-street for the Allegheny and Smithfield-street for the Monongahela. The state subscribes $40,000 of stock for each bridge. A bill is about being reported for establishing a horse and cattle market in the vicinity of Pittsburgh. The bill for erecting Pittsburgh into a city has passed the senate and is before the house, where it is expected to pass through without opposition. The bill for erecting a new county out of parts of Allegheny, Westmoreland, Washington, and Fayette, is reported. This bill will throw off the greater part of Elizabeth township from Allegheny county. There have been no remonstrances against it received from this county; but we understand that some have been received from the other counties concerned. The bill for erecting a Poorhouse for Allegheny county, it is expected will pass. Mercury, Feb. 24, 1816. We regret to say that neither from our correspondent at Harrisburg nor from the papers printed there, have we been enabled to procure an account of the legislative proceedings. We take two papers published at the seat of government, but from some unaccountable reason they do not contain the intelligence our readers require. We are reduced to the necessity of picking up here and there from letters to editors—from information derived from travellers—or from some other like inconclusive sources of information, that intelligence with which Journals published at the seat of government should
- 85. supply us. They ought to be the fountains of information to the mass of the community: Instead of dabbling in politics and abusing or eulogizing party leaders, they should deal in facts. The National Intelligencer we look upon as the best model with which we are acquainted of a national journal. Commonwealth, Dec. 24, 1816.
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