# Encyclopedia of Social Network Analysis and Mining

Living Edition
| Editors: Reda Alhajj, Jon Rokne

# Spatial Networks

• Marc Barthelemy
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7163-9_40-1

## Keywords

Planar Graph Degree Distribution Power Grid Betweenness Centrality Transportation Network
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Glossary

Graph

(or Network) A set of vertices connected by edges

A matrix A which represents the structure of a graph. The element A ij is either 0 if i and j are not connected or A ij = 1 if there is an edge from i to j. For a spatial network, the position of the nodes {x i } is needed in order to completely characterize the network

Betweenness Centrality

The betweenness centrality of a vertex (or an edge) x is defined as $$BC(x)={\sum}_{s, t\in V}\frac{\sigma_{s t}(x)}{\sigma_{s t}}$$ where σ st (x) is the number of shortest paths between s and t using x and σ st is the number of all shortest paths between s and t

Betweenness Centrality Impact

Measures how a new link affects the average betweenness centrality of a graph. This quantity can help in characterizing the different types of new links during the evolution of a (spatial) network

Cell

Also called face for planar network is a region bounded by edges. The Euler formula relates the number of nodes, edges, and cells (faces)

Diameter

The diameter of a graph is defined as the maximum value of all ℓ(i, j), is the distance between i and j, and is used to measure the “size” of it. For most real-world spatial network, the diameter scales as the number of nodes to the power 1/d where d is the dimension of the embedding space

Planar Graph

A planar graph can be drawn in 2-D such that none of its edges are crossing

Organic Ratio

Measures the proportion of degree 1 (“dead ends”) and degree 3 nodes (“T-shaped intersections”). If the organic ratio is small, the corresponding spatial network is very close to a regular rectangular lattice

Alpha Index

Also called the meshedness, it measures the ratio of observed circuits to the maximum number of elementary circuits which can exist in the network

Gamma Index

Ratio of the number of edges to the maximum number possible for a planar graph with the same number of nodes

Shape Factor

Ratio of the area of a cell to the area of the circumscribed circle

Route Distance

Distance between two nodes measured by the length of the shortest path connecting them

Detour Index

Ratio of the route distance between two nodes and the Euclidean distance between them

Network Cost

Ratio of the total length of the network to the total length of the minimum spanning tree constructed on the same set of nodes

Network Performance

Ratio of the average shortest path of the network to the average shortest path of the minimum spanning tree constructed on the same set of nodes

## Definition

More generally, the term “spatial network” has come to be used to describe any network in which the nodes are located in a space equipped with a metric (Barthelemy 2011). For most practical applications, the space is the two-dimensional space and the metric is the usual Euclidean distance. For these networks we thus need both the topological information about the graph (given by the adjacency matrix) and the spatial information about the nodes (given by the position of the nodes).

Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, and neural networks are all examples where space is relevant and where topology alone does not contain all the information.

Characterizing and understanding the structure and the evolution of spatial networks is crucial for many different fields ranging from urban-ism to epidemiology. An important consequence of space on networks is that there is usually a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks. Indeed, a long link will be very costly and can exist if this cost if balanced with another good reason (economical or connection to a hub, …). For most real-world spatial networks, we indeed observe that the probability of finding a link between two nodes will decrease with the distance. Spatial constraints affect not only the structure and properties of these networks but also processes which take place on these networks such as phase transitions, random walks, synchronization, navigation, resilience, and disease spread.

All planar graphs can be embedded in a two-dimensional space and can be represented as spatial networks, but the converse is not necessarily true: there are some spatial and nonplanar graphs. In general, however, most spatial networks are, to a good approximation, planar graphs (Clark and Holton 1991), such as road or railway networks, but there are some important exceptions such as the airline network (Barrat et al. 2004): in this case the nodes are airports and there is a link connecting two nodes if there is at least one direct connection. For many infrastructure networks, however, planarity is unavoidable. Power grids, roads, rail, and other transportation networks are to a very good accuracy planar networks. For many applications, planar spatial networks are the most important and most studies have focused on these examples.

Also, the above definition does not imply that the links are necessarily embedded in space. Indeed, in social networks, individuals are connected through a friendship relation which is a virtual network of relations. There is however a strong spatial component in these networks as the probability that individuals located in space are friends generally decreases with the distance between them (Liben-Nowell et al. 2005).

## Introduction

For many critical infrastructures, communication or biological networks, space is relevant: most of the people have their friends and relatives in their neighborhood, power grids and transportation networks depend obviously on distance, many communication network devices have short radio range, the length of axons in a brain has a cost, and the spread of contagious diseases is not uniform across territories. In particular, in the important case of the brain, regions that are spatially closer have a larger probability of being connected than remote regions as longer axons are more costly in terms of material and energy (Bullmore and Sporns 2009). Wiring costs depending on distance are thus certainly an important aspect of brain networks, and we can probably expect spatial networks to be very relevant in this rapidly evolving topic. Another particularly important example of such a spatial network is the Internet which is defined as the set of routers linked by physical cables with different lengths and latency times. More generally, the distance could be another parameter such as a social distance measured by salary, socio-professional category differences, or any quantity which measures the cost associated with the formation of a link.

All these examples show that these networks have nodes and edges which are constrained by some geometry and are usually embedded in a two- or three-dimensional space, and this has important effects on their topological properties and consequently on processes which take place on them. If there is a cost associated to the edge length, longer links must be compensated by some advantage, for example, being connected to a well-connected node - that is, a hub. The topo-logical aspects of the network are then correlated to spatial aspects such as the location of the nodes and the length of edges.

## Tools for Characterizing Spatial Networks

Graphs are usually characterized by the adjacency matrix A where the elements are A ij = 1 if nodes i and j are connected (see, e.g., a graph textbook Clark and Holton 1991). This matrix completely characterizes the topology of the graph and is enough for most applications. This is however not the case for spatial networks where the spatial information is contained in the location of the nodes x i . Two topologically identical graphs can then have completely different spatial properties, and this is at the heart of the richness and complexity of spatial networks.

In this section, we will discuss some tools which can be helpful to characterize some aspects of spatial networks.

## Degree Distribution, Clustering, and Average Shortest Path Length

### Degree Distribution

In complex networks, the degree distribution, the clustering spectrum, and the average shortest distance are of utmost importance (Albert and Barabasi 2002). Their knowledge already gives a useful picture of the graph under study. In contrast, in spatial networks, physical constraints impose some of the properties. In particular, there is usually a sharp cutoff on the degree distribution P(k) which is therefore not broad. This is true for most spatial and planar networks such as power grids or transportation networks, for example. For a spatial, nonplanar network such as the airline network, the cutoff can be large enough and the degree distribution could be characterized as broad.

### Clustering

The clustering coefficient of a node counts how its neighbors are connected with each other. For spatial networks, the dominant mechanism is usually to minimize cost associated with length, and nodes have a tendency to connect to their nearest neighbors, independently from their degree. This in general implies that the clustering spectrum C(k) is relatively flat for spatial networks. The same argument can be used to show that the assortativity “spectrum” defined as the function k nn (k) is also approximately constant in general when spatial constraints are very strong (see Barthelemy 2011 for more details).

### Average Shortest Distance

Usually, there are many paths between two nodes in a connected network, and the shortest one defines a distance on the network:
$$\ell \left( i, j\right)=\underset{\mathrm{paths}\left( i\to j\right)}{ \min}\left|\mathrm{path}\right|$$
(1)

where the length ǀpathǀ of the path is defined as its number of edges. This quantity is infinity when there are no paths between the nodes and is equal to one for the complete graph (for which (i, j) = 1). For weighted graphs, we assign to each link e a weight w e and the length of a path is given by |path| = ∑ e ∈ Path w e .

In most complex networks, one observes a small-world behavior (Watts and Strogatz 1998) of the form
$$\left\langle \ell \right\rangle \sim \log N$$
(2)
In contrast, for a real-world spatial network embedded in a d-dimensional space, we usually observe the very different behavior:
$$\left\langle \ell \right\rangle \sim {N}^{1/ d}$$
(3)

which also means that to go from one node to another one, one has to cross a path of length of the order of the diameter (which is not the case when shortcuts exist). The measure of the average shortest path length could thus be a first indication whether a network is close to a lattice or if long-range links are important.

## Organic Ratio

We note that more recently, other interesting indices were proposed in order to characterize specifically road networks (Xie and Levinson 2007). Indeed, the degree distribution is very peaked around 3–4, and an interesting information is given by the ratio
$${r}_N=\frac{N(1)+ N(3)}{\sum_{k\ne 2} N(k)}$$
(4)

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 regular crossing with k = 4, signalling a more organized city. In the opposite case of large r N ≃ 1, there is dominance of k = 1 and k = 3 nodes which signals a more “organic” city.

## Betweenness Centrality

### Anomalies

The betweenness centrality (BC) of a vertex (Freeman 1977) is determined by its ability to provide a path between separated regions of the network. Hubs are natural crossroads for paths, and it is natural to observe a marked correlation between the average $$\Big( g(k)={\sum}_{i/{k}_i= k} g(i)/ N(k)$$ and k as expressed in the following relation:
$$g(k)\sim {k}^{\eta}$$
(5)
where η depends on the characteristics of the network. We expect this relation to be altered when spatial constraints become important, and in order to understand this effect, we consider a one-dimensional lattice which is the simplest case of a spatially ordered network. For this lattice the shortest path between two nodes is simply the Euclidean geodesic, and for two points lying far from each other, the probability that the shortest path passes near the barycenter of the network is very large. In other words, the barycenter (and its neighbors) will have a large centrality as illustrated in Fig. 1a. In contrast, in a purely topological network with no underlying geography, this consideration does not apply anymore, and if we rewire more and more links (as illustrated in Fig. 1b), we observe a progressive decorrelation of centrality and space while the correlation with degree increases. In a lattice, it is easy to show that the BC depends on space and is maximum at the barycenter, while in a network the BC of a node depends on its degree. When the network is constituted of long links superimposed on a lattice, we then expect the appearance of “anomalies” characterized by large deviations around the behavior g ~ k η .

### Betweenness Centrality Impact

When studying the time evolution of networks, it is important to be able to characterize quantitatively new links. This is particularly true for spatial networks, but what follows could also be applied to general, complex networks.

We consider a time-evolving graph G t described by a set of nodes V t and edges E t at time t. In order to evaluate the impact of a new link on the overall distribution of the betweenness centrality in the graph at time t, we first compute the average betweenness centrality of all the links of G t as
$$\overline{b}\left({G}_t\right)=\frac{1}{\left( N(t)-1\right)\left( N(t)-2\right)}\sum_{e\in {E}_t} b(e)$$
(6)
where b (e) is the betweenness centrality of the edge e in the graph Gt. Then, for each new link e* added in the time window [t − 1, t], we consider the new graph obtained by removing the link e* from G t , denoted by Gt\{e*}. The impact δ b (e*) of edge e* on the betweenness centrality of the network at time t is then defined as (Strano et al. 2012 )
$${\delta}_b\left({e}^{\ast}\right)=\frac{\left[\overline{b}\left({G}_t\right)-\overline{b}\left({G}_t\backslash \left\{{e}^{\ast}\right\}\right)\right]}{\overline{b}\left({G}_t\right)}$$
(7)

The betweenness centrality impact is thus the relative variation of the graph average betweenness due to the removal of the link e* and can thus help to characterize quantitatively the various mechanisms at play during the evolution of the network (Strano et al. 2012).

## Mixing Space and Topology

All the previous indicators describe essentially the topology of the network, but are not specifically designed to characterize spatial networks. We will here briefly review other indicators which provide useful information about the spatial structure of networks. Different indices were defined a long time ago mainly by scientists working in quantitative geography since the 1960s and can be found in Haggett and Chorley (1969) (see also the more recent paper by Xie and Levinson (2007)). Most of these indices are relatively simple but still give important information about the structure of the network in particular if we are interested in planar networks. These indices were used so far to characterize transportation networks such as highways or railway systems.

### Alpha and Gamma Indices

The most important indices are called the “alpha” and the “gamma” indices. The simplest index is called the gamma index and is simply defined by
$$\gamma =\frac{E}{E_{\max }}$$
(8)
where E is the number of edges and E max is the maximal number of edges (for a given number of nodes N). For nonplanar networks, E max is given by N(N − 1)/2 for nondirected graphs and for planar graphs E max = 3 N − 6 leading to
$${\gamma}_P=\frac{E}{3 N-6}$$
(9)
The gamma index is a simple measure of the density of the network, but one can define a similar quantity by counting not the edges but the number of elementary cycles. The number of elementary cycle for a network is known as the cyclomatic number (see, e.g., Clark and Holton 1991) and is equal to
$$\Gamma = E-\mathrm{N}+1$$
(10)
For a planar graph this number is always less or equal to 2 N − 5 which leads naturally to the definition of the alpha index (also coined as meshedness in Buhl et al. 2006)
$$\alpha =\frac{E- N+1}{2 N-5}$$
(11)

This index belongs to [0,1] and is equal to 0 for a tree and equal to 1 for a maximal planar graph.

### Cell Area and Shape

For planar spatial networks, we have faces or cells which have a certain area and shape. In certain conditions, it can be interesting to characterize statistically these shapes, and various indicators were developed in this perspective (see Haggett and Chorley 1969 for a list of these indicators).

The first, simple important information is the distribution of the area P(A) which for many cases follows a power law (Lammer et al. 2006; Barthelemy and Flammini 2008):
$$P(A)\sim {A}^{-\tau}$$
(12)

where τ ≈ 2. We can note here that a simple argument on node density fluctuation leads indeed to this value τ = 2 and further empirical analysis is needed to test the universality of this result.

In addition to the area of the cell, its shape distribution is also interesting and contains a large part of the information about the structure of the network. A simple way to characterize the shape is given by the form factor ϕ. If we denote by L the major axis, the shape ratio is defined as A/L 2 (or equivalently, we can define the elongation ratio √A/L). In the paper (Lammer et al. 2006) on the road network structure, Lämmer et al. use another definition of the form factor and define it as
$$\varphi =\frac{4 A}{\pi {D}^2}$$
(13)

where π D N is the area of the circumscribed circle. If this ratio is small, the cell is very anisotropic, while on the contrary if ϕ is closer to one, the corresponding cell is almost circular. In many cases where rectangles and squares predominate (Lammer et al. 2006; Strano et al. 2012), we have ϕ ≈ 0:5–0:6.

## 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 d E (i, j) which can also be seen as the “as crow flies” distance. There is also the total “route” distance d R (i, j) from i to j by computing the sum of lengths of segments belonging to the shortest path between i and j. The detour index – also called the route factor – for this pair of nodes (i, j) is then given by (see Fig. 2 for an example)
$$Q\left( i, j\right)=\frac{d_R\left( i, j\right)}{d_E\left( i, j\right)}$$
(14)
This ratio is always larger than one, and the closer to one, the more efficient the network. From this quantity, we can derive another one for a single node defined by
$$\left\langle Q(i)\right\rangle =\frac{1}{N-1}\sum_j Q\left( i, j\right)$$
(15)
which measures the “accessibility” for this specific node i. Indeed the smaller it is, the easier it is to reach the node i. This quantity is related to the quantity called “straightness centrality” (Crucitti et al. 2006):
$${C}^S(i)=\frac{1}{N-1}\sum_{j\ne i}\frac{d_E\left( i, j\right)}{d_R\left( i, j\right)}$$
(16)
And if one is interested in assessing the global efficiency of the network, one can compute the average over all pairs of nodes:
$$\left\langle Q\right\rangle =\frac{1}{N\left( N-1\right)}\sum_{i\ne j} Q\left( i, j\right)$$
(17)
The average 〈Q〉 or the maximum Q max, and more generally the statistics of Q(i, j), is important and contains a lot of information about the spatial network under consideration (see Aldous and Shun 2010 for a discussion on this quantity for various networks). For example, one can define the interesting quantity Aldous and Shun (2010)
$$\rho (d)=\frac{1}{N_d}\sum_{ij/{d}_E\left( i, j\right)= d} Q\left( i, j\right)$$
(18)

(where N d is the number of nodes such that d E (i, j) = d) whose shape can help in characterizing combined spatial and topological properties.

## Cost and Efficiency

The minimum number of links to connect N nodes is E = N − 1 and the corresponding network is then a tree. We can also look for the tree which minimizes the total length given by the sum of the lengths of all links:
$${\ell}_T=\sum_{e\in E}{d}_E(e)$$
(19)
where d E (e) denotes the length of the link e. This procedure leads to the minimum spanning tree (MST) which has a total length $${\ell}_T^{\mathrm{MST}}$$ (see, e.g., Clark and Holton 1991). Obviously the tree is not a very efficient network (e.g., from the point of view of transportation), and usually more edges are added to the network, leading to an increase of accessibility but also of ℓ T. A natural measure of the “cost” of the network is then given by
$$C=\frac{\ell_T}{\ell_T^{\mathrm{MST}}}$$
(20)
We note here that we easily estimate the total length if the segment length distribution is peaked around its average ℓ1, and if the node distribution is uniform, ℓ1 ~ 1 = √ρ where ρ = N/A is the average node density (A is the area of the system). In this case, the total length is given by ℓ T = E1 leading to
$${\ell}_T=\frac{\left\langle k\right\rangle }{2}\sqrt{AN}$$
(21)
where 〈k〉 is the average degree of the graph. Adding links thus increases the cost but improves accessibility or the transport performance P of the network which can be measured as the minimum distance between all pairs of nodes, normalized by the same quantity computed for the minimum spanning tree:
$$P=\frac{\left\langle \ell \right\rangle }{\left\langle {\ell}_{\mathrm{MST}}\right\rangle }$$
(22)
Another measure of efficiency was also proposed in Latora and Marchiori (2001) and is defined as
$$E=\frac{1}{N\left( N-1\right)}\sum_{i\ne j}\frac{1}{\ell \left( i, j\right)}$$
(23)

where (i, j) is the shortest path distance from i to j. 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, e.g., Buhl et al. 2006).

Finally, adding links improves the resilience of the network to attacks or dysfunctions. A way to quantify this is by using fault tolerance (FT) (see, e.g., Tero et al. 2010) measured as the probability of disconnecting parts of the network with the failure of a single link. The benefit/cost ratio could then be estimated by the quantity $$FT/{\ell}_T^{\mathrm{MST}}$$ which is a quantitative characterization of the trade-off between cost and efficiency (Tero et al. 2010).

## Future Directions

In this final section, we discuss briefly two directions for future research which seem very promising. Both directions come from the fact that ever more data are available, opening the path for new measures, new models, and new understanding of the formation and evolution of spatial networks.

### Measuring and Modeling the Time Evolution of Spatial Networks

Thanks to the efforts of GIS scientists (Batty 2005), we now have digitalized maps, combined with data from remote sensing, which allows for studying the time evolution of spatial networks such as roads and streets over long periods. Understanding the evolution of transportation networks (Xie and Levinson 2009) is important from a fundamental point of view but also sheds some light on the crucial problem of understanding the time evolution of a city. Recent studies (Xie and Levinson 2009; Strano et al. 2012; Barthelemy et al. 2013) started to quantify the evolution of spatial networks, and more empirical results are certainly to come (Fig. 3).

At the time this article is written, we are still in the process of collecting data, processing them, and extracting stylized facts. The next important step will be the modeling of the evolution of these systems. There are already some simplified models, but we will now be able to confront theoretical models with stylized fact and hopefully converge to simple realistic models of spatial network evolution. In particular, all these studies will have to address the issue of self-organization versus centralized planning for different time scales, a crucial problem in the modeling of urban systems.

### Connecting Spatial Networks with Socioeconomical Indicators

Revealing the relationships of network topology to socioeconomical features is not a new project. There is indeed a wealth of papers in quantitative geography of the 1960s–1970s (see, e.g., Haggett and Chorley 1969; Radke 1977 and references therein). In 1969, for example, (Kissling 1969) concludes that the analysis of the network structure is “likely to reveal probable growth points in the system.” However, the recent availability of spatial data on networks and on socioeconomical indicators reinvigorates this direction of research. This can even be done at various scales. At large scales, for example, one can try to understand the relation between population, activity densities, and the structure of transportation networks. At a smaller scale, one can try to understand crime rates and activity density fluctuation in terms of topological properties of the transportation network.

This problem will also require a lot of efforts from the modeling side. In particular, we know that there is strong coupling between the population density and the network structure, but we still need a modeling framework for describing such a coupling and coevolution. From a longer time scale perspective, these studies on spatial networks belong to the more general problem of understanding the time evolution of a city. So far, modeling a city has mostly been done in the field of spatial economics (Fujita et al. 1999). However most of these studies consider monocentric structures and static properties, and their predictions are not compared with empirical data. Gathering various data, proposing simple dynamical models integrating the most relevant economical ingredients, and confronting their prediction to data will certainly lead in some future to a wealth of new and original results about this very complex system that is a city.

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