Spatial Networks
Keywords
Planar Graph Degree Distribution Power Grid Betweenness Centrality Transportation NetworkSynonyms
Glossary
 Graph

(or Network) A set of vertices connected by edges
 Adjacency Matrix

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 realworld 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 2D such that none of its edges are crossing
 Organic Ratio

Measures the proportion of degree 1 (“dead ends”) and degree 3 nodes (“Tshaped 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 twodimensional 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 urbanism 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 realworld 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 twodimensional 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 (LibenNowell 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, socioprofessional 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 threedimensional 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 wellconnected node  that is, a hub. The topological 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
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 ℓ(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 }.
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 longrange links are important.
Organic Ratio
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
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.
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
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).
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.
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
(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
where ℓ(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 tradeoff 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
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 selforganization 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.
CrossReferences
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