A spatial network is a network of spatial elements. In physical space (which typically includes urban or building space) spatial networks are derived from maps of open space within the urban context or building. One might think of the 'space map' as being the negative image of the standard map, with the open space cut out of the background buildings or walls. The space map is then broken into units; most simply, these might be road segments. The road segments (the nodes of the graph) can be linked into a network via their intersections (the edges of a graph). A common instance of a spatial network, the transportation network analysis, reverses this and treats the road segments as edges and the street intersections as nodes in the graph.
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. For most practical applications, the space is the two-dimensional space and the metric is the usual Euclidean distance. This definition implies in general that the probability of finding a link between two nodes will decrease with the distance. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, 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 a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks. 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.
Characterizing spatial networks
The following aspects are some of the characteristics to examine a spatial network:
- Planar networks
In many applications, such as rail, roads, and other transportation networks the network is assumed to be planar. Planar networks build up an important group out of the spatial networks, but not all spatial networks are planar. Indeed, the airline passenger networks is an non-planar example: All airports in the world are connected through direct flights.
- The way it is embedded in space
There are examples of networks, which seem to be not "directly" embedded in space. Social networks for instance connect individuals through friendship relations. But in this case, space intervenes in the fact that the connection probability between two individuals usually decreases with the distance between them.
- Voronoi tessellation
A spatial network can be represented by a Voronoi diagram, which is a way of dividing space into a number of regions. The dual graph for a Voronoi diagram corresponds to the Delaunay triangulation for the same set of points. Voronoi tessellations are interesting for spatial networks in the sense that they provide a natural representation model to which one can compare a real world network.
- Mixing space and topology
Examining the topology of the nodes and edges itself is another way to characterize networks. The distribution of degree of the nodes is often considered, regarding the structure of edges it is useful to find the Minimum spanning tree, or the generalization, the Steiner tree and the relative neighborhood graph
Probability and spatial networks
In the "real" world many aspects of networks are not deterministic - randomness plays an important role. For example new links, representing friendships, in social networks are in a certain manner random. Modelling spatial networks in respect of stochastic operations is consequent. In many cases the spatial Poisson process used to approximate data sets of processes on spacial networks. Other stochastic aspects of interest are:
Approach from the theory of space syntax
Another definition of spatial network derives from the theory of space syntax. It can be notoriously difficult to decide what a spatial element should be in complex spaces involving large open areas or many interconnected paths. The originators of space syntax, Bill Hillier and Julienne Hanson use axial lines and convex spaces as the spatial elements. Loosely, an axial line is the 'longest line of sight and access' through open space, and a convex space the 'maximal convex polygon' that can be drawn in open space. Each of these elements is defined by the geometry of the local boundary in different regions of the space map. Decomposition of a space map into a complete set of intersecting axial lines or overlapping convex spaces produces the axial map or overlapping convex map respectively. Algorithmic definitions of these maps exist, and this allows the mapping from an arbitrary shaped space map to a network amenable to graph mathematics to be carried out in a relatively well defined manner. Axial maps are used to analyse urban networks, where the system generally comprises linear segments, whereas convex maps are more often used to analyse building plans where space patterns are often more convexly articulated, however both convex and axial maps may be used in either situation.
Currently, there is a move within the space syntax community to integrate better with geographic information systems (GIS), and much of the software they produce interlinks with commercially available GIS systems.
While networks and graphs were already for a long time the subject of many studies in mathematics, mathematical sociology, computer science, spatial networks have been studied intensively during the 1970s in quantitative geography. Objects of studies in geography are inter alia locations, activities and flows of individuals, but also networks evolving in time and space. Most of the important problems such as the location of nodes of a network, the evolution of transportation networks and their interaction with population and activity density are addressed in these earlier studies. On the other side, many important points still remain unclear, partly because at that time datasets of large networks and larger computer capabilities were lacking. Recently, spatial networks have been the subject of studies in Statistics, to connect probabilities and stochastic processes with networks in the real world.
- Hillier B, Hanson J, 1984, The social logic of space (Cambridge University Press, Cambridge, UK).
- M. Barthelemy, "Spatial Networks", Physics Reports 499:1-101 (2011) ( http://arxiv.org/abs/1010.0302 ).
- P. Haggett and R.J. Chorley. Network analysis in geog- raphy. Edward Arnold, London, 1969.