Spatial network

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A random geometric graph, one of the simplest models of spatial network. Random geometric graph.svg
A random geometric graph, one of the simplest models of spatial network.

A spatial network (sometimes also geometric graph ) is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e., the nodes are located in a space equipped with a certain metric. [1] [2] The simplest mathematical realization of spatial network is a lattice or a random geometric graph (see figure in the right), where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and biological neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.

Contents

Examples

An urban spatial network can be constructed by abstracting intersections as nodes and streets as links, which is referred to as a transportation network.

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. [3]

Characterizing spatial networks

The following aspects are some of the characteristics to examine a spatial network: [1]

In many applications, such as railways, 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 a non-planar example: Many large airports in the world are connected through direct flights.

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.

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.

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 "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 is used to approximate data sets of processes on spatial 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.

History

While networks and graphs were already for a long time the subject of many studies in mathematics, physics, mathematical sociology, computer science, spatial networks have been also 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. [4] 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. [5]

See also

Related Research Articles

<span class="mw-page-title-main">Voronoi diagram</span> Type of plane partition

In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane. For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation.

<span class="mw-page-title-main">Tessellation</span> Tiling of a plane in mathematics

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.

<span class="mw-page-title-main">Random graph</span> Graph generated by a random process

In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.

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Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Point processes can be used for spatial data analysis, which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others.

Spatial network analysis software packages are analytic software used to prepare graph-based analysis of spatial networks. They stem from research fields in transportation, architecture, and urban planning. The earliest examples of such software include the work of Garrison (1962), Kansky (1963), Levin (1964), Harary (1969), Rittel (1967), Tabor (1970) and others in the 1960s and 70s. Specific packages address to suit their domain-specific needs, including TransCAD for transportation, GIS for planning and geography, and Axman for Space syntax researchers.

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<span class="mw-page-title-main">Unit disk graph</span> Intersection graph of unit disks in the plane

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<span class="mw-page-title-main">Stochastic geometry</span>

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Edgar Nelson Gilbert was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories whose accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Elliott model of bursty errors in signal transmission, and the Erdős–Rényi model for random graphs.

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<span class="mw-page-title-main">Poisson point process</span> Type of random mathematical object

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Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell has exactly the same number of nearest, next nearest, nearest of next nearest etc. neighbors and hence they are called regular lattice. Often physicists and mathematicians study phenomena which require disordered lattice where each cell do not have exactly the same number of neighbors rather the number of neighbors can vary wildly. For instance, if one wants to study the spread of disease, viruses, rumors etc. then the last thing one would look for is the square lattice. In such cases a disordered lattice is necessary. One way of constructing a disordered lattice is by doing the following.

References

  1. 1 2 Barthelemy, M. (2011). "Spatial Networks". Physics Reports. 499 (1–3): 1–101. arXiv: 1010.0302 . Bibcode:2011PhR...499....1B. doi:10.1016/j.physrep.2010.11.002. S2CID   4627021.
  2. M. Barthelemy, "Morphogenesis of Spatial Networks", Springer (2018).
  3. Hillier B, Hanson J, 1984, The social logic of space (Cambridge University Press, Cambridge, UK).
  4. P. Haggett and R.J. Chorley. Network analysis in geog- raphy. Edward Arnold, London, 1969.
  5. "Spatial Networks". Archived from the original on 2014-01-10. Retrieved 2014-01-10.