943:, 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. 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
27:
60:
908:
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
907:
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.
814:
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
863:
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
806:. 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.
899:. 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
770:
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.
831:
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.
873:
576:
786:
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.
738:(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
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An urban spatial network can be constructed by abstracting intersections as nodes and streets as links, which is referred to as a
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where space patterns are often more convexly articulated, however both convex and axial maps may be used in either situation.
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827:, which is a way of dividing space into a number of regions. The dual graph for a Voronoi diagram corresponds to the
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Examining the topology of the nodes and edges itself is another way to characterize networks. The distribution of
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In many applications, such as railways, roads, and other transportation networks, the network is assumed to be
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is used to approximate data sets of processes on spatial networks. Other stochastic aspects of interest are:
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912:, where the system generally comprises linear segments, whereas convex maps are more often used to analyse
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Hillier B, Hanson J, 1984, The social logic of space (Cambridge
University Press, Cambridge, UK).
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of the nodes is often considered, regarding the structure of edges it is useful to find the
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1216:. Washington, DC: Mathematical Association of America. pp. 174–194. Archived from
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While networks and graphs were already for a long time the subject of many studies in
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Currently, there is a move within the space syntax community to integrate better with
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947:, to connect probabilities and stochastic processes with networks in the real world.
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The following aspects are some of the characteristics to examine a spatial network:
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are all examples where the underlying space is relevant and where the graph's
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objects, i.e., the nodes are located in a space equipped with a certain
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A random geometric graph, one of the simplest models of spatial network
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162:
1086:
M. Barthelemy, "Morphogenesis of
Spatial Networks", Springer (2018).
59:
1194:. Contemporary Mathematics, no. 342, American Mathematical Society.
913:
767:
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they produce interlinks with commercially available GIS systems.
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Another definition of spatial network derives from the theory of
730:. The simplest mathematical realization of spatial network is a
1200:
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858:
789:
1226:
1142:Bandelt, Hans-JĂĽrgen; Chepoi, Victor (2008).
742:is smaller than a given neighborhood radius.
677:
1214:Geometry at Work: Papers in Applied Geometry
1144:"Metric graph theory and geometry: a survey"
1210:"Bridges between geometry and graph theory"
1034:Barthelemy, M. (2011). "Spatial Networks".
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823:A spatial network can be represented by a
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891:Approach from the theory of space syntax
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1192:Towards a Theory of Geometric Graphs
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744:Transportation and mobility networks
16:Network representing spatial objects
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939:, physics, mathematical sociology,
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962:Spatial network analysis software
859:Probability and spatial networks
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1125:from the original on 2014-01-10
810:The way it is embedded in space
790:Characterizing spatial networks
1108:. Edward Arnold, London, 1969.
921:geographic information systems
1:
1106:Network analysis in geography
1104:P. Haggett and R.J. Chorley.
1066:10.1016/j.physrep.2010.11.002
1017:
847:, or the generalization, the
1153:. Contemporary Mathematics.
7:
1255:Application-specific graphs
950:
853:relative neighborhood graph
774:
760:social and contact networks
10:
1276:
1212:. In Gorini, C. A. (ed.).
957:Hyperbolic geometric graph
930:
764:biological neural networks
18:
878:Stochastic geometry: the
835:Mixing space and topology
537:Exponential random (ERGM)
204:Informational (computing)
1250:Environmental psychology
997:Topological graph theory
224:Scientific collaboration
19:Not to be confused with
1012:Interdependent networks
923:(GIS), and much of the
866:spatial Poisson process
653:Category:Network theory
173:Preferential attachment
1235:Geometric graph theory
1190:; et al. (2004).
1163:10.1090/conm/453/08795
829:Delaunay triangulation
781:transportation network
736:random geometric graph
542:Random geometric (RGG)
31:
21:Spatial neural network
987:Modularity (networks)
845:Minimum spanning tree
752:mobile phone networks
658:Category:Graph theory
29:
1245:Environmental design
1240:Architectural theory
874:Poisson line process
819:Voronoi tessellation
1058:2011PhR...499....1B
1002:Small-world network
462:Degree distribution
113:Community structure
1119:"Spatial Networks"
982:Percolation theory
885:Percolation theory
740:Euclidean distance
646:Network scientists
572:Soft configuration
32:
967:Cascading failure
880:Erdős–Rényi graph
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522:Bianconi–Barabási
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234:Artificial neural
209:Telecommunication
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1175:. Archived from
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722:associated with
720:spatial elements
700:(sometimes also
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557:Stochastic block
547:Hyperbolic (HGN)
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972:Complex network
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825:Voronoi diagram
798:Planar networks
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703:geometric graph
698:spatial network
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593:Boolean network
567:Maximum entropy
517:Barabási–Albert
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128:Controllability
93:Complex network
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1042:(1–3): 1–101.
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910:urban networks
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1260:Graph theory
1218:the original
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1177:the original
1154:
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1127:. Retrieved
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897:space syntax
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862:
849:Steiner tree
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719:
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552:Hierarchical
507:Random graph
355: /
344: /
326:Neighborhood
248:
168:Transitivity
148:Optimization
1188:Pach, János
937:mathematics
901:axial lines
756:power grids
598:agent based
512:Erdős–Rényi
153:Reciprocity
118:Percolation
103:Small-world
1229:Categories
1129:2014-01-10
1018:References
945:Statistics
625:Categories
482:Efficiency
477:Modularity
457:Clustering
442:Centrality
430:Algorithms
254:Dependency
229:Biological
108:Scale-free
1157:: 49–86.
1049:1010.0302
724:geometric
374:Bipartite
296:Component
214:Transport
163:Homophily
123:Evolution
98:Contagion
1208:(2000).
1123:Archived
951:See also
925:software
851:and the
775:Examples
768:topology
748:Internet
712:vertices
641:Software
603:Epidemic
585:Dynamics
499:Topology
472:Distance
409:Weighted
384:Directed
379:Complete
283:Features
244:Semantic
39:a series
37:Part of
1074:4627021
1054:Bibcode
931:History
732:lattice
706:) is a
425:Metrics
394:Labeled
264:on-Chip
249:Spatial
158:Closure
1169:
1072:
841:degree
804:planar
728:metric
636:Topics
490:Models
447:Degree
404:Random
357:matrix
346:matrix
336:Vertex
291:Clique
273:Graphs
219:Social
76:Theory
1180:(PDF)
1147:(PDF)
1070:S2CID
1044:arXiv
734:or a
716:edges
708:graph
622:Lists
452:Motif
399:Multi
389:Hyper
366:Types
306:Cycle
88:Graph
1167:ISBN
903:and
872:The
864:the
762:and
718:are
331:Path
321:Loop
316:Edge
259:Flow
1159:doi
1155:453
1062:doi
1040:499
714:or
607:SIR
301:Cut
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
