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A '''spatial network''' (sometimes also '''[[Geometric graph theory|geometric graph]]''') is a [[Graph (discrete mathematics)|graph]] in which the [[Vertex (graph theory)|vertices]] or [[Edge (graph theory)|edges]] are ''spatial elements'' associated with [[Geometry|geometric]] objects, i.e., the nodes are located in a space equipped with a certain [[Metric (mathematics)|metric]].<ref name="Bart">{{cite journal | last1 = Barthelemy | first1 = M. | year = 2011| title = Spatial Networks | arxiv = 1010.0302 | journal = Physics Reports | volume = 499 | issue = 1–3 | pages = 1–101 | doi=10.1016/j.physrep.2010.11.002 | bibcode = 2011PhR...499....1B| s2cid = 4627021 }}</ref><ref name="Bart2">M. Barthelemy, "Morphogenesis of Spatial Networks", Springer (2018).</ref> The simplest mathematical realization of spatial network is a [[Lattice graph|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. [[Transport network|Transportation and mobility networks]], [[Internet]], [[cellular network|mobile phone networks]], [[electrical grid|power grids]], [[social network|social and contact networks]] and [[neural network|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.
A '''spatial network''' (sometimes also '''[[Geometric graph theory|geometric graph]]''') is a [[Graph (discrete mathematics)|graph]] in which the [[Vertex (graph theory)|vertices]] or [[Edge (graph theory)|edges]] are ''spatial elements'' associated with [[Geometry|geometric]] objects, i.e., the nodes are located in a space equipped with a certain [[Metric (mathematics)|metric]].<ref name="Bart">{{cite journal | last1 = Barthelemy | first1 = M. | year = 2011| title = Spatial Networks | arxiv = 1010.0302 | journal = Physics Reports | volume = 499 | issue = 1–3 | pages = 1–101 | doi=10.1016/j.physrep.2010.11.002 | bibcode = 2011PhR...499....1B| s2cid = 4627021 }}</ref><ref name="Bart2">M. Barthelemy, "Morphogenesis of Spatial Networks", Springer (2018).</ref> The simplest mathematical realization of spatial network is a [[Lattice graph|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. [[Transport network|Transportation and mobility networks]], [[Internet]], [[cellular network|mobile phone networks]], [[electrical grid|power grids]], [[social network|social and contact networks]] and [[neural network|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.
'''空间网络 spatial network'''（也被称为'''[[几何图]] geometric graph'''）是一种图，其中顶点或边是与几何对象关联的空间元素，例如节点位于有特定度量的空间中，具有空间位置信息。<ref name="Bart" /><ref name="Bart2" /> 空间网络最简单的数学形式是[[晶格]]或[[随机几何图]]（见右图），其中节点随机均匀分布在二维平面上；如果一对节点之间[[欧氏距离]]小于给定的邻域半径，则将该对节点相连。[[交通和移动网络]]、[[互联网]]、[[移动电话网络]]、[[电网]]、[[社交网络]]以及[[生物神经网络]]都是图具有空间相关性的示例，并且这些图的拓扑性质本身并不包含关于网络的所有信息。表征和理解空间网络的结构、适应力和演化过程对于如城市化、流行病学等的不同领域都至关重要。
'''空间网络 spatial network'''（也被称为'''[[几何图]] geometric graph'''）是一种图，其中[[顶点 Vertex|顶点]]或[[连边]]是与几何对象关联的空间元素，例如节点位于有特定度量的空间中，具有空间位置信息。<ref name="Bart" /><ref name="Bart2" /> 空间网络最简单的数学形式是[[晶格]]或[[随机几何图]]（见右图），其中节点随机均匀分布在二维平面上；如果一对节点之间[[欧氏距离]]小于给定的邻域半径，则将该对节点相连。[[交通和移动网络]]、[[互联网]]、[[移动电话网络]]、[[电网]]、[[社交网络]]以及[[生物神经网络]]都是图具有空间相关性的示例，并且这些图的拓扑性质本身并不包含关于网络的所有信息。表征和理解空间网络的结构、适应力和演化过程对于如城市化、流行病学等的不同领域都至关重要。
==Examples 例子==
==Examples 例子==
An urban spatial network can be constructed by abstracting intersections as nodes and streets as links, which is referred to as a [[Transportation network (graph theory)|transportation network]].
An urban spatial network can be constructed by abstracting intersections as nodes and streets as links, which is referred to as a [[Transportation network (graph theory)|transportation network]].