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此词条由城市科学读书会词条梳理志愿者（Yukiweng）翻译审校，未经专家审核，带来阅读不便，请见谅。

[[文件:Random geometric graph.svg|链接=link=Special:FilePath/Random_geometric_graph.svg|替代=|缩略图|201x201像素|随机几何图，最简单的空间网络模型之一。]]

[[文件:Random geometric graph.svg|链接=link=Special:FilePath/Random_geometric_graph.svg|缩略图|201x201像素|随机几何图，最简单的空间网络模型之一。]]

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="Bart3">{{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="Bart22">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>M. Barthelemy, "Morphogenesis of Spatial Networks", Springer (2018).<nowiki></ref></nowiki> 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="Bart3" /><ref name="Bart22" /> 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="Bart3" /><ref name="Bart22" /> 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。

==Characterizing spatial networks 空间网络的表征==

==Characterizing spatial networks 空间网络的表征==

The following aspects are some of the characteristics to examine a spatial network:<ref name="Bart" />

The following aspects are some of the characteristics to examine a spatial network:<ref name="Bart" />

*Planar networks

* Planar networks

In many applications, such as railways, roads, and other transportation networks, the network is assumed to be [[planar graph|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.

In many applications, such as railways, roads, and other transportation networks, the network is assumed to be [[planar graph|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.

以下几个方面是可用于检验空间网络的一些特征：<ref name="Bart" />

以下几个方面是可用于检验空间网络的一些特征：<ref name="Bart" />

* 平面网络

*平面网络

在许多应用场景中，例如铁路、公路和其他运输网络，网络被假定为平面的。[[平面网络]]是空间网络中的一个重要分类，但并非所有空间网络都是平面的。例如航空客运网络即是一个非平面网络的例子：世界上许多大型机场都是通过直飞航班连接起来的。

在许多应用场景中，例如铁路、公路和其他运输网络，网络被假定为平面的。[[平面网络]]是空间网络中的一个重要分类，但并非所有空间网络都是平面的。例如航空客运网络即是一个非平面网络的例子：世界上许多大型机场都是通过直飞航班连接起来的。

*The way it is embedded in space

*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.

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.

* 网络与空间相关的方式

*网络与空间相关的方式

有一些网络的例子，它们似乎没有“直接”与空间联系起来。例如，社交网络通过朋友关系将个人联系起来，但在这种情况下，两个人之间的联系概率通常随着他们之间的距离而减小。

有一些网络的例子，它们似乎没有“直接”与空间联系起来。例如，社交网络通过朋友关系将个人联系起来，但在这种情况下，两个人之间的联系概率通常随着他们之间的距离而减小。

*Voronoi tessellation

* 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.

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.

* 沃罗诺伊镶嵌 Voronoi tessellation

* 沃罗诺伊镶嵌 Voronoi tessellation

*Mixing space and topology

*Mixing space and topology

Examining the topology of the nodes and edges itself is another way to characterize networks.  The distribution of [[Degree (graph theory)|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]].

Examining the topology of the nodes and edges itself is another way to characterize networks.  The distribution of [[Degree (graph theory)|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]].

* 混合空间和拓扑

*混合空间和拓扑

检验节点和连边本身的拓扑是表征网络的另一种方法。通常我们会关注节点的度分布，而对于边的结构，找到[[最小生成树]]  Minimum spanning tree或泛化、[[斯坦纳树]] Steiner tree和[[相对邻域图]] relative neighborhood graph是有用的。

检验节点和连边本身的拓扑是表征网络的另一种方法。通常我们会关注节点的度分布，而对于边的结构，找到[[最小生成树]]  Minimum spanning tree或泛化、[[斯坦纳树]] Steiner tree和[[相对邻域图]] relative neighborhood graph是有用的。

==Probability and spatial networks 概率和空间网络==

== Probability and spatial networks 概率和空间网络==

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:

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:

*The [[Poisson line process]]

* The [[Poisson line process]]

*Stochastic geometry: the [[Erdős–Rényi model|Erdős–Rényi graph]]

*Stochastic geometry: the [[Erdős–Rényi model|Erdős–Rényi graph]]

*[[Percolation theory]]

*[[Percolation theory]]

在“真实”世界中，网络的许多内容是不确定的，随机性起着重要作用。例如，社交网络中代表友谊的新连边在某种程度上是随机的。因此，通常我们会通过一些随机操作来对空间网络进行建模。在很多情况下，[[空间泊松过程]] spatial Poisson process被用于近似生成空间网络过程的数据集。其他常用的随机操作包括：

在“真实”世界中，网络的许多内容是不确定的，随机性起着重要作用。例如，社交网络中代表友谊的新连边在某种程度上是随机的。因此，通常我们会通过一些随机操作来对空间网络进行建模。在很多情况下，[[空间泊松过程]] spatial Poisson process被用于近似生成空间网络过程的数据集。其他常用的随机操作包括：

* [[泊松线过程]] Poisson line process

*[[泊松线过程]] Poisson line process

* 随机几何：[[Erdős–Rényi 图|Erdős–Rényi 随机图]]模型

*随机几何：[[Erdős–Rényi 图|Erdős–Rényi 随机图]]模型

* [[渗流理论]] Percolation theory

* [[渗流理论]] Percolation theory

==Approach from the theory of space syntax 从空间句法理论入手==

==Approach from the theory of space syntax 从空间句法理论入手==