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文件:Random geometric graph.svg

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.

空间网络 spatial network(也被称为几何图 geometric graph)是一种图,其中顶点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。[1][2] 空间网络最简单的数学形式是晶格随机几何图(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

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 空间网络的表征

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

  • Planar networks

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.


  • 平面网络


  • 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.

  • 网络与空间相关的方式


  • 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.

  • 沃罗诺伊镶嵌 Voronoi tessellation

空间网络可以用 Voronoi 图表示,这是一种将空间划分为多个区域的方法。 Voronoi 图的对偶图对应于同一组点的 Delaunay 三角剖分。 Voronoi 镶嵌之于空间网络很有趣,因为它提供了一种自然的表示模型,使得人们可以将其与现实世界的网络进行比较。

  • 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.

  • 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。

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:

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:

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

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.

空间网络的另一个定义源自空间句法 space syntax理论。在涉及大片开放区域或具有许多互连路径的复杂空间中,决定空间元素应该是什么是非常困难的。空间句法的发起人Bill Hillier和Julienne Hanson使用轴线和凸空间作为空间元素。简单地说,轴线是穿过开放空间“最长的视野和通道”,凸空间是在开放空间中可以绘制的“最大凸多边形”。这些元素中的每一个都是由空间地图不同区域中局部边界的几何形状定义的,将空间映射分解为一组完整的相交轴线,或重叠凸空间,将分别产生轴向映射或重叠凸映射。这些映射的算法定义是存在的,并且这允许以相对明确的方式执行从任意形状的空间映射到适合图形数学的网络的映射。轴图用于分析城市网络,其中系统通常由线性部分组成,而凸图更常用于分析建筑平面图, 其中空间模式通常更凸地铰接,但是凸图和轴图都可以在任何一种情况下使用。

目前,空间句法界正在努力与地理信息系统 (GIS) 更好地合作,他们生产的许多软件都与商用 GIS 系统相互链接。

History 历史

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]

网络和图长期以来一直是数学、物理学、数学社会学、计算机科学等许多研究的主题,而在 1970 年代,空间网络在定量地理学中也得到了深入研究。地理学研究的对象主要是个人的位置、活动和流动,但也包括随时间和空间演化的网络。[4]大多数重要的问题,如网络节点的位置、交通网络的演化及其与人口和活动密度的相互作用,都在这些早期的研究中得到解决。但另一方面,许多重要问题仍然不清楚,部分是由于当时缺乏大型网络的数据集和更强大的计算机能力。近期空间网络已成为统计学研究的主题,将概率和随机过程与现实世界中的网络联系起来。[5]


  1. 1.0 1.1 1.2 1.3 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. 2.0 2.1 M. Barthelemy, "Morphogenesis of Spatial Networks", Springer (2018).
  3. 3.0 3.1 Hillier B, Hanson J, 1984, The social logic of space (Cambridge University Press, Cambridge, UK).
  4. 4.0 4.1 P. Haggett and R.J. Chorley. Network analysis in geog- raphy. Edward Arnold, London, 1969.
  5. 5.0 5.1 "Spatial Networks". Archived from the original on 2014-01-10. Retrieved 2014-01-10.