具有无标度分布的长边几何随机图中的化学距离
Chemical distance in geometric random graphs with long edges and scale-free degree distribution
摘要：
本文研究了定义在$d$维空间中泊松过程的点上的几何随机图，这些点上还带有独立的随机标记。边是利用端点的标记和点之间的距离灵活地随机建立的。我们的框架包括软布尔模型（其中标记扮演以顶点为中心的球的半径的角色），空间优先连接的版本（其中标记扮演出生时间的角色），以及一系列具有无标度度分布和跨越大距离的边缘的其他图模型。在这个通用的框架中，我们给出了尖锐的标准，没有ultrasmallness的图，并在ultrasmall政权建立一个极限定理的化学距离的两个点。除了在平均场无标度网络模型中，超小区域的边界不仅取决于度分布的幂律指数，而且还取决于图的空间嵌入，由连接典型点的边缘的概率的衰减率量化。
英文摘要：
We study geometric random graphs defined on the points of a Poisson process in $d$-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centred in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.
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