Abstract

In a generalized random graph with random vertex weights, we investigate the asymptotic behaviors for two crucial empirical measures: The empirical pair measure, which represents the number of edges connecting each pair of weights, and the empirical neighborhood measure, which interprets the number of vertices of a given weight connected to a given number of vertices of each weight. By some mixing approaches, we obtain the large deviation principles for these empirical measures in the weak topology. Through these large deviation results, the large deviation principle for the number of edges in a generalized random graph is obtained, as well as the large deviation principle for the empirical degree distribution.

摘要

在具有随机顶点权重的广义随机图中，我们研究了两个关键经验度量的渐近行为：经验对度量，表示连接每对权重的边的数量，以及经验邻域度量，它解释顶点的数量连接到每个权重的给定数量的顶点的给定权重。通过一些混合方法，我们获得了弱拓扑中这些经验测量的大偏差原则。通过这些大偏差结果，得到了广义随机图中边数的大偏差原理，以及经验度分布的大偏差原理。