Global Clustering Coefficient in Scale-Free Weighted and Unweighted Networks

Q3 Mathematics
Liudmila Ostroumova Prokhorenkova
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引用次数: 3

Abstract

Abstract In this article, we present a detailed analysis of the global clustering coefficient in scale-free graphs. Many observed real-world networks of diverse nature have a power-law degree distribution. Moreover, the observed degree distribution usually has an infinite variance. Therefore, we are especially interested in such degree distributions. In addition, we analyze the clustering coefficient for both weighted and unweighted graphs. There are two well-known definitions of the clustering coefficient of a graph: the global and the average local clustering coefficients. There are several models proposed in the literature for which the average local clustering coefficient tends to a positive constant as a graph grows. However, there are no models of scale-free networks with an infinite variance of the degree distribution and with an asymptotically constant global clustering coefficient. Models with constant global clustering and finite variance were also proposed. Therefore, in this work we focus only on the most interesting case: we analyze the global clustering coefficient for graphs with an infinite variance of the degree distribution. For unweighted graphs, we prove that the global clustering coefficient tends to zero with high probability and we also estimate the largest possible clustering coefficient for such graphs. On the contrary, for weighted graphs, the constant global clustering coefficient can be obtained even for the case of an infinite variance of the degree distribution.
无标度加权和非加权网络的全局聚类系数
摘要本文对无标度图的全局聚类系数进行了详细的分析。许多观察到的具有不同性质的现实世界网络具有幂律度分布。而且,观测到的度分布通常具有无限大的方差。因此,我们对这种度分布特别感兴趣。此外,我们还分析了加权图和未加权图的聚类系数。图的聚类系数有两种众所周知的定义:全局聚类系数和平均局部聚类系数。文献中提出了几种模型,随着图的增长,平均局部聚类系数趋于正常数。然而,没有一个无标度网络的模型,其度分布的方差是无穷大的,全局聚类系数是渐近常数的。还提出了具有恒定全局聚类和有限方差的模型。因此,在这项工作中,我们只关注最有趣的情况:我们分析了具有无限方差度分布的图的全局聚类系数。对于未加权图,我们证明了全局聚类系数大概率趋于零,并估计了这类图的最大可能聚类系数。相反,对于加权图,即使在度分布方差无穷大的情况下,也能得到恒定的全局聚类系数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Internet Mathematics
Internet Mathematics Mathematics-Applied Mathematics
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