Journal of Applied Probability | VOL. 59
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On local weak limit and subgraph counts for sparse random graphs

Publication Date Jul 21, 2022

Abstract

AbstractWe use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics, such as the clustering coefficient and assortativity, is determined by the local weak limit.As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton–Watson branching process.

Concepts

Sparse Random Graphs Sparse Graphs Subgraph Counts Local Limit Clustering Coefficient Models Of Random Graphs Random Graphs Sequences Of Graphs Global Moments Behaviour Of Statistics

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