On local weak limit and subgraph counts for sparse random graphs

TL;DR

This paper establishes a connection between local and global subgraph counts in sparse random graphs using Sidorenko's inequality, providing criteria to determine when local limits influence global graph statistics.

Contribution

It introduces an optimal criterion linking local weak limits to global subgraph counts and applies it to analyze models of sparse random intersection graphs.

Findings

Derived a relation between local and global subgraph counts

Identified conditions where local weak limits determine global graph statistics

Applied results to models with random clique tree limits

Abstract

We 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.