Large deviations of empirical neighborhood distribution in sparse random graphs

TL;DR

This paper investigates the probabilities of large deviations from typical local structures in sparse Erdős-Rényi and other random graphs, using entropy-based rate functions and introducing new models for sampling and analysis.

Contribution

It introduces a new configuration model for sampling graphs with prescribed neighborhood distributions and generalizes Galton-Watson trees for analyzing unimodular random trees.

Findings

Large deviations are characterized by an entropy-based rate function supported on trees.

The new configuration model enables sampling graphs with specific neighborhood distributions.

Generalized Galton-Watson trees are useful for analyzing unimodular random trees.

Abstract

Consider the Erd\H{o}s-Renyi random graph on n vertices where each edge is present independently with probability c/n, with c>0 fixed. For large n, a typical random graph locally behaves like a Galton-Watson tree with Poisson offspring distribution with mean c. Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right.