Large deviations for subgraphs in inhomogeneous random graphs

TL;DR

This paper investigates the probabilities of rare large deviations in subgraph counts within inhomogeneous random graphs, especially focusing on heavy-tailed degree distributions and the emergence of large hubs.

Contribution

It introduces a framework for analyzing large deviations of subgraph counts in inhomogeneous graphs, including sharp results for clique counts under sublinear expectations.

Findings

Derived sharp large deviation results for subgraph counts

Identified conditions for polynomially large deviations

Analyzed the impact of heavy-tailed degree distributions

Abstract

Inhomogeneous random graphs are fundamental models for real-world networks, where prescribed degrees are imposed as soft constraints. A common assumption in such models is that the degree distribution follows a power-law, capturing the heavy-tailed nature observed in many contexts. While various graph functionals have been studied in this setting, inhomogeneity makes their analysis significantly more challenging. The goal of this paper is to investigate the large deviations of subgraph counts in inhomogeneous random graphs. Rare events concerning these functionals translate into quantifying the probability that extremely large hubs appear in the graph. This can be achieved by defining a specific optimization problem that captures the most likely way to generate numerous additional subgraphs. When the expected number of subgraphs is sublinear in the graph size, polynomially large deviations are possible, and in this case, we can derive sharp results on clique counts.