Upper Tail For Homomorphism Counts In Constrained Sparse Random Graphs

TL;DR

This paper derives explicit sharp upper tail decay rates for homomorphism counts in various sparse random graph models, extending known results to regular, uniform, inhomogeneous graphs, and multiple graph counts.

Contribution

It provides the first explicit sharp upper tail bounds for homomorphism counts in regular, uniform, and inhomogeneous sparse random graphs, including joint tail probabilities.

Findings

Established sharp upper tail decay rates for regular sparse graphs.

Extended upper tail results to multiple homomorphism counts.

Bounded upper tail probabilities in inhomogeneous graph models like stochastic block models.

Abstract

Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+\delta$. Going beyond the Erd\H{o}s-R\'enyi model, we establish here explicit, sharp upper tail decay rates for sparse random $d_n$-regular graphs (provided $H$ has a regular $2$-core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs $H_1,\ldots, H_k$ (extending the known results for $k=1$), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erd\H{o}s-R\'enyi graphs.