The large deviation principle for inhomogeneous Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper extends the large deviation principle for inhomogeneous Erdős-Rényi graphs by relaxing conditions on the underlying graphon and providing a more tractable rate function, with applications to eigenvalue analysis.

Contribution

It generalizes previous LDP results for ERRGs by weakening assumptions on the graphon and identifying a simpler rate function, also applying to eigenvalue large deviations.

Findings

Relaxed conditions on the reference graphon to $ ext{log } r, ext{log }(1-r) ext{ in } L^1$.

Proved the rate function equals a more tractable form under these conditions.

Applied results to large deviations of the largest eigenvalue, weakening prior assumptions.

Abstract

Consider the inhomogeneous Erd\H{o}s-R\'enyi random graph (ERRG) on $n$ vertices for which each pair $i,j\in\{1,\ldots,n\}$, $i\neq j$ is connected independently by an edge with probability $r_n(\frac{i-1}{n},\frac{j-1}{n})$, where $(r_n)_{n\in\mathbb{N}}$ is a sequence of graphons converging to a reference graphon $r$. As a generalization of the celebrated LDP for ERRGs by Chatterjee and Varadhan (2010), Dhara and Sen (2019) proved a large deviation principle (LDP) for a sequence of such graphs under the assumption that $r$ is bounded away from 0 and 1, and with a rate function in the form of a lower semi-continuous envelope. We further extend the results by Dhara and Sen. We relax the conditions on the reference graphon to $\log r,\log(1-r)\in L^1([0,1]^2)$. We also show that, under this condition, their rate function equals a different, more tractable rate function. We then apply these results to the large deviation principle for the largest eigenvalue of inhomogeneous ERRGs and weaken the conditions for part of the analysis of the rate function by Chakrabarty, Hazra, Den Hollander and Sfragara (2020).