# Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs

## Article Type

Research Article

## Publication Title

Journal of Theoretical Probability

## Abstract

We consider an inhomogeneous Erdős-Rényi random graph GN with vertex set [N] = { 1 , ⋯ , N} for which the pair of vertices i, j∈ [N] , i≠ j, is connected by an edge with probability r(iN,jN), independently of other pairs of vertices. Here, r:[0,1]2→(0,1) is a symmetric function that plays the role of a reference graphon. Let λN be the maximal eigenvalue of the adjacency matrix of GN. It is known that λN/ N satisfies a large deviation principle as N→ ∞. The associated rate function ψr is given by a variational formula that involves the rate function Ir of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of ψr, specially when the reference graphon is of rank 1.

## First Page

2413

## Last Page

2441

## DOI

10.1007/s10959-021-01138-w

## Publication Date

12-1-2022

## Recommended Citation

Chakrabarty, Arijit; Hazra, Rajat Subhra; Hollander, Frank den; and Sfragara, Matteo, "Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs" (2022). *Journal Articles*. 2883.

https://digitalcommons.isical.ac.in/journal-articles/2883

## Comments

Open Access, Green