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

Comments

Open Access, Green

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