Functional Central Limit Theorem for the principal eigenvalue of dynamic Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper establishes a functional central limit theorem for the principal eigenvalue of a dynamic Erdős-Rényi graph, showing it reflects the edge dynamics as the number of vertices increases.

Contribution

It introduces a functional CLT for the principal eigenvalue in a dynamic Erdős-Rényi graph, linking eigenvalue fluctuations to edge dynamics.

Findings

Principal eigenvalue follows a Gaussian process in the limit.

Eigenvalue dynamics mirror individual edge behavior.

Results hold as the number of vertices tends to infinity.

Abstract

In this paper we consider a dynamic version of the Erd\H{o}s-R\'{e}nyi random graph, in which edges independently appear and disappear in time, with the on- and off times being exponentially distributed. The focus lies on the evolution of the principle eigenvalue of the adjacency matrix in the regime that the number of vertices grows large. The main result is a functional central limit theorem, which displays that the principal eigenvalue essentially inherits the characteristics of the dynamics of the individual edges.