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

TL;DR

This paper proves a functional central limit theorem for the joint evolution of subgraph counts in dynamic Erdős-Rényi graphs, showing convergence to a Gaussian process under mild conditions.

Contribution

It establishes the first functional CLT for subgraph counts in dynamic Erdős-Rényi graphs, extending previous finite-dimensional results to a joint process.

Findings

Joint subgraph count processes converge to a Gaussian process

Results hold under mild Lipschitz-type conditions on edge processes

Provides a theoretical foundation for analyzing dynamic random graphs

Abstract

In this paper we consider a dynamic Erd\H{o}s-R\'{e}nyi random graph with independent identically distributed edge processes. Our aim is to describe the joint evolution of the entries of a subgraph count vector. The main result of this paper is a functional central limit theorem: we establish, under an appropriate centering and scaling, the joint functional convergence of the vector of subgraph counts to a specific multidimensional Gaussian process. The result holds under mild assumptions on the edge processes, most notably a Lipschitz-type condition.