A Central Limit Theorem for Functions on Weighted Sparse Inhomogeneous Random Graphs

TL;DR

This paper establishes a central limit theorem for a class of functions on sparse inhomogeneous random graphs with weights, extending previous results on Erdős-Rényi graphs.

Contribution

It introduces a new CLT for weighted sparse inhomogeneous graphs using Stein's method and local graph analysis, broadening the scope of prior work.

Findings

Proves a CLT for functions on weighted sparse inhomogeneous graphs.

Extends previous CLT results from Erdős-Rényi graphs to inhomogeneous models.

Uses perturbative Stein's method and local structure analysis.

Abstract

We prove a central limit theorem for a certain class of functions on sparse rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and vertex weights. Our proof of the central limit theorem uses a perturbative form of Stein's method and relies on a careful analysis of the local structure of the underlying sparse inhomogeneous random graphs (as the number of vertices in the graph tends to infinity), which may be of independent interest, as well as a local approximation property of the function, which is satisfied for a number of combinatorial optimisation problems. These results extend recent work by Cao (2021) for Erd\H{o}s-R\'enyi random graphs and additional i.i.d. weights only on the edges.