Normal approximation for generalized U-statistics and weighted random graphs

TL;DR

This paper develops bounds for normal approximation of weighted U-statistics and applies them to analyze the distribution of weighted subgraph counts in Erdős-Rényi random graphs, extending existing graph counting results.

Contribution

It introduces a general stochastic framework for normal approximation of weighted U-statistics and applies it to weighted graph substructure counts, broadening previous unweighted analyses.

Findings

Derived Wasserstein distance bounds for weighted U-statistics

Extended graph counting results to weighted subgraphs

Provided a new analytical framework for functionals of independent variables

Abstract

We derive normal approximation bounds in the Wasserstein distance for sums of weighted U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to normal approximation for the combined weights of subgraphs in the Erd\H{o}s-R\'enyi random graph, extending the graph counting results of [1] to the setting of graph weighting. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences.