Product Measure Approximation of Symmetric Graph Properties

TL;DR

This paper develops tools to approximate complex graph properties under uniform measures with simpler product measures, bridging the gap between realistic models and tractable analysis in random graph theory.

Contribution

It introduces conditions under which the uniform measure on graph sets can be approximated by product measures, enhancing analysis of random structures without assuming independence.

Findings

Provided a framework for measure approximation in graph properties

Established sufficient conditions for measure approximation

Applied results to various graph property sets

Abstract

In the study of random structures we often face a trade-off between realism and tractability, the latter typically enabled by assuming some form of independence. In this work we initiate an effort to bridge this gap by developing tools that allow us to work with independence without assuming it. Let $\mathcal{G}_{n}$ be the set of all graphs on $n$ vertices and let $S$ be an arbitrary subset of $\mathcal{G}_{n}$, e.g., the set of graphs with $m$ edges. The study of random networks can be seen as the study of properties that are true for most elements of $S$, i.e., that are true with high probability for a uniformly random element of $S$. With this in mind, we pursue the following question: What are general sufficient conditions for the uniform measure on a set of graphs $S \subseteq \mathcal{G}_{n}$ to be approximable by a product measure?