Random Graphs by Product Random Measures

TL;DR

This paper introduces a comprehensive framework for representing and analyzing random graphs using product random measures, capturing various graph types and properties through measure transformations and test functions.

Contribution

It develops a general mathematical framework for weighted random graphs via product random measures, including properties and potential applications.

Findings

Mean measures encode edge and vertex densities.

Mean edge weight relates to spectral and Sobol representations.

Framework applies to directed, undirected, labeled, and unlabeled graphs.

Abstract

A natural representation of random graphs is the random measure. The collection of product random measures, their transformations, and non-negative test functions forms a general representation of the collection of non-negative weighted random graphs, directed or undirected, labeled or unlabeled, where (i) the composition of the test function and transformation is a non-negative edge weight function, (ii) the mean measures encode edge density/weight and vertex degree density/weight, and (iii) the mean edge weight, when square-integrable, encodes generalized spectral and Sobol representations. We develop a number of properties of these random graphs, and we give simple examples of some of their possible applications.