Infinitely exchangeable random graphs generated from a Poisson point process on monotone sets and applications to cluster analysis for networks

TL;DR

This paper introduces a novel infinitely exchangeable process for generating random graphs from a Poisson point process on monotone sets, with applications in network cluster analysis and Bayesian inference.

Contribution

It constructs a new exchangeable graph model based on Poisson processes on monotone sets, linking set systems to graph structures for statistical applications.

Findings

Defines a Poisson-based process on monotone set collections

Establishes a mapping from sets to graphs inducing exchangeability

Suggests applications in clustering, machine learning, and Bayesian methods

Abstract

We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers $\mathbb{N}$ via a Poisson point process with mean measure $\Lambda$ on the power set of $\mathbb{N}$. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, and every monotone subset maps to an undirected graph $G\in\catg$, the space of undirected graphs with vertex set $\mathbb{N}$. We show a natural mapping $\cate\rightarrow\catf\rightarrow\catg$ which induces an infinitely exchangeable measure on the projective system $\catg^{\rest}$ of graphs $\catg$ under permutation and restriction mappings given an infinitely exchangeable family of measures on the projective system $\cate^{\rest}$ of subsets with permutation and restriction maps. We show potential connections of this process to applications in cluster analysis, machine learning, classification and Bayesian inference.