A measure-on-graph-valued diffusion: a particle system with collisions, and their applications

TL;DR

This paper introduces a graph-valued diffusion process modeled by stochastic differential equations, explores its stationary states via dual Markov chains, and applies it to graph algorithms and Bayesian model selection.

Contribution

It develops a novel measure-on-graph diffusion framework, analyzes its stationary states, and demonstrates applications in graph algorithms and Bayesian inference.

Findings

Support of extremal stationary states forms independent sets

Diffusion with linear drift relates to a generalized Dirichlet distribution

Applications include graph independent set algorithms and Bayesian graph selection

Abstract

A diffusion taking value in probability measures on a graph with a vertex set $V$, $\sum_{i\in V}x_i\delta_i$, is studied. The masses on each vertices satisfy the stochastic differential equation of the form $dx_i=\sum_{j\in N(i)}\sqrt{x_ix_j}dB_{ij}$ on the simplex, where $\{B_{ij}\}$ are independent standard Brownian motions with skew symmetry and $N(i)$ is the neighbour of the vertex $i$. A dual Markov chain on integer partitions to the Markov semigroup associated with the diffusion is used to show that the support of an extremal stationary state of the adjoint semigroup is an independent set of the graph. We also investigate the diffusion with a linear drift, which gives a killing of the dual Markov chain on a finite integer lattice. The Markov chain is used to study the unique stationary state of the diffusion, which generalizes the Dirichlet distribution. Two applications of the diffusions are discussed: analysis of an algorithm to find an independent set of a graph, and a Bayesian graph selection based on computation of probability of a sample by using coupling from the past.