Controlled Markovian dynamics of graphs: unbiased generation of random graphs with prescribed topological properties

TL;DR

This paper develops a method to generate random graphs with specific topological properties using controlled Markov chains based on edge switchings, ensuring unbiased sampling of graphs.

Contribution

It introduces exact formulas for graph mobility and a framework for controlled Markov chains that converge to desired graph distributions.

Findings

Exact formulas for graph mobility based on adjacency matrices

A controlled Markov chain framework for unbiased graph sampling

Ability to generate graphs with prescribed topological properties

Abstract

We analyze the properties of degree-preserving Markov chains based on elementary edge switchings in undirected and directed graphs. We give exact yet simple formulas for the mobility of a graph (the number of possible moves) in terms of its adjacency matrix. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process).