An MCMC sampling of densest $k$-sub-graphs of regular graphs with connected complement graph

TL;DR

This paper introduces an MCMC sampling method for finding densest $k$-sub-graphs in regular graphs with connected complements, using an exclusion process interpreted as a Markov chain to analyze convergence.

Contribution

It presents a novel exclusion process-based MCMC approach with theoretical bounds on convergence speed tailored to the graph's geometry.

Findings

Quantitative bounds on Markov chain convergence depending on graph structure.

Algorithm leveraging particle view to reduce memory use.

Sharp bounds in the case of complete graphs.

Abstract

We present an exclusion process based approach for sampling densest $k$-sub-graphs from regular graphs $L$ with connected complement. By interpreting an exclusion process as a Markov chain on a corresponding Token Graph $\mathfrak{L}_k$, we make use of classical Markov chain theory to obtain quantitative bounds on the convergence speed depending on the geometry of $L$, which are sharp in the sens that variations in the valency lead to an equality, which is $1$, for the geometric bounds in the case of a complete graph. We propose an algorithm which makes use of this particle view to avoid excessive memory use due to the state space size and discuss the regularity and connectivity condition on $L$ and $L^c$ in the Outlook.