Triangle processes on graphs with given degree sequence

TL;DR

This paper introduces a modified Markov chain for generating graphs with a given degree sequence that favors higher triangle counts, analyzes its mixing properties, and studies triangle distributions under different probability measures.

Contribution

It develops a new Metropolis-based switch chain that biases towards graphs with more triangles and proves rapid mixing under certain conditions.

Findings

The modified chain is rapidly mixing if the standard chain is.

The triangle-restricted switch chain is irreducible for degrees ≥ 3.

Imposing a triangle count cutoff does not affect chain behavior significantly.

Abstract

The switch chain is a well-studied Markov chain which generates random graphs with a given degree sequence and has uniform stationary distribution. Motivated by the high number of triangles seen in some real-world networks, we study a variant of the switch chain which is more likely to produce graphs with higher numbers of triangles. Specifically, we apply a Metropolis scheme designed to have the following stationary distribution: graph $G$ has probability proportional to $\lambda^{\min\{t(G),\nu\}}$, where $t(G)$ is the number of triangles in $G$ and $\nu$ is a cut-off value introduced to moderate the impact of graphs with a very high number of triangles. We assume that the "activity" $\lambda$ satisfies $\lambda\geq 1$, and call the resulting chain the modified Metropolis switch chain. We prove that the modified Metropolis switch chain is rapidly mixing whenever the (standard) switch chain is rapidly mixing, provided that the activity and maximum degree are not too large. The triangle switch (or "$\triangle$-switch") chain is a restriction of the switch chain which only performs switches that change the set of triangles in the graph. We prove that the $\triangle$-switch chain is irreducible for any degree sequence with minimum degree at least 3, and prove a rapid mixing result for the modified Metropolis $\triangle$-switch chain. Finally, we investigate the distribution of triangles in random graphs with given degrees, under both the uniform distribution and the distribution in which graph $G$ has probability proportional to $\lambda^{t(G)}$. Our analysis implies that the imposition of the cut-off $\nu$ does not significantly impact the behaviour of these modified Metropolis chains over polynomially many steps