Switch chain mixing times through triangle counts

TL;DR

This paper introduces a new sampling method for uniform simple graphs with power-law degree distributions, analyzing the impact of small subgraphs like triangles on the mixing times of Markov Chain algorithms.

Contribution

It combines a constrained configuration model with a Markov Chain switching method and compares triangle counts in different graph models, providing new insights into their structural differences.

Findings

The proposed method effectively samples uniform simple graphs with specified degree distributions.

Simulations suggest the number of triangles in the erased configuration model exceeds that in uniform graphs for large sizes.

Heuristic analysis supports the conjecture about triangle count differences between models.

Abstract

Sampling uniform simple graphs with power-law degree distributions with degree exponent $\tau\in(2,3)$ is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large.