Approximate Sampling of Graphs with Near-$P$-stable Degree Intervals

TL;DR

This paper extends the understanding of Markov chain mixing times for sampling graphs with degree intervals, demonstrating rapid mixing when intervals are centered at P-stable degree sequences, which broadens applicability in network analysis.

Contribution

It generalizes previous results by proving rapid mixing of the degree interval Markov chain for intervals centered at P-stable degree sequences, beyond thin interval cases.

Findings

Proves rapid mixing of the degree interval Markov chain for P-stable degree sequences.

Extends previous results to broader classes of degree intervals.

Enhances methods for uniform sampling of graph realizations in network analysis.

Abstract

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and M\"uller-Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well-studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently Amanatidis and Kleer published a beautiful paper (arXiv:2110.09068), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper we extend substantially their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centred at P-stable degree sequences.