Any fully graphic region of degree sequences can be sampled rapidly

TL;DR

This paper proves that certain regions of degree sequences are rapidly sampleable using Markov chains if all sequences are graphic, and explores the inverse relation between stability conditions and the proportion of graphic sequences.

Contribution

It establishes the $3n^{13}$-stability of degree sequence regions with all sequences graphic and links stability conditions to the prevalence of graphic sequences.

Findings

Degree sequence regions are $3n^{13}$-stable if all sequences are graphic.

Rapid mixing of switch Markov chains on these regions.

Majority of sequences are graphic if stability conditions hold.

Abstract

Let $n>c_1\ge c_2$ and $\Sigma$ be positive integers with $n\cdot c_1\ge \Sigma \ge n\cdot c_2.$ Let $\mD=\dds{n}{\Sigma}{c_1}{c_2}$ denote the set of all degree sequences of length $n$ with the even sum $\Sigma$ and satisfying $c_1\ge d_i\ge c_2.$ We show that if all degree sequences in $\mD$ are graphic, then $\mD$ is $3n^{13}$-stable. (The concept of $P$-stability was introduced by Jerrum and Sinclair in 1990.) In particular, this implies that the switch Markov-chain mixes rapidly on all such degree sequences. In this paper we also study the inverse direction. We show the following: if all graphic sequences of a degree sequence region satisfy the $p(n)$-stability condition then the overwhelming majority of the sequences in the region is graphic. This answers affirmatively a question raised in the paper \DOI{10.1016/j.aam.2024.102805}.