Any fully graphic region of degree sequences can be sampled rapidly

TL;DR

The paper studies when fast, uniform sampling of graphs with a fixed degree sequence is guaranteed by $P$-stability. It develops a general framework connecting $P$-stability to the fully graphic property of simple degree-sequence regions, using witness trails, hinge-flip operations, and hostile configurations to structure proofs. The authors prove a $3\cdot n^{13}$-bound on the perturbation map for fully graphic simple regions, establishing $P$-stability for their union, while also showing that not-fully-graphic regions are typically not $P$-stable under mild assumptions; they further provide a concrete application and discuss extensions to bipartite and directed settings. Overall, the work both broadens the classes of degree sequences known to yield polynomial-time mixing for the switch Markov chain and clarifies the relationship between full graphicity and $P$-stability, guiding future investigations into broader graph families.

Abstract

Let $n>c_1\ge c_2$ and $Σ$ be positive integers with $n\cdot c_1\ge Σ\ge n\cdot c_2.$ Let $\mD=\dds{n}Σ{c_1}{c_2}$ denote the set of all degree sequences of length $n$ with the even sum $Σ$ 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}.

Figures (7)

• Figure 1: The thick dashed lines indicate no edge at all, the solid thick lines indicate all edges.
• Figure 2: The dashed lines indicate non-edges, the solid lines indicate edges.
• Figure 3: We provide the complete witness trail.
• Figure 4: In $R_N$ there is no edge. The vertices $x_j, z_j\in R_0$ and $y_j\in R_N$. The curved, loosely dotted arrow indicates a hinge-flip operation. As earlier: the thick dashed lines indicate no edge at all, the solid thick lines indicate all edges.
• Figure 5: In $Y\cup R_\infty$ there is no edge. The solid lines are existing edges, the dashed line are missing edges The thick lines note all edges / non-edges in the subgraph.

Theorems & Definitions (56)

• Theorem 1.1: fully
• Theorem 1.2: fully
• Theorem 1.3
• Corollary 1.4
• Lemma 1.5
• Theorem 1.6
• Remark 1.7
• Definition 2.1
• Proposition 2.2
• proof