Efficiently sampling the realizations of bounded, irregular degree sequences of bipartite and directed graphs

Abstract

Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. All of these results assume some kind of regularity in the corresponding degree sequences. Recently, Greenhill and Sfragara published a breakthrough paper about irregular normal and directed degree sequences for which rapid mixing of the swap Markov chain is proved. In this paper we present two groups of results. An example from the first group is the following theorem: let be a directed degree sequence on n vertices. Denote by Δ the maximum value among all in- and out-degrees and denote by the number of edges in the realization. Assume furthermore that . Then the swap Markov chain on the realizations of is rapidly mixing. This result is a slight improvement on one of the results of Greenhill and Sfragara. An example from the second group is the following: let d be a bipartite degree sequence on the vertex set U ⊎ V, and let 0 < c1 ≤ c2 < |U| and 0 < d1 ≤ d2 < |V| be integers, where c1 ≤ d(v) ≤ c2: ∀v ∈ V and d1 ≤ d(u) ≤ d2: ∀u ∈ U. Furthermore assume that (c2 − c1 − 1)(d2 − d1 − 1) < max{c1(|V| − d2), d1(|U| − c2)}. Then the swap Markov chain on the realizations of d is rapidly mixing. A straightforward application of this latter result shows that when a random bipartite or directed graph is generated under the Erdős—Rényi G(n, p) model with mild assumptions on n and p then the degree sequence of the generated graph has, with high probability, a rapidly mixing swap Markov chain on its realizations.

Generating graphs randomly

“…Erdős et al [42] gave new conditions on bipartite and directed degree sequences which guarantee rapid mixing of the augmented switch chain. In particular, suppose that a bipartite degree sequence has degrees s = (s 1 , .…”

“…, t b ) in the other, where a + b = n. Let s max , s min , t max , t min be the maximum and minimum degrees on each side. If all degrees are positive and (s max − s min − 1)(t max − t min − 1) ≤ max{s min (a − t max ), t min (b − s max )} then the augmented switch chain on the set of bipartite graphs with bipartite degree sequence (s, t) is rapidly mixing [42,Theorem 3]. They applied this result to the analysis of the bipartite Erdős-Rényi model G(a, b, p), with a vertices in one side of the bipartition, b vertices on the other and each possible edge between the two parts is included with probability p. Erdős et al [42,Corollary 13] proved that if p is not too close to 0 or 1 then the augmented switch chain is rapidly mixing for the degree sequence arising from G(a, b, p), with high probability as n → ∞, where n = a + b.…”

The mixing time of the switch Markov chains: a unified approach

“…Power-law densitybound, γ > 2.5 [18] (∆ − δ + 1) 2 ≤ (∆ − δ) 2 ≤ similar to bipartite case [10,11] [ 10,11] proof in [1] (or Corollary 18 in [1])…”

“…Bipartite Erds-Rnyi [10,11] similar to bipartite case p, 1 − p ≥ 4 2 log n n [10,11] strongly stable degree sequence classes [1] Table 1: Some classes of degree sequences for which the switch Markov chain is rapidly mixing.…”

Half-Graphs, Other Non-stable Degree Sequences, and the Switch Markov Chain

“…There is a long line of results where the rapid mixing of the switch Markov chain is proven for certain degree sequences, see [2,19,13,6,7,12]. Some of these results were unified, first by Amanatidis and Kleer [1], who established rapid mixing for so-called strongly stable classes of degree sequences of unconstrained and bipartite graphs (definition given in Section 7).…”