Sampling hypergraphs with given degrees

Dyer, Martin; Greenhill, Catherine; Kleer, Pieter; Ross, James; Stougie, Leen

doi:10.1016/j.disc.2021.112566

Cited by 18 publications

(9 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We call the t-th operation, flip t , significant if {a t , b t } ∈ E t−1 (S, S). 8 Observe that nonsignificant operations do not change the size of cut {S, S}. Let t i denote the time of the ith significant operation for i ≥ 1, and let t 0 = 0.…”

Section: Analysis Of a Single Cutmentioning

confidence: 99%

“…Both the switch-chain and the flip-chain have been studied extensively, for various families of graphs. Here, we focus on results for d-regular (undirected) graphs; for results on graphs with general degree sequences and directed graphs see [14,10,15,2,8,9,11,27]. A restriction of the switch-chain on regular bipartite graphs was first analyzed in [19], using a canonical path argument.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Expanders via local edge flips in quasilinear time

Giakkoupis

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

Mahlmann and Schindelhaue [24] proposed the following simple process, called flip-chain, for transforming any given connected d-regular graph into a d-regular expander: In each step, a random 3-path abcd is selected, and edges ab and cd are replaced by two new edges ac and bd, provided that ac and bd do not exist already. A motivation for the study of the flip-chain arises in the design of overlay networks, where it is common practice that adjacent nodes periodically exchange random neighbors, to maintain good connectivity properties. It is known that the flipchain converges to the uniform distribution over connected d-regular graphs, and it is conjectured that an expander graph is obtained after O(nd log n) steps, w.h.p., where n is the number of vertices. However, the best known upper bound on the number of steps is, and the best bound on the mixing time of the chain is O(n 16 d 36 log n) [11,6].We provide a new analysis of a natural flip-chain instantiation, which shows that starting from any connected d-regular graph, for d = Ω(log 2 n), an expander is obtained after O(nd log 2 n) steps, w.h.p. This result is tight within logarithmic factors, and almost matches the conjectured bound. Moreover, it justifies the use of edge flip operations in practice: for any d-regular graph with d = poly(log n), an expander is reached after each vertex participates in at most poly(log n) operations, w.h.p. Our analysis is arguably more elementary than previous approaches. It uses the novel notion of the strain of a cut, a value that depends both on the crossing edges and their adjacent edges. By keeping track of the cut strains, we form a recursive argument that bounds the time before all sets of a given size have large expansion, after all smaller sets have already attained large expansion.

show abstract

Section: Analysis Of a Single Cutmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Expanders via local edge flips in quasilinear time

Giakkoupis

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

show abstract

“…, m. It follows that the modified log-Sobolev constant of this chain satisfies ρ N ≥ 1/n. A simple Markov chain comparison argument (as described in Appendix A.2), in combination with (18), then yields that the modified log-Solev constant ρ of the Markov chain in which we use the original quantities |S(α)|, instead of the approximation φ(α), satisfies ρ ≥ 1 2n . Of course, the algorithmic problem is now that we cannot compute the quantities w α = |S(α)| exactly in polynomial time, so one step of this base-exchange Markov chain cannot be implemented efficiently.…”

Section: Sampling From the Gibbs Distributionmentioning

confidence: 99%

“…That is, formally speaking, we define it to be −∞ outside of L(k, u)28 For an overview of algorithms that can be used to (approximately) sample a bipartite graph with a given degree sequence, see, e.g.,[18].…”

mentioning

confidence: 99%

Sampling from the Gibbs Distribution in Congestion Games

Kleer¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Logit dynamics is a form of randomized game dynamics where players have a bias towards strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converges to the so-called Gibbs distribution over the set of all strategy profiles, when interpreted as a Markov chain. In general, logit dynamics might converge slowly to the Gibbs distribution, but beyond that, not much is known about their algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: i) Is there an efficient algorithm for sampling from the Gibbs distribution? ii) If yes, do there also exist natural randomized dynamics that converges quickly to the Gibbs distribution?We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics (in which we in addition are also allowed to randomly interchange the strategies of two players) that converges quickly to the Gibbs distribution in such games. This answers both questions above affirmatively. We also address the first question for the class of so-called capacitated k-uniform congestion games.To prove our results, we rely on the recent breakthrough work of Anari, Liu, Oveis-Gharan and Vinzant (2019) concerning the approximate sampling of the base of a matroid according to strongly log-concave probability distribution.

show abstract

“…Dyer et al . [ 24 ] introduced a rejection sampling method that randomly and uniformly samples hypergraphs with a prescribed degree sequence. They need a strong condition on the degree sequence to ensure that the rejection sampling be efficient.…”

Section: Introductionmentioning

confidence: 99%

Constructing and sampling partite, 3-uniform hypergraphs with given degree sequence

Hubai,

Mezei,

Béres

et al. 2024

PLoS ONE

View full text Add to dashboard Cite

Partite, 3-uniform hypergraphs are 3-uniform hypergraphs in which each hyperedge contains exactly one point from each of the 3 disjoint vertex classes. We consider the degree sequence problem of partite, 3-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is k or k − 1 for some fixed k, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, 3-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for χ2 testing of contingency tables. We have shown that this hypergraph-based χ2 test is more sensitive than the standard χ2 test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.

show abstract

Sampling hypergraphs with given degrees

Cited by 18 publications

References 26 publications

Expanders via local edge flips in quasilinear time

Expanders via local edge flips in quasilinear time

Sampling from the Gibbs Distribution in Congestion Games

Constructing and sampling partite, 3-uniform hypergraphs with given degree sequence

Contact Info

Product

Resources

About