Edge flipping in the complete graph

Butler, Steve; Chung, Fan; Cummings, J. R.; Graham, Ron

doi:10.1016/j.aam.2015.06.002

Cited by 6 publications

(10 citation statements)

References 3 publications

(4 reference statements)

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“…This clearly holds for m = 0 when ≥ 0. Suppose next that it is valid for m = r − 1, and let us verify it for m = r. Via Lemma 3.1 and (18), we have Finally, we note that the following recursive formula for the stationary distribution of K m,n can be obtained by the inductive method in Theorem 2 of [BCCG15], which the authors used to identify the stationary distribution of the edge flipping on K n .…”

Section: Stationary Distribution For Bipartite Graphsmentioning

confidence: 94%

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Mixing time bounds for edge flipping on regular graphs

Demirci¹,

Işlak²,

Özdemir³

2022

Preprint

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This paper studies two related Markov chains on a given graph, edge flipping and its dual, vertex flipping. The edge flipping is defined by Chung and Graham in [CG12]. Its eigenvalues and stationary distributions for some cases are identified in the same paper. We further study its spectral properties to show a lower bound for the rate of convergence in the case of regular graphs. In particular, we show a cutoff of order n 4 log n for the edge flipping on the complete graph. Besides, some recursive formulas and explicit expressions are obtained for stationary distributions of both processes on bipartite graphs.

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Section: Stationary Distribution For Bipartite Graphsmentioning

confidence: 94%

“…Stationary distributions for some graph classes are already studied in [CG12,BCCG15], but various other important classes remain to be considered. We also study the stationary behavior of a closely related process called vertex flipping, which is defined in the same paper [CG12].…”

Section: Introductionmentioning

confidence: 99%

Mixing time bounds for edge flipping on regular graphs

Demirci¹,

Işlak²,

Özdemir³

2022

Preprint

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“…This generating function describes the stationary distributions of a red/blue color-swapping algorithm on the complete graphs K r , where the coefficient of x r y s is proportional to the probability that s of the vertices of K r are blue in the stationary distribution, rescaled by a factor proportional to 2r−2 r . For more details, see [BCCG15]. Because each y term is attached to an x of equal or greater power, the power series expansion of F will have no terms where the power of y is larger than the power of x.…”

Section: Complete Graph Coloringmentioning

confidence: 99%

Asymptotics of bivariate analytic functions with algebraic singularities

Greenwood

2018

Journal of Combinatorial Theory, Series A

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In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Then, we apply these results to a generating function encoding information about the stationary distributions of a graph coloring algorithm studied by Butler, Chung, Cummings, and Graham (2015). Historically, Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996Hwang ( , 1998

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“…This process was implicitly used in a paper of Butler et al [2] for the complete graph, and explicitly introduced in a paper by Berikkyzy et al [1] where some basic properties were established and the probabilities for complete bipartite graphs were determined. We summarize these results here.…”

Section: Introductionmentioning

confidence: 99%

Graphs with at most two trees in a forest-building process

Butler

Hamanaka

Hardt

2019

Involve

Self Cite

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Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with k trees.We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are non-isomorphic but produce the same probabilities.

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Edge flipping in the complete graph

Cited by 6 publications

References 3 publications

Mixing time bounds for edge flipping on regular graphs

Mixing time bounds for edge flipping on regular graphs

Asymptotics of bivariate analytic functions with algebraic singularities

Graphs with at most two trees in a forest-building process

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