Preferential duplication graphs

Cohen, Netta; Jordan, Jonathan; Voliotis, Margaritis

doi:10.1239/jap/1276784910

Cited by 6 publications

(3 citation statements)

References 11 publications

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“…Polynomial decay is stated for the limit of the mean of degree distribution. We also mention the somewhat different model of Jordan [12], and the duplication model of Cohen et al [7], where the duplicated vertex is chosen not uniformly, but with probabilities proportional to the actual degrees.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Properties of a Random Graph with Duplications

Backhausz

Móri

2015

J. Appl. Probab.

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We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number c d > 0 almost surely as the number of steps goes to ∞, and c d ∼ (eπ) 1/2 d 1/4 e −2 √ d holds as d → ∞.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Properties of a Random Graph with Duplications

Backhausz

Móri

2015

J. Appl. Probab.

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“…The model differs from duplication graphs, for example those considered in [3,5], in that in those models only one vertex, chosen at random, duplicates at any one time step, whereas in the models considered here and in [2,10,11,12] all vertices simultaneously duplicate.…”

Section: Introductionmentioning

confidence: 99%

Randomised Reproducing Graphs

Jordan

2011

Electron. J. Probab.

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We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, α, β and γ, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If (1 + α)(1 + γ) < 1 then the degree distribution converges to a stationary distribution, which in most cases has an approximately power law tail with an index which depends on α and γ. If (1 + α)(1 + γ) > 1 then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree d tends to zero. We also give some results on the number of edges and on the spectral gap.

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“…This mechanism was formulated into a network growth model by [11]. Since then, many variations and extensions have been proposed and studied [12][13][14][15][16][17][18]. We refer to this family of network growth models as duplication and divergence (DD) models.…”

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Degree distribution of large networks generated by the partial duplication model

Choi

2013

Theoretical Computer Science

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In this paper, we present a rigorous analysis on the limiting behavior of the degree distribution of the partial duplication model, a random network growth model in the duplication and divergence family that is popular in the study of biological networks. We show that for each non-negative integer k, the expected proportion of nodes of degree k approaches a limit as the network becomes large. This fills in a gap in previous studies. In addition, we prove that p=1/2, where p is the selection probability of the model, is the phase transition for the expected proportion of isolated nodes converging to 1, and hence answer a question raised in Bebek et al. [G. Bebek, P. Berenbrink, C. Cooper, T. Friedetzky, J. Nadeau, S.C. Sahinalp, The degree distribution of the generalized duplication model, Theoret. Comput. Sci. 369 (2006) 239–249]. We also obtain asymptotic bounds on the convergence rates of degree distribution. Since the observed networks typically do not contain isolated nodes, we study the subgraph consisting of all non-isolated nodes contained in the networks generated by the partial duplication model, and show that p=1/2 is again a phase transition for the limiting behavior of its degree distribution

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Preferential duplication graphs

Cited by 6 publications

References 11 publications

Asymptotic Properties of a Random Graph with Duplications

Asymptotic Properties of a Random Graph with Duplications

Randomised Reproducing Graphs

Degree distribution of large networks generated by the partial duplication model

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