2015
DOI: 10.1017/s0021900200012523
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Asymptotic Properties of a Random Graph with Duplications

Abstract: 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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Cited by 7 publications
(16 citation statements)
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“…For instance, multiplying (19) by k(k + 1) and summing over all k ≥ 0 gives, after some straightforward steps,…”
Section: A Sparse Regimementioning
confidence: 99%
“…For instance, multiplying (19) by k(k + 1) and summing over all k ≥ 0 gives, after some straightforward steps,…”
Section: A Sparse Regimementioning
confidence: 99%
“…With probability 1 − p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri [4,5] and Thörnblad [17]. We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls.…”
mentioning
confidence: 83%
“…Let us mention a few known results about the urn model, coming from [4,5,17]. These results were originally proved in the random graph version, but transfer immediately to the urn version.…”
Section: Introductionmentioning
confidence: 99%
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