Asymptotic Properties of a Random Graph with Duplications

We introduce a growing network model-the copying model-in which a new node attaches to a randomly selected target node and, in addition, independently to each of the neighbors of the target with copying probability p. When p < 1 2 , this algorithm generates sparse networks, in which the average node degree is finite. A power-law degree distribution also arises, with a non-universal exponent whose value is determined by a transcendental equation in p. In the sparse regime, the network is "normal", e.g., the relative fluctuations in the number of links are asymptotically negligible. For p ≥ 1 2 , the emergent networks are dense (the average degree increases with the number of nodes N ) and they exhibit intriguing structural behaviors. In particular, the N -dependence of the number of m-cliques (complete subgraphs of m nodes) undergoes m − 1 transitions from normal to progressively more anomalous behavior at a m-dependent critical values of p. Different realizations of the network, which start from the same initial state, exhibit macroscopic fluctuations in the thermodynamic limit-absence of self averaging. When linking to second neighbors of the target node can occur, the number of links asymptotically grows as N 2 as N → ∞, so that the network is effectively complete as N → ∞.

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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%

Densification and structural transitions in networks that grow by node copying

et al. 2016

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“…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%

“…If the last ball of a certain colour is recoloured, then this colour will never appear again in the urn. It is equivalent to the following random graph model studied in [4,5,17]. Let G 0 be the graph with a single isolated vertex.…”

Section: Introductionmentioning

confidence: 99%

“…Nn , where U j,n is the number of colours at time n with j balls, and N n is the number of balls at time n. This was done for p = 1/2 in [4] and for the remaining cases in [17]. Both exact and asymptotic results were found.…”

Section: Introductionmentioning

confidence: 99%

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The dominating colour of an infinite Pólya urn model

Thörnblad

2016

J. Appl. Probab.

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ABSTRACT. We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n ≥ 1, choose a ball from the urn uniformly at random. With probability 1/2 < p < 1, return the ball to the urn along with another ball of the same colour. 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. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1 − p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p = 1).

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Preferential Attachment Random Graphs with Edge-Step Functions

2019

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In this work we investigate a preferential attachment model whose parameter is a function f : N → [0, 1] that drives the asymptotic proportion between the numbers of vertices and edges of the graph. We investigate topological features of the graphs, proving general bounds for the diameter and the clique number. Our results regarding the diameter are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation −γ. For this particular class, we prove a characterization for the diameter that depends only on −γ. More specifically, we prove that the diameter of such graphs is of order 1/γ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained. The almost sure convergence for the properly normalized logarithm of the clique number of the graphs generated by slowly varying functions is also proved.

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Asymptotic Properties of a Random Graph with Duplications

Cited by 7 publications

References 24 publications

Densification and structural transitions in networks that grow by node copying

Densification and structural transitions in networks that grow by node copying

The dominating colour of an infinite Pólya urn model

Preferential Attachment Random Graphs with Edge-Step Functions

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