Growth of preferential attachment random graphs via continuous-time branching processes

Athreya, Krishna B.; Ghosh, Arka P.; Sethuraman, Sunder

doi:10.1007/s12044-008-0036-2

Cited by 38 publications

(59 citation statements)

References 19 publications

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“…The deterministic model, that is, the model given above, is when P(γ 1 = 1) ≡ 1 and the directions on edges are not taken in account. We note that the randomized model is LDP for leaves in random trees 851 similar to the one in [3]. See also [16] for a more general randomized model, and also [19] and [32] for other random edge schemes.…”

Section: Applications To Random Graph Modelsmentioning

confidence: 98%

Large deviations for the leaves in some random trees

2009

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Section: Applications To Random Graph Modelsmentioning

confidence: 98%

Large deviations for the leaves in some random trees

2009

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“…, n ≥ 1} into a process constructed from pairs of SBI processes, as specified in Section 3. The embedding idea is proposed in [1] and has been used in [27] to model two different undirected linear preferential attachment models.…”

Section: Embedding Processmentioning

confidence: 99%

“…(·) jumps first . Denote the index of the (I, O)-pair that jumps at T 3 by L 3 and write P F (1) T 2 (·) := P(·|F (1)…”

Section: 1mentioning

confidence: 99%

“…with T being an exponential random variable with unit mean, independent of J, B (1) , B (2) , B (1) and B (2) .…”

Section: 1mentioning

confidence: 99%

“…This paper only considers the directed model where at each stage, a new node is born and either it points to one of the existing nodes or one of the existing nodes attaches to the new node. Results on the degree growth in the undirected case are investigated in [1,27].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Degree growth rates and index estimation in a directed preferential attachment model

Wang

Resnick

2020

Stochastic Processes and their Applications

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Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. In a directed preferential attachment model, despite the well-known marginal power-law degree distributions, not much investigation has been done on the joint behavior of the in-and out-degree growth. Also, statistical estimates of the marginal tail exponent of the power-law degree distribution often use the Hill estimator as one of the key summary statistics, even though no theoretical justification has been given. This paper focuses on convergence of the joint empirical measure for in-and out-degrees and proves the consistency of the Hill estimator. To do this, we first derive the asymptotic behavior of the joint degree sequences by embedding the in-and out-degrees of a fixed node into a pair of switched birth processes with immigration and then establish the convergence of the joint tail empirical measure. From these steps, the consistency of the Hill estimators is obtained.MSC Classes: 60G70, 60B10, 60G55, 60G57, 05C80, 62E20.

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A scaling limit for the degree distribution in sublinear preferential attachment schemes

Choi

Sethuraman

Venkataramani

2015

Random Struct. Alg.

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ABSTRACT:We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures.In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network.A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C 0 -semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

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Growth of preferential attachment random graphs via continuous-time branching processes

Abstract: Some growth asymptotics of a version of 'preferential attachment' random graphs are studied through an embedding into a continuous-time branching scheme. These results complement and extend previous work in the literature.

Cited by 38 publications

References 19 publications

Large deviations for the leaves in some random trees

Large deviations for the leaves in some random trees

Degree growth rates and index estimation in a directed preferential attachment model

A scaling limit for the degree distribution in sublinear preferential attachment schemes

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