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Directed preferential attachment models: Limiting degree distributions and their tails

Part of: Special processes Graph theory

Published online by Cambridge University Press:  04 May 2020

Tom Britton
Tom Britton*
Affiliation:
Stockholm University
*
*Postal address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. Email address: tom.britton@math.su.se
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Abstract

The directed preferential attachment model is revisited. A new exact characterization of the limiting in- and out-degree distribution is given by two independent pure birth processes that are observed at a common exponentially distributed time T (thus creating dependence between in- and out-degree). The characterization gives an explicit form for the joint degree distribution, and this confirms previously derived tail probabilities for the two marginal degree distributions. The new characterization is also used to obtain an explicit expression for tail probabilities in which both degrees are large. A new generalized directed preferential attachment model is then defined and analyzed using similar methods. The two extensions, motivated by empirical evidence, are to allow double-directed (i.e. undirected) edges in the network, and to allow the probability of connecting an ingoing (outgoing) edge to a specified node to also depend on the out-degree (in-degree) of that node.

Keywords

Preferential attachmentdirected networkbirth processestail distribution.

MSC classification

Primary: 60K40: Other physical applications of random processes
Secondary: 05C80: Random graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 122 - 136
Copyright
© Applied Probability Trust 2020

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