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Random networks with preferential growth and vertex death
Stockholm University, Faculty of Science, Department of Mathematics.
2010 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 47, no 4, p. 1150-1163Article in journal (Refereed) Published
##### Abstract [en]

A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function b of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function d of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {pk} is derived and analyzed for a number of specific choices of b and d. When b(i) = i + α and d(i) = β, that is, linear preferential attachment for the newborn and random deaths, then pkk-(2+α). When b(i) = i + 1 and d(i) = β(i + 1), with β < 1, then pk ∼ (1 + β)-k, that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, when b(i) = i + 1 and d(i) = β(i + 1)γ, with β, γ < 1, then logpk ∼ -kγ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.

##### Place, publisher, year, edition, pages
2010. Vol. 47, no 4, p. 1150-1163
##### Keywords [en]
Branching process, random network, preferential attachment, power-law distribution, degree distribution
Mathematics
##### Identifiers
OAI: oai:DiVA.org:su-55144DiVA, id: diva2:401466
Available from: 2011-03-02 Created: 2011-03-02 Last updated: 2017-12-11Bibliographically approved

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Deijfen, Maria
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Journal of Applied Probability
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