Random Networks with Preferential Growth and Vertex Death

Abstract

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 pk ∼ k-(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.