RANDOM NETWORKS WITH PREFERENTIAL GROWTH AND VERTEX DEATH

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 {p(k)} is derived and analyzed for a number of specific choices of b and d. When b(i) = i + alpha and d(i) = beta, that is, linear preferential attachment for the newborn and random deaths, then P-k similar to k(-(2+alpha)). When b(i) = i + 1 and d(i) = beta(i + 1), with beta < 1, then p(k) similar to (1 + beta)(-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) = beta(i + 1)(gamma), with beta, gamma < 1, then log p(k) similar to -k(gamma), 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.