Generalized Becker–Döring equations modeling the time evolution of a process of preferential attachment with fitness

Abstract

We introduce an infinite system of equations modeling the time evolution of the growth process of a network. The nodes are characterized by their degree k ∈ ℕ and a fitness parameter f ∈ [0, h]. Every new node which emerges becomes a fitness f′ according to a given distribution P and attaches to an existing node with fitness f and degree k at rate f A_k, where A_k are positive coefficients, growing sublinearly in k. If the parameter f takes only one value, the dynamics of this process can be described by a variant of the Becker–Döring equations, where the growth of the size of clusters of size k occurs only with increment 1. In contrast to the established Becker–Döring equations, the system considered here is nonconservative, since mass (i.e. links) is continuously added. Nevertheless, it has the property of linearity, which is a natural consequence of the process which is being modeled. The purpose of this paper is to construct a solution of the system based on a stochastic approximation algorithm, which allows also a numerical simulation in order to get insight into its qualitative behaviour. In particular we show analytically and numerically the property of Bose–Einstein condensation, which was observed in the literature on random graphs.