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 E N and a fitness parameter f E [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 fA_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.