Scale-Free Graph with Preferential Attachment and Evolving Internal Vertex Structure

Abstract

We extend the classical Barabási-Albert preferential attachment procedure to graphs with internal vertex structure given by weights of vertices. In our model, weight dynamics depends on the current vertex degree distribution and the preferential attachment procedure takes into account both weights and degrees of vertices. We prove that such a coupled dynamics leads to scale-free graphs with exponents depending on parameters of the weight dynamics.