An enumeration approach to network evolution

Abstract

A simple theoretical model of network evolution is discussed here. In each step, we add a new vertex to the graph and it is allowed to connect it to maximum degree vertices (hubs) only. Given a constant p, the probability of such a connection is p for any hub. The initial (non-random) graph G 1 is arbitrary but here we investigate mostly the case when G 1 has one vertex.We solve here some particular cases of the problem, using enumeration methods. We obtain not limit theorems but exact results for the parameters discussed.