Formula omitted]-variable first-order logic of preferential attachment random graphs

Abstract

We study logical limit laws for preferential attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time 1, we start with vertices 0,1 and m edges between them. At step n+1 the vertex n+1 is introduced together with m edges joining the new vertex with m vertices chosen from 1,…,n independently with probabilities proportional to their degrees plus a positive parameter δ. We prove that this random graph obeys the convergence law for first-order sentences with at most m−2 variables.