Canonical fitness model for simple scale-free graphs

Abstract

We consider a fitness model assumed to generate simple graphs with a power-law heavy-tailed degree sequence, P(k)∝k−1−α with 0<α<1, in which the corresponding distributions do not possess a mean. We discuss the situations in which the model is used to produce a multigraph and examine what happens if the multiple edges are merged to a single one and thus a simple graph is built. We give the relation between the (normalized) fitness parameter r and the expected degree ν of a node and show analytically that it possesses nontrivial intermediate and final asymptotic behaviors. We show that the model produces P(k)∝k−2 for large values of k independent of α. Our analytical findings are confirmed by numerical simulations.