Towards Degree Distribution of a Duplication-Divergence Graph Model

Abstract

We present a rigorous and precise analysis of degree distribution in a dynamic graph model introduced by Solé, Pastor-Satorras et al. in which nodes are added according to a duplication-divergence mechanism. This model is discussed in numerous publications with only very few recent rigorous results, especially for the degree distribution. In this paper we focus on two related problems: the expected value and variance of the degree of a given node over the evolution of the graph and the expected value and variance of the average degree over all nodes. We present exact and precise asymptotic results showing that both quantities may decrease or increase over time depending on the model parameters. Our findings are a step towards a better understanding of the graph behaviors such as degree distributions, symmetry, power law, and structural compression.