Degree Distribution for Duplication-Divergence Graphs: Large Deviations

We present a rigorous and precise analysis of the degree distribution in a dynamic graph model introduced by Solé et al. in which nodes are added according to a duplication-divergence mechanism, i.e. by iteratively copying a node and then randomly inserting and deleting some edges for a copied node. This graph model finds many applications since it well captures the growth of some real-world processes e.g. biological or social networks. However, there are only a handful of rigorous results concerning this model. In this paper we present rigorous results concerning the degree distribution. We focus on two related problems: the expected value and large deviation for the degree of a fixed node through the evolution of the graph and the expected value and large deviation of the average degree in the graph. We present exact and 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 overall graph behaviors, especially, degree distribution, symmetry, and compression, important open problems in this area.