The degree distribution of the generalized duplication model

Abstract

We study and generalize the duplication model of Pastor-Satorras et al. [Evolving protein interaction networks through gene duplication, J. Theor. Biol. 222 (2003) 199–210]. This model generates a graph by iteratively “duplicating” a randomly chosen node as follows: we start at t 0 with a fixed graph G ( t 0 ) of size t 0 . At each step t > t 0 a new node v t is added. The node v t selects an existing node u from V ( G ( t - 1 ) ) = { v 1 , … , v t - 1 } uniformly at random (uar). The node v t then connects to each neighbor of the node u in G ( t - 1 ) independently with probability p . Additionally, v t connects uar to every node of V ( G ( t - 1 ) ) independently with probability r / t , and parallel edges are merged. Unlike other copy-based models, the degree of the node v t in this model is not fixed in advance; rather it depends strongly on the degree of the original node u it selected. Our main contributions are as follows: we show that (1) the duplication model of Pastor-Satorras et al. does not generate a truncated power-law degree distribution as stated in Pastor-Satorras et al. [Evolving protein interaction networks through gene duplication, J. Theor. Biol. 222 (2003) 199–210]. (2) The special case where r = 0 does not give a power-law degree distribution as stated in Chung et al. [Duplication models for biological networks, J. Comput. Biol. 10 (2003) 677–687]. (3) We generalize the Pastor-Satorras et al. duplication process to ensure (if required) that the minimum degree of all vertices is positive. We prove that this generalized model has a power-law degree distribution.