The degree distribution of the generalized duplication model

We study and generalize the duplication model of Pastor-Satorras et al. [25]. This model generates a graph by iteratively “duplicating” a randomly chosen node as follows: We start at t0 with a fixed graph G(t0) of size t0. At each step t > t0 a new node vt is added. The node vt selects an existing node u from V (G(t − 1)) = {v1, ..., vt−1} uniformly at random (uar). The node vt then connects to each neighbour of the node u in G(t − 1) independently with probability p. Additionally vt 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 vt 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 [25]. (2) The special case where r = 0 does not give a power law degree distribution as stated in [12]. (3) We generalize the Pastor-Satorras 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.