Analytical results for the in-degree and out-degree distributions of directed random networks that grow by node duplication

Abstract

We present exact analytical results for the degree distribution in a directed network model that grows by node duplication (ND). Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability p . In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. The corresponding undirected ND model was studied before and was found to exhibit a power-law degree distribution. We obtain analytical results for the in-degree distribution , and for the out-degree distribution , of the corded DND network at time t. It is found that the in-degrees follow a shifted power-law distribution, so the network is asymptotically scale free. In contrast, the out-degree distribution is a narrow distribution, that converges to a Poisson distribution in the limit of and to a Gaussian distribution in the limit of . Such distinction between a broad in-degree distribution and a narrow out-degree distribution is common in empirical networks such as scientific citation networks. Using these distributions we calculate the mean degree , which converges to in the large network limit, for the whole range of 0 < p < 1. This is in contrast to the corresponding undirected network, which exhibits a phase transition at p = 1/2 such that for p > 1/2 the mean degree diverges in the large network limit. We also present analytical results for the distribution of the number of upstream nodes, , and for the distribution of the number of downstream nodes, , from a random node. We show that the mean values scale logarithmically with the network size. This means that in the large network limit only a diminishing fraction of pairs of nodes are connected by directed paths, unlike the corded undirected ND network that consists of a single connected component. Therefore, the corded DND network is not a small-world network.