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

We present exact 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, 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. We obtain analytical results for the in-degree distribution $P_t(K_{in}=k)$, and for the out-degree distribution $P_t(K_{out}=k)$ at time t. The in-degrees follow a shifted power-law, so the network is asymptotically scale free. In contrast, the out-degree distribution is narrow, and converges to a Poisson distribution in the sparse network limit and to a Gaussian distribution in the dense network limit. 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 this we calculate the mean degree $\langle K_{in}\rangle_t=\langle K_{out}\rangle_t$, which converges to $1/(1-p)$ 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 and downstream nodes from a random node. The mean values $\langle N_{up}\rangle_t=\langle N_{down}\rangle_t$ scale logarithmically with the network size, implying that only a small fraction of pairs of nodes are connected by directed paths, unlike the undirected ND case that consists of a single component, hence not a small-world network.