Growing directed networks: stationary in-degree probability for arbitrary out-degree one

Abstract

We compute the stationary in-degree probability, P(kin), for a growing network model with directed edges and arbitrary out-degree probability. In particular, under preferential linking, we find that if the nodes have a light tail (finite variance) out-degree distribution, then the corresponding in-degree one behaves as kin-3. Moreover, for an out-degree distribution with a scale invariant tail, , the corresponding in-degree distribution has exactly the same asymptotic behavior only if 2 < < 3 (infinite variance). Similar results are obtained when attractiveness is included. We also present some results on descriptive statistics measures such as the correlation between the number of in-going links, Kin, and outgoing links, Kout, and the conditional expectation of Kin given Kout, and we calculate these measures for the WWW network. Finally, we present an application to the scientific publications network. The results presented here can explain the tail behavior of in/out-degree distribution observed in many real networks.