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

We compute the stationary in-degree probability, P(k<Subscript>in</Subscript>), 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 k<Subscript>in</Subscript> <Superscript>-3</Superscript>. Moreover, for an out-degree distribution with a scale invariant tail, P(k<Subscript>out</Subscript>)∼k<Subscript>out</Subscript> <Superscript>-α</Superscript>, the corresponding in-degree distribution has exactly the same asymptotic behavior only if 2 > α > 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, K<Subscript>in</Subscript>, and outgoing links, K<Subscript>out</Subscript>, and the conditional expectation of K<Subscript>in</Subscript> given K<Subscript>out</Subscript>, 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. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2008