We compute the stationary in-degree probability, $P_{ in}(k)$, 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 $k^{ -3}$. Moreover, for an out-degree distribution with a scale invariant tail, $P_{ out}(k)\sim k^{ -\alpha}$, the corresponding in-degree distribution has exactly the same asymptotic behavior only if $2<\alpha<3$ (infinite variance). Similar results are obtained when attractiveness is included. We also present some results on descriptive statistics measures %descriptive statistics such as the correlation between the number of in-going links, $D_{ in}$, and outgoing links, $D_{ out}$, and the conditional expectation of $D_{ in}$ given $D_{ out}$, 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.
Categories: physics.soc-ph physics.data-an