Directed scale-free graphs.

Abstract We introduce a model for directed scale-free graphs that grow with preferential attachment depending in a natural way on the in-and out-degrees. We show that the resulting in-and out-degree distributions are power laws with different exponents, reproducing observed properties of the worldwide web. We also derive exponents for the distribution of in-(out-) degrees among vertices with fixed out-(in-) degree. We conclude by suggesting a corresponding model with hidden variables

Convergence of rank based degree-degree correlations in random directed networks

We introduce, and analyze, three measures for degree-degree dependencies, also called degree assortativity, in directed random graphs, based on Spearman's rho and Kendall's tau. We proof statistical consistency of these measures in general random graphs and show that the directed configuration model can serve as a null model for our degree-degree dependency measures. Based on these results we argue that the measures we introduce should be preferred over Pearson's correlation coefficients, when studying degree-degree dependencies, since the latter has several issues in the case of large networks with scale-free degree distributions

Random Networks Tossing Biased Coins

In statistical mechanical investigations on complex networks, it is useful to employ random graphs ensembles as null models, to compare with experimental realizations. Motivated by transcription networks, we present here a simple way to generate an ensemble of random directed graphs with, asymptotically, scale-free outdegree and compact indegree. Entries in each row of the adjacency matrix are set to be zero or one according to the toss of a biased coin, with a chosen probability distribution for the biases. This defines a quick and simple algorithm, which yields good results already for graphs of size n ~ 100. Perhaps more importantly, many of the relevant observables are accessible analytically, improving upon previous estimates for similar graphs