Directed random graphs with given degree distributions

Abstract

Given two distributions $F$ and $G$ on the nonnegative integers we propose an algorithm to construct in- and out-degree sequences from samples of i.i.d. observations from $F$ and $G$, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that $F$ and $G$ have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in- and out-degree distributions are (approximately) $F$ and $G$ for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.