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Degree distributions are arguably the most important property of real worldnetworks. The classic edge configuration model or Chung-Lu model can generatean undirected graph with any desired degree distribution. This serves as a goodnull model to compare algorithms or perform experimental studies. Furthermore,there are scalable algorithms that implement these models and they areinvaluable in the study of graphs. However, networks in the real-world areoften directed, and have a significant proportion of reciprocal edges. Astronger relation exists between two nodes when they each point to one another(reciprocal edge) as compared to when only one points to the other (one-wayedge). Despite their importance, reciprocal edges have been disregarded by mostdirected graph models.  We propose a null model for directed graphs inspired by the Chung-Lu modelthat matches the in-, out-, and reciprocal-degree distributions of the realgraphs. Our algorithm is scalable and requires $O(m)$ random numbers togenerate a graph with $m$ edges. We perform a series of experiments on realdatasets and compare with existing graph models.

A Scalable Null Model for Directed Graphs Matching All Degree Distributions: In, Out, and Reciprocal