Reciprocity in Directed Hypergraphs: Measures, Findings, and Generators

Group interactions are prevalent in a variety of areas. Many of them, including email exchanges, chemical reactions, and bitcoin transactions, are directional, and thus they are naturally modeled as directed hypergraphs, where each hyperarc consists of the set of source nodes and the set of destination nodes. For directed graphs, which are a special case of directed hypergraphs, reciprocity has played a key role as a fundamental graph statistic in revealing organizing principles of graphs and in solving graph learning tasks. For general directed hypergraphs, however, even no systematic measure of reciprocity has been developed.In this work, we investigate the reciprocity of 11 real-world hypergraphs. To this end, we first introduce eight axioms that any reasonable measure of reciprocity should satisfy. Second, we propose HYPERREC, a principled measure of hypergraph reciprocity that satisfies all the axioms. Third, we develop FERRET, a fast and exact algorithm for computing the measure, whose search space is up to $10 ^{147} \times$ smaller than that of naive computation. Fourth, using them, we examine 11 real-world hypergraphs and discover patterns that distinguish them from random hypergraphs. Lastly, we propose REDI, an intuitive generative model for directed hypergraphs exhibiting the patterns. The code and the datasets are available at https://github.com/kswoo97/hyprec.