  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 family of principled measures 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}x 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.
