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Recent work studying triadic closure in undirected graphs has drawn attentionto the distinction between measures that focus on the "center" node of a wedge(i.e., length-2 path) vs. measures that focus on the "initiator," a distinctionwith considerable consequences. Existing measures in directed graphs,meanwhile, have all been center-focused. In this work, we propose a family ofeight directed closure coefficients that measure the frequency of triadicclosure in directed graphs from the perspective of the node initiating closure.The eight coefficients correspond to different labelled wedges, where theinitiator and center nodes are labelled, and we observe dramatic empiricalvariation in these coefficients on real-world networks, even in cases when theinduced directed triangles are isomorphic. To understand this phenomenon, weexamine the theoretical behavior of our closure coefficients under a directedconfiguration model. Our analysis illustrates an underlying connection betweenthe closure coefficients and moments of the joint in- and out-degreedistributions of the network, offering an explanation of the observedasymmetries. We also use our directed closure coefficients as predictors in twomachine learning tasks. We find interpretable models with AUC scores above 0.92in class-balanced binary prediction, substantially outperforming models thatuse traditional center-focused measures.

Measuring Directed Triadic Closure with Closure Coefficients