Measuring directed triadic closure with closure coefficients

Abstract

Recent work studying triadic closure in undirected graphs has drawn attention to the distinction between measures that focus on the “center” node of a wedge (i.e., length-2 path) versus measures that focus on the “initiator,” a distinction with considerable consequences. Existing measures in directed graphs, meanwhile, have all been center-focused. In this work, we propose a family of eight directed closure coefficients that measure the frequency of triadic closure in directed graphs from the perspective of the node initiating closure. The eight coefficients correspond to different labeled wedges, where the initiator and center nodes are labeled, and we observe dramatic empirical variation in these coefficients on real-world networks, even in cases when the induced directed triangles are isomorphic. To understand this phenomenon, we examine the theoretical behavior of our closure coefficients under a directed configuration model. Our analysis illustrates an underlying connection between the closure coefficients and moments of the joint in- and out-degree distributions of the network, offering an explanation of the observed asymmetries. We also use our directed closure coefficients as predictors in two machine learning tasks. We find interpretable models with AUC scores above 0.92 in class-balanced binary prediction, substantially outperforming models that use traditional center-focused measures.

Directed closure coefficient and its patterns

“…In this section, we summarise some additional related work that also measure directed triangle formation from an end-node perspective. Similar to our work, Yin et al [ 60 ] extended the local closure coefficient in directed networks by proposing a family of eight coefficients. Their definition of the local directed closure coefficients of node i are eight scalars with x , y , z ∈ { i , o } ( i and o represent edge direction, incoming or outgoing).…”

Weighted directed clustering: interpretations and requirements for heterogeneous, inferred, and measured networks

“…For directed networks, contrary to [25], we only consider the extension of the undirected measure to directed walks and therefore define only four variants of the directed closure, two for outgoing walks (cycle-out, CO, and fan-out, FO) and two for incoming walks (cycle-in, CI, and fan-in, FI).…”

The Strength of Structural Diversity in Online Social Networks

“…To directly compare these measures, we further adopt L 1 -regularized linear regression, which is also called least absolute shrinkage and selection operator (LASSO) regression. LASSO regression is a standard model in sparse regression and has been widely used for simultaneous estimation and variable selection [48][49][50][51][52] (see Methods for more information on LASSO regression). Note that, with the increase of the regularization level, LASSO would continuously shrink the coefficients of less important features to be zero [49,50].…”