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Measuring directed triadic closure with closure coefficients

Published online by Cambridge University Press:  01 June 2020

Hao Yin Open the ORCID record for Hao Yin [Opens in a new window] ,
Austin R. Benson  and
Johan Ugander
Hao Yin
Affiliation:
Institute for Computation and Mathematical Engineering, Stanford University, Stanford, CA, USA (e-mail: [email protected])
Austin R. Benson
Affiliation:
Department of Computer Science, Cornell University, Ithaca, NY, USA (e-mail: [email protected])
Johan Ugander*
Affiliation:
Department of Management Science and Engineering, Stanford University, Stanford, CA, USA
*
*Corresponding author. Email: [email protected]
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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.

Keywords

directed networkstriadic closureclosure coefficientsconfiguration model
Type
Research Article
Information
Network Science , Volume 8 , Issue 4 , December 2020 , pp. 551 - 573
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

Action Editor: Ulrik Brandes

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