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.

中文翻译：

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用闭合系数测量定向三元闭合

最近在无向图中研究三元闭包的工作引起了人们对关注楔形“中心”节点（即长度为 2 路径）的度量与关注“发起者”的度量之间的区别的关注，这种区别具有相当大的影响. 与此同时，有向图中的现有度量都以中心为中心。在这项工作中，我们提出了一个八口之家有向闭合系数从节点启动闭合的角度测量有向图中三元闭合的频率。这八个系数对应于不同的标记楔，其中起始节点和中心节点被标记，我们观察到这些系数在现实世界网络中的显着经验变化，即使在诱导有向三角形是同构的情况下也是如此。为了理解这种现象，我们检查了在有向配置模型下我们的闭合系数的理论行为。我们的分析说明了网络的联合进出度分布的闭合系数和矩之间的潜在联系，提供了对观察到的不对称性的解释。我们还在两个机器学习任务中使用我们的定向闭合系数作为预测因子。