  The study of triangles in graphs is a standard tool in network analysis,
leading to measures such as the \emph{transitivity}, i.e., the fraction of
paths of length $2$ that participate in triangles. Real-world networks are
often directed, and it can be difficult to "measure" this network structure
meaningfully. We propose a collection of \emph{directed closure values} for
measuring triangles in directed graphs in a way that is analogous to
transitivity in an undirected graph. Our study of these values reveals much
information about directed triadic closure. For instance, we immediately see
that reciprocal edges have a high propensity to participate in triangles. We
also observe striking similarities between the triadic closure patterns of
different web and social networks. We perform mathematical and empirical
analysis showing that directed configuration models that preserve reciprocity
cannot capture the triadic closure patterns of real networks.

Social and Information Networks;Data Structures and Algorithms;Physics and Society