Directed closure measures for networks with reciprocity

The study of triangles in graphs is a standard tool in network analysis. Real-world networks are often directed and have a significant fraction of reciprocal edges. Direction leads to many complexities, and it can be difficult to “measure” this structure meaningfully. We propose a collection of directed closure values that are analogues of the classic transitivity measure (the fraction of wedges that participate in triangles). Our study of these values reveals a wealth of information of the nature of direction. For instance, we immediately see the importance of reciprocal edges in forming triangles and can measure the power of transitivity. Surprisingly, the chance that a wedge is closed depends heavily on its directed structure. We also observe striking similarities between the triadic closure patterns of different web and social networks.