The importance of directed triangles with reciprocity: patterns and algorithms

The computation and study of triangles in graphs is a standard tool in the analysis of real-world networks. Yet most of this work focuses on undirected graphs. Real-world networks are often directed and have a significant fraction of reciprocal edges. While there is much focus on directed triadic patterns in the social sciences community, most data mining and graph analysis studies ignore direction. But how to we make sense of this complex directed structure? We propose a collection of directed closure values that are analogues of the classic transitivity measure (the fraction of wedges that participate in triangles). We perform an extensive set of triadic measurements on a variety of massive real-world networks. Our study of these values reveal 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. Together with these observations, we also present the first sampling based algorithm for fast estimation of directed triangles. Previous estimation methods were targeted towards undirected triangles and could not be extended to directed graphs. Our method, based on wedge sampling, gives orders of magnitude speedup over state of the art enumeration. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. ∗This work was funded by the GRAPHS Program at DARPA and the Laboratory Directed Research and Development program at Sandia National Laboratories.