Graphs and networks are used to model interactions in a variety of contexts. There is a growing need to quickly assess the characteristics of a graph in order to understand its underlying structure. Some of the most useful metrics are triangle-based and give a measure of the connectedness of mutual friends. This is often summarized in terms of clustering coefficients, which measure the likelihood that two neighbors of a node are themselves connected. Computing these measures exactly for large-scale networks is prohibitively expensive in both memory and time. However, a recent wedge-sampling algorithm has proved successful in efficiently and accurately estimating clustering coefficients. In this paper, we describe how to implement this approach in MapReduce to deal with massive graphs. We show results on publicly available networks, the largest of which is 132M nodes and 4.7B edges, as well as artificially generated networks (using the Graph500 benchmark), the largest of which has 240M nodes and 8.5B edges...
Graphs are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the structure of a graph. Some of the most useful graph metrics, especially those measuring social cohesion, are based on triangles. Despite the importance of these triadic measures, associated algorithms can be extremely expensive. We propose a new method based on wedge sampling. This versatile technique allows for the fast and accurate approximation of all current variants of clustering coefficients and enables rapid uniform sampling of the triangles of a graph. Our methods come with provable and practical time-approximation tradeoffs for all computations. We provide extensive results that show our methods are orders of magnitude faster than the state-of-the-art, while providing nearly the accuracy of full enumeration. Our results will enable more wide-scale adoption of triadic measures for analysis of extremely large graphs, as demonstrated on several real-world examples. Since its introduction in the year 1998 by Watts and Strogatz, the clustering coecient has become a frequently used tool for analyzing graphs. In 2002 the transitivity was proposed by Newman, Watts and Strogatz as an alternative to the clustering coecient. As many networks considered in complex systems are huge, the ecient computation of such network parameters is crucial. Several algorithms with polynomial running time can be derived from results known in graph theory. The main contribution of this work is a new fast approximation algorithm for the weighted clustering coecient which also gives very ecient approximation algorithms for the clustering coecient and the transitivity. We namely present an algorithm with running time in O(1) for the clustering coecient, respectively with running time in O(n) for the transitivity. By an experimental study we demonstrate the performance of the proposed algorithms on real-world data as well as on generated graphs. Moreover we give a simple graph generator algorithm that works according to the preferential attachment rule but also generates graphs with adjustable clustering coecient.