Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model [12]. The key idea of our algorithm is to combine the sampling algorithm of [31,32] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [1], treating each set appropriately. We obtain a running time \(O \left( m + \frac{m^{3/2} \Delta \log{n} }{t \epsilon^2} \right)\) and an e approximation (multiplicative error), where n is the number of vertices, m the number of edges and Δ the maximum number of triangles an edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage \(O\left(m^{1/2}\log{n} + \frac{m^{3/2} \Delta \log{n}}{t \epsilon^2} \right)\) and a constant number of passes (three) over the graph stream. We apply our methods in various networks with several millions of edges and we obtain excellent results. Finally, we propose a random projection based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.