Doulion

Abstract

Counting the number of triangles in a graph is a beautiful algorithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the transitivity ratio involve the execution of a triangle counting algorithm. Furthermore, several interesting graph mining applications rely on computing the number of triangles in the graph of interest.In this paper, we focus on the problem of counting triangles in a graph. We propose a practical method, out of which all triangle counting algorithms can potentially benefit. Using a straightforward triangle counting algorithm as a black box, we performed 166 experiments on real-world networks and on synthetic datasets as well, where we show that our method works with high accuracy, typically more than 99% and gives significant speedups, resulting in even ≈ 130 times faster performance.

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

“…Our algorithm combines two approaches that have been taken on triangle counting: sparsify the graph by keeping a random subset of the edges [34,35] followed by a triple sampling using the idea of vertex partitioning due to Alon, Yuster and Zwick [2].…”

“…The key observation is that since counting triangles reduces to computing the intersection of two sets, namely the induced neighborhoods of two adjacent nodes, ideas from locality sensitivity hashing [6] are applicable to the problem. In [34] an algorithm which tosses a coin independently for each edge with probability p to keep the edge and probability q = 1 − p to throw it away is proposed. It was shown later by Tsourakakis, Kolountzakis and Miller [35] using a powerful theorem due to Kim and Vu [22] that under mild conditions on the triangle density the method results in a strongly concentrated estimate on the number of triangles.…”

“…The following method was introduced in [34] and was shown to perform very well in practice: keep each edge with probability p independently. Then for each triangle, the probability of it being kept is p 3 .…”

“…The key idea of the method is to combine the sampling scheme introduced by Tsourakakis et al in [34,35] with the partitioning idea of Alon, Yuster and Zwick [2] in order to obtain a more efficient sampling scheme. Furthermore, we show that this method can be adapted to the semistreaming model with a constant number of passes and O m 1/2 log n + m 3/2 ∆ log n tǫ 2 space.…”

“…In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model [15]. The key idea of our algorithm is to combine the sampling algorithm of [34,35] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [2], treating each set appropriately. We obtain a running time O m + m 3/2 ∆ log n tǫ 2 and an ǫ approximation (multiplicative error), where n is the number of vertices, m the number of edges and ∆ the maximum number of triangles an edge is contained.…”