Efficient semi-streaming algorithms for local triangle counting in massive graphs

Information Processing Letters

2012

157

124

Abstract. In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, let G be a graph with n vertices, t triangles and let ∆ be the maximum number of triangles an edge of G is contained in. Also, let N = 1/p the number of colors we use in our randomized algorithm. We show that if p ≥ max ( ∆ log n t , log n t ) then for any constant > 0 our unbiased estimate T is concentrated around its expectation, i.e.,. Finally, our algorithm is amenable to being parallelized. We present a simple MapReduce implementation of our algorithm.

Section: Introductionmentioning

confidence: 99%

Colorful triangle counting and a MapReduce implementation

Pagh

Information Processing Letters

2012

157

124

“…Semi-streaming model Recently, Becchetti, Boldi, Castillo and Gionis introduced the semi-streaming model in [5] to solve the local triangle counting problem. Their method relies on the locality sensitivity hashing concept.…”

Section: Exact Counting Methodsmentioning

confidence: 99%

“…Besides the significance of triangles in network analysis statistics, they also play an important role in graph mining applications: Eckmann and Moses showed how one can use triangles in order to uncover the hidden thematic structure of the web [11] and Beccheti et al in [5] used the local distribution of triangles and the clustering coefficient to detect spamming activity. Furthermore, triangle-related power laws [21] can be used to define outliers in a graph with respect to triangles.…”

Section: Introductionmentioning

confidence: 99%

Spectral Counting of Triangles in Power-Law Networks via Element-Wise Sparsification

2009 International Conference on Advances in Social Network Analysis and Mining

Drineas

Michelakis

et al. 2009

Triangle counting is an important problem in graph mining. The clustering coefficient and the transitivity ratio, two commonly used measures effectively quantify the triangle density in order to quantify the fact that friends of friends tend to be friends themselves. Furthermore, several successful graph mining applications rely on the number of triangles.In this paper, we study the problem of counting triangles in large, power-law networks. Our algorithm, SPARSI-FYINGEIGENTRIANGLE , relies on the spectral properties of power-law networks and the Achlioptas-McSherry sparsification process. SPARSIFYINGEIGENTRIANGLE is easy to parallelize, fast and accurate.We verify the validity of our approach with several experiments in real-world graphs, where we achieve at the same time high accuracy and important speedup versus a straight-forward exact counting competitor.

“…This algorithm can be viewed as a special case of a streaming algorithm, since there exist algorithms, e.g., [29], that perform a constant number of passes over the non-zero elements of the matrix to produce a good low rank matrix approximation. In [5] the semi-streaming model for counting triangles is introduced, which allows log n passes over the edges. 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.…”

Section: Existing Workmentioning

confidence: 99%

“…and is one of the main reasons which gave rise to the definitions of the transitivity ratio and the clustering coefficients of a graph in complex network analysis [27]. Triangles are used in several applications such as uncovering the hidden thematic structure of the web [13], as a feature to assist the classification of web activity [5] and for link recommendation in online social networks [36]. Furthermore, triangles are used as a network statistic in the exponential random graph model [14].…”

Section: Introductionmentioning

confidence: 99%

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

Kolountzakis

Peng

Algorithms and Models for the Web-Graph

2010

Abstract. The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. 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. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage O m 1/2 log n + m 3/2 ∆ log n tǫ 2 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.