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

Kolountzakis, Mihail N.; Peng, Richard; Tsourakakis, Charalampos E.

doi:10.1007/978-3-642-18009-5_3

Cited by 51 publications

(48 citation statements)

References 33 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, Björklund, Pagh, Williams and Zwick gave refined algorithms which are output sensitive algorithms [14]. Finally a wealth of approximate triangle counting methods exist [35,41,44,51].…”

Section: Related Workmentioning

confidence: 99%

The K-clique Densest Subgraph Problem

Tsourakakis

2015

Proceedings of the 24th International Conference on World Wide Web

215

160

View full text Add to dashboard Cite

Numerous graph mining applications rely on detecting subgraphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem, the poly-time solvable densest subgraph problem which maximizes the average degree over all possible subgraphs "lies at the core of large scale data mining" [10]. However, frequently the densest subgraph problem fails in detecting large near-cliques in networks.In this work, we introduce the k-clique densest subgraph problem, k ≥ 2. This generalizes the well studied densest subgraph problem which is obtained as a special case for k = 2. For k = 3 we obtain a novel formulation which we refer to as the triangle densest subgraph problem: given a graph G(V, E), find a subset of vertices S * such that, where t(S) is the number of triangles induced by the set S.On the theory side, we prove that for any k constant, there exist an exact polynomial time algorithm for the kclique densest subgraph problem. Furthermore, we propose an efficient 1 k -approximation algorithm which generalizes the greedy peeling algorithm of Asahiro and Charikar [8,18] for k = 2. Finally, we show how to implement efficiently this peeling framework on MapReduce for any k ≥ 3, generalizing the work of Bahmani, Kumar and Vassilvitskii for the case k = 2 [10]. On the empirical side, our two main findings are that (i) the triangle densest subgraph is consistently closer to being a large near-clique compared to the densest subgraph and (ii) the peeling approximation algorithms for both k = 2 and k = 3 achieve on real-world networks approximation ratios closer to 1 rather than the pessimistic 1 kguarantee. An interesting consequence of our work is that triangle counting, a well-studied computational problem in the context of social network analysis can be used to detect large near-cliques. Finally, we evaluate our proposed method on a popular graph mining application.

show abstract

Section: Related Workmentioning

confidence: 99%

The K-clique Densest Subgraph Problem

Tsourakakis

2015

Proceedings of the 24th International Conference on World Wide Web

215

160

View full text Add to dashboard Cite

show abstract

“…For a nonexhaustive sample of the extensive body of work in this direction, including theoretical and engineering work, cf. [108,92,111,68,29,64,107,36,3,47,73,97,74,88,18,45,24,91,96,63,93,4,25,90]. Our work differs from these works in that we seek a proof-of-concept implementation for simultaneous delegatability and errortolerance.…”

Section: Counting and Enumerating Subgraphsmentioning

confidence: 99%

Engineering a Delegatable and Error-Tolerant Algorithm for Counting Small Subgraphs

Kaski

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

View full text Add to dashboard Cite

We study the problem of counting the number of occurrences of a given six-vertex pattern graph S in an n-vertex host graph H. We engineer an open-source GPU implementation of a distributed algorithm design of Björklund and Kaski [PODC 2016] where (i) the execution of the algorithm can be delegated [Goldwasser, Kalai, and Rothblum, J. ACM 2015] to produce a noninteractive probabilistically checkable proof of correctness, and (ii) the execution of the algorithm when preparing the proof tolerates a controllable number of adversarial errors. Experiments with NVIDIA Tesla K80 and Tesla P100 Accelerators demonstrate that the framework is practical for inputs of up to 512 vertices, with proof checking being several orders of magnitude more efficient than preparing the proof; however, proof preparation still carries at least one order of magnitude overhead compared with just solving the problem.

show abstract

“…Several authors presented sampling based algorithms for the estimation of the global number of triangles in a (semi)-streaming setting [5,9,12,16,21,24,28,29]. Building upon results from linear algebra, researchers proposed techniques for approximate triangle counting not relying on sampling [4,27].…”

Section: Previous Workmentioning

confidence: 99%

On the streaming complexity of computing local clustering coefficients

Kutzkov

Pagh

2013

Proceedings of the Sixth ACM International Conference on Web Search and Data Mining

View full text Add to dashboard Cite

Due to a large number of applications, the problem of estimating the number of triangles in graphs revealed as a stream of edges, and the closely related problem of estimating the graph's clustering coefficient, have received considerable attention in the last decade. Both efficient algorithms and impossibility results have shed light on the computational complexity of the problem. Motivated by applications in Web mining, Becchetti et al. presented new algorithms for the estimation of the local number of triangles, i.e., the number of triangles incident to individual vertices. The algorithms are shown, both theoretically and experimentally, to efficiently handle the problem. However, at least two passes over the data are needed and thus the algorithms are not suitable for real streaming scenarios.In the present work, we consider the problem of estimating the clustering coefficient of individual vertices in a graph over n vertices revealed as a stream of m edges. As a first result we show that any one pass randomized streaming algorithm that can distinguish a graph with no triangles from a graph having a vertex of degree d with clustering coefficient > 1/2 must use Ω(m/d) bits of space in expectation.Our second result is a new randomized one pass algorithm estimating the local clustering coefficient of each vertex with degree at least d. The space requirement of our algorithm is within a logarithmic factor of the lower bound, thus our approach is close to optimal. We also extend the algorithm to local triangle counting and report experimental results on its performance on real-life graphs.

show abstract

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

Cited by 51 publications

References 33 publications

The K-clique Densest Subgraph Problem

The K-clique Densest Subgraph Problem

Engineering a Delegatable and Error-Tolerant Algorithm for Counting Small Subgraphs

On the streaming complexity of computing local clustering coefficients

Contact Info

Product

Resources

About