Counting Triangles in Massive Graphs with MapReduce

Kolda, Tamara G.; Pınar, Ali; Plantenga, Todd; Seshadhri, C.; Task, Christine

doi:10.1137/13090729x

Cited by 83 publications

(38 citation statements)

References 60 publications

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“…Tsourakakis et al [2009a] started the use of sparsification methods, the most important of which is Doulion [Tsourakakis et al 2009b]. Various analyses of this algorithm (and its variants) have been proposed [Kolountzakis et al 2010;Tsourakakis et al 2011;Yoon and Kim 2011;Pagh and Tsourakakis 2012]. Algorithms based on wedge sampling provide provable accurate estimations on various triadic measures on graphs [Schank and Wagner 2005a;Seshadhri et al 2013a].…”

Section: Previous Workmentioning

confidence: 99%

A Space-Efficient Streaming Algorithm for Estimating Transitivity and Triangle Counts Using the Birthday Paradox

Jha

Seshadhri

Pınar

2015

ACM Trans. Knowl. Discov. Data

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We design a space-efficient algorithm that approximates the transitivity (global clustering coefficient) and total triangle count with only a single pass through a graph given as a stream of edges. Our procedure is based on the classic probabilistic result, the birthday paradox. When the transitivity is constant and there are more edges than wedges (common properties for social networks), we can prove that our algorithm requires O( √ n) space (n is the number of vertices) to provide accurate estimates. We run a detailed set of experiments on a variety of real graphs and demonstrate that the memory requirement of the algorithm is a tiny fraction of the graph. For example, even for a graph with 200 million edges, our algorithm stores just 40,000 edges to give accurate results. Being a single pass streaming algorithm, our procedure also maintains a real-time estimate of the transitivity/number of triangles of a graph by storing a minuscule fraction of edges. ACM Reference Format:Madhav Jha, C. Seshadhri, and Ali Pinar. 2015. A space-efficient streaming algorithm for estimating transitivity and triangle counts using the birthday paradox. ACM Trans. Knowl. Discov. Data 9, 3, Article 15 (February 2015), 21 pages.

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Section: Previous Workmentioning

confidence: 99%

A Space-Efficient Streaming Algorithm for Estimating Transitivity and Triangle Counts Using the Birthday Paradox

Jha

Seshadhri

Pınar

2015

ACM Trans. Knowl. Discov. Data

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“…Testing showed that in networks with high modularity, internal and global degree and clustering coefficients were nearly identical, except for high degree nodes. Recent work [28,46] has shown CCPD distribution can be estimated via sampling and scales to large networks. Future research is needed to uncover explicit relationships and network properties that would allow for autonomous and scalable generation of EGBTER graphs without the direct reliance on the measured inputs from a seed graph.…”

Section: Discussionmentioning

confidence: 99%

EGBTER: Capturing Degree Distribution, Clustering Coefficients, and Community Structure in a Single Random Graph Model

El-daghar

Lundberg

Bridges

2018

2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM)

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Random graph models are important constructs for data analytic applications as well as pure mathematical developments, as they provide capabilities for network synthesis and principled analysis. Several models have been developed with the aim of faithfully preserving important graph metrics and substructures. With the goal of capturing degree distribution, clustering coefficient, and communities in a single random graph model, we propose a new model to address shortcomings in a progression of network modeling capabilities. The Block Two-Level Erdős-Rényi (BTER) model of Seshadhri et al., designed to allow prescription of expected degree and clustering coefficient distributions, neglects community modeling, while the Generalized BTER (GBTER) model of Bridges et al., designed to add community modeling capabilities to BTER, struggles to faithfully represent all three characteristics simultaneously. In this work, we fit BTER and two GBTER configurations to several real-world networks and compare the results with that of our new model, the Extended GBTER (EGBTER) model. Our results support that EBGTER adds a community-modeling flexibility to BTER, while retaining a satisfactory level of accuracy in terms of degree and clustering coefficient. Our insights and empirical testing of previous models as well as the new model are novel contributions to the literature.

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“…Within each pattern, vertices are present in different "roles" or orbits. In some patterns like the 5-cycle (H 15 ) and 5-clique (H 29 ), there is just one orbit. In contrast, H 10 has four different orbits, indicated by the different colors.…”

Section: Problem Descriptionmentioning

confidence: 99%

Efficiently Counting Vertex Orbits of All 5-vertex Subgraphs, by EVOKE

Pashanasangi

Seshadhri

2020

Proceedings of the 13th International Conference on Web Search and Data Mining

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Subgraph counting is a fundamental task in network analysis. Typically, algorithmic work is on total counting, where we wish to count the total frequency of a (small) pattern subgraph in a large input data set. But many applications require local counts (also called vertex orbit counts) wherein, for every vertex v of the input graph, one needs the count of the pattern subgraph involving v. This provides a rich set of vertex features that can be used in machine learning tasks, especially classification and clustering. But getting local counts is extremely challenging. Even the easier problem of getting total counts has received much research attention. Local counts require algorithms that get much finer grained information, and the sheer output size makes it difficult to design scalable algorithms.We present EVOKE, a scalable algorithm that can determine vertex orbits counts for all 5-vertex pattern subgraphs. In other words, EVOKE exactly determines, for every vertex v of the input graph and every 5-vertex subgraph H , the number of copies of H that v participates in. EVOKE can process graphs with tens of millions of edges, within an hour on a commodity machine. EVOKE is typically hundreds of times faster than previous state of the art algorithms, and gets results on datasets beyond the reach of previous methods.Theoretically, we generalize a recent "graph cutting" framework to get vertex orbit counts. This framework generate a collection of polynomial equations relating vertex orbit counts of larger subgraphs to those of smaller subgraphs. EVOKE carefully exploits the structure among these equations to rapidly count. We prove and empirically validate that EVOKE only has a small constant factor overhead over the best (total) 5-vertex subgraph counter.

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Counting Triangles in Massive Graphs with MapReduce

Cited by 83 publications

References 60 publications

A Space-Efficient Streaming Algorithm for Estimating Transitivity and Triangle Counts Using the Birthday Paradox

A Space-Efficient Streaming Algorithm for Estimating Transitivity and Triangle Counts Using the Birthday Paradox

EGBTER: Capturing Degree Distribution, Clustering Coefficients, and Community Structure in a Single Random Graph Model

Efficiently Counting Vertex Orbits of All 5-vertex Subgraphs, by EVOKE

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