Efficient Graphlet Counting for Large Networks

Ahmed, Nesreen K.; Neville, Jennifer; Rossi, Ryan A.; Duffield, Nick

doi:10.1109/icdm.2015.141

Cited by 181 publications

(212 citation statements)

References 32 publications

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“…Many of our experiments use triangle motifs, which have long been studied for their frequency in social networks [19]. The algorithmic problem of counting or estimating the frequency of various higher-order patterns has also drawn a large amount of attention [1, 6, 12, 23]. …”

Section: Relatedworkmentioning

confidence: 99%

Local Higher-Order Graph Clustering

Yin

Benson

Leskovec

et al. 2017

Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

461

236

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Local graph clustering methods aim to find a cluster of nodes by exploring a small region of the graph. These methods are attractive because they enable targeted clustering around a given seed node and are faster than traditional global graph clustering methods because their runtime does not depend on the size of the input graph. However, current local graph partitioning methods are not designed to account for the higher-order structures crucial to the network, nor can they effectively handle directed networks. Here we introduce a new class of local graph clustering methods that address these issues by incorporating higher-order network information captured by small subgraphs, also called network motifs. We develop the Motif-based Approximate Personalized PageRank (MAPPR) algorithm that finds clusters containing a seed node with minimal motif conductance, a generalization of the conductance metric for network motifs. We generalize existing theory to prove the fast running time (independent of the size of the graph) and obtain theoretical guarantees on the cluster quality (in terms of motif conductance). We also develop a theory of node neighborhoods for finding sets that have small motif conductance, and apply these results to the case of finding good seed nodes to use as input to the MAPPR algorithm. Experimental validation on community detection tasks in both synthetic and real-world networks, shows that our new framework MAPPR outperforms the current edge-based personalized PageRank methodology.

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Section: Relatedworkmentioning

confidence: 99%

Local Higher-Order Graph Clustering

Yin

Benson

Leskovec

et al. 2017

Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

461

236

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: 88%

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)

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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.

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“…The graph G (k) = (V (k) , E (k) ) defined above is called the k-state graph of G. Note that G = (V, E) can also be seen as G (1) = (V (1) , E (1) ) for the case of k = 1.…”

Section: Terminology and Problem Definitionmentioning

confidence: 99%

“…or equivalently, sampling k-subgraphs uniformly at random, which is a challenge by itself. As a result, the problem of uniform sampling k-subgraphs, has been extensively studied in data mining, statistics, and theoretical computer science [1,4,6,7,10,14,22,25,27].In this paper, we study the problem of sampling uniformly at random k-subgraphs from a given input graph. Among the different methodologies that have been proposed, we focus on the Markov chain Monte Carlo (MCMC) approach [20], and in particular on the Metropolis-Hastings algorithm (MH) [11].…”

mentioning

confidence: 99%

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Improved mixing time for k-subgraph sampling

Matsuno¹,

Gionis

2020

Proceedings of the 2020 SIAM International Conference on Data Mining

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Understanding the local structure of a graph provides valuable insights about the underlying phenomena from which the graph has originated. Sampling and examining k-subgraphs is a widely used approach to understand the local structure of a graph. In this paper, we study the problem of sampling uniformly k-subgraphs from a given graph. We analyse a few different Markov chain Monte Carlo (MCMC) approaches, and obtain analytical results on their mixing times, which improve significantly the state of the art. In particular, we improve the bound on the mixing times of the standard MCMC approach, and the state-of-the-art MCMC sampling method PSRW, using the canonical-paths argument. In addition, we propose a novel sampling method, which we call recursive subgraph sampling RSS, and its optimized variant RSS+. The proposed methods, RSS and RSS+, are provably faster than the existing approaches. We conduct experiments and verify the uniformity of samples and the time efficiency of RSS and RSS+. or equivalently, sampling k-subgraphs uniformly at random, which is a challenge by itself. As a result, the problem of uniform sampling k-subgraphs, has been extensively studied in data mining, statistics, and theoretical computer science [1,4,6,7,10,14,22,25,27].In this paper, we study the problem of sampling uniformly at random k-subgraphs from a given input graph. Among the different methodologies that have been proposed, we focus on the Markov chain Monte Carlo (MCMC) approach [20], and in particular on the Metropolis-Hastings algorithm (MH) [11]. The high-level idea is to sample from the stationary distribution of a Markov chain, whose set of states is the set of k-subgraphs, by performing a random walk. The MH algorithm [11] is used to ensure that the stationary distribution, and thus, the sampling, is uniform. An important theoretical question is to upper bound the mixing time of the random walk, which is the time needed for the empirical sampling distribution to be close enough to the stationary distribution.We present improved results for the mixing time of Markov chains designed for uniform sampling of k-subgraphs. Our starting point is the recent work of Bressan et al. [7], who analyze a MCMC method and show that it mixes in timeÕ((k!) 2 ∆ 2k |V | 2 ), where |V | is the number of nodes in the input graph, ∆ is a maximum node degree, andÕ(·) is used to suppress logarithmic and other lower-term factors.Our first result is to analyze the Markov chain with the MH algorithm using the technique of canonical paths [23] and obtain an upper bound on the mixing time ofÕ(k!∆ k (D + k)|V |), where D is a diameter of the graph. Our bound is a significant improvement of the bound of Bressan et al. [7].Next, we proceed to improve this bound even further, by introducing a novel Markov chain to perform the random walk. In particular, we propose a technique based on recursive subgraph sampling (RSS), and a further improvement called RSS+, which exploits the fact that we can easily sample 2-subgraph uniformly at random in time O(1): sam...

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Efficient Graphlet Counting for Large Networks

Cited by 181 publications

References 32 publications

Local Higher-Order Graph Clustering

Local Higher-Order Graph Clustering

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

Improved mixing time for k-subgraph sampling

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