Higher-order clustering in networks

Yin, Hao; Benson, Austin R.; Leskovec, Jure

doi:10.1103/physreve.97.052306

Cited by 90 publications

(63 citation statements)

References 47 publications

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“…The formal theory rests on the idea of higher-order clustering coefficients, which we developed in recent work [49] and briefly review below. As an extreme case of our theory, consider a graph with clustering coefficient 1.…”

Section: Motif Conductance Minimizationmentioning

confidence: 99%

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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

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461

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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: Motif Conductance Minimizationmentioning

confidence: 99%

“…al [49]. The classical clustering coefficient is the fraction of wedges (length-2 paths) in the graph that are closed (i.e., induce a 3-clique, or a triangle).…”

Section: Motif Conductance Minimizationmentioning

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

Self Cite

461

236

View full text Add to dashboard Cite

show abstract

“…We also presented an extended variational principle for general subgraphs to investigate the self-averaging properties of clustering. This method can also be applied to investigate higher order clustering [56,6].…”

Section: Discussionmentioning

confidence: 99%

Scale-free network clustering in hyperbolic and other random graphs

Stegehuis¹,

Hofstad²,

Leeuwaarden³

2019

J. Phys. A: Math. Theor.

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Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the largenetwork limit. We also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model.

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“…• Orbits that we can count in time O(W (G) + D(G) + m + n): For θ 66 and θ 70 , we need to enumerate 4-cliques, which takes time O(W (G) + D(G) + m + n) [41]. Getting counts of orbit θ 71 requires enumeration of triangles, but for each triangle t at hand, we need to get K 4 (t), which is overall possible in time O(W (G) + D(G) +m + n).…”

Section: B Getting 5-vocsmentioning

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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Higher-order clustering in networks

Cited by 90 publications

References 47 publications

Local Higher-Order Graph Clustering

Local Higher-Order Graph Clustering

Scale-free network clustering in hyperbolic and other random graphs

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

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