Solving Linear Systems with Boundary Conditions Using Heat Kernel Pagerank

Chung, Fan; Simpson, Olivia

doi:10.1007/978-3-319-03536-9_16

Cited by 21 publications

(23 citation statements)

References 28 publications

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“…This paper instead focuses on local clustering methods that examine local regions of the network. There are several methods for finding clusters containing a seed node beyond the personalized PageRank method considered here, including other graph diffusions [8, 26], local spectral methods [32, 33], local modularity maximization [10], flow-based algorithms [36], and minimum degree maximization [11, 40]. All of these local methods optimize edge-based criteria to find clusters, whereas we are focused on finding clusters based on motifs.…”

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

446

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

446

236

View full text Add to dashboard Cite

show abstract

“…There are several variations and extensions of PageRank, which can be used to deal with information networks arising in a variety of scenarios. For example, the Kronecker PageRank can be used to treat networks with multiple attributes [14], [30], the connection PageRank is useful for geometrical graphs in high dimensions and the heat kernel PageRank which is expressed as an exponential sum of random walks, leads to improved local partitioning algorithms [12], [13], [16]. Since many real-world information networks are directed graphs, a modified PageRank algorithm for directed graphs can be found in [4].…”

Section: Applications and Generalization Of Pagerankmentioning

confidence: 98%

A Brief Survey of PageRank Algorithms

Chung

2014

IEEE Trans. Netw. Sci. Eng.

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“…Based on a continuous-time Markov chain, the heat kernel diffusion involves the exponential of a generator matrix, which may be approximated via a series of expansion. Chung et al have proposed a local graph partitioning based on the heat kernel diffusion [14,15], and a Monte Carlo algorithm to estimate the heat kernel process [16]. Another approach is described in [27], where the authors estimate the heat kernel diffusion via coordinate relaxation on an implicit linear system; their approach uncovers smaller communities with substantially higher F 1 measures than those found through the personalized PageRank diffusion.…”

Section: Local Community Detectionmentioning

confidence: 99%

Krylov Subspace Approximation for Local Community Detection in Large Networks

Shi

Bindel

et al. 2019

ACM Trans. Knowl. Discov. Data

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Community detection is an important information mining task to uncover modular structures in large networks. For increasingly common large network data sets, global community detection is prohibitively expensive, and attention has shifted to methods that mine local communities, i.e. identifying all latent members of a particular community from a few labeled seed members. To address such semi-supervised mining task, we systematically develop a local spectral subspace-based community detection method, called LOSP. We define a family of local spectral subspaces based on Krylov subspaces, and seek a sparse indicator for the target community via an ℓ 1 norm minimization over the Krylov subspace. Variants of LOSP depend on type of random walks with different diffusion speeds, type of random walks, dimension of the local spectral subspace and step of diffusions. The effectiveness of the proposed LOSP approach is theoretically analyzed based on Rayleigh quotients, and it is experimentally verified on a wide variety of real-world networks across social, production and biological domains, as well as on an extensive set of synthetic LFR benchmark datasets.

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Solving Linear Systems with Boundary Conditions Using Heat Kernel Pagerank

Cited by 21 publications

References 28 publications

Local Higher-Order Graph Clustering

Local Higher-Order Graph Clustering

A Brief Survey of PageRank Algorithms

Krylov Subspace Approximation for Local Community Detection in Large Networks

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