Community Detection in Sparse Networks Using the Symmetrized Laplacian Inverse Matrix (SLIM)

Jing, Bing-Yi; Li, Ting; Ying, Ningchen; Yu, Xianshi

doi:10.5705/ss.202020.0094

Cited by 15 publications

(24 citation statements)

References 7 publications

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“…The more homogeneous the degrees are, the more cancellation c −1 n could deal on b n and weaker assumption is needed. One possible remedy for networks with well-clustered node degree sequences, one can estimate for subnetworks consisting of high-and low-degree nodes, respectively, exactly similar to a classical strategy for community detection in sparse networks that simply eliminates high-degree nodes (Jing et al, 2021). But considering many real-world networks, such as those we present in Section 5, do not have very high degree hubs (namely, we can think c −1 n 1), but they have rather continuous degree distribution, the result in Theorem 1's type might be more meaningful than one for the aforementioned divide-and-conquer strategy.…”

Section: Main Theoretical Results (1): Consistencymentioning

confidence: 99%

L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

Wang¹,

Zhang²,

Yan³

et al. 2021

Preprint

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The β-model is a powerful tool for modeling network generation driven by node degree heterogeneity. Its simple yet expressive nature particularly well-suits large and sparse networks, where many network models become infeasible due to computational challenge and observation scarcity. However, existing estimation algorithms for β-model do not scale up; and theoretical understandings remain limited to dense networks. This paper brings several major improvements to the method and theory of β-model to address urgent needs of practical applications. Our contributions include: 1. method: we propose a new 2 penalized MLE scheme; we design a novel algorithm that can comfortably handle sparse networks of millions of nodes, much faster and more memory-parsimonious than any existing algorithm; 2. theory: we present new error bounds on beta-models under much weaker assumptions; we also establish new lower-bounds and new asymptotic normality results; distinct from existing literature, our results cover both small and large regularization scenarios and reveal their distinct asymptotic dependency structures; 3. application: we apply our method to large COVID-19 network data sets and discover meaningful results.

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Section: Main Theoretical Results (1): Consistencymentioning

confidence: 99%

L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

Wang¹,

Zhang²,

Yan³

et al. 2021

Preprint

View full text Add to dashboard Cite

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“…Many methods are well developed to detect communities. In these studies some may focus on the (non-mixed membership) community detection problem in which one node/individual only belongs to one community in a network, such as Holland et al (1983), Jin (2015), Papadopoulos et al (2012), Qin & Rohe (2013), Jing et al (2021). Some may be interested in the mixed membership community detection in which some vertices can belong to many communities (Airoldi et al 2008, Goldenberg et al 2010, Jin et al 2017, Mao et al 2020, Zhang et al 2020, and such case is more realistic.…”

Section: Introductionmentioning

confidence: 99%

Impact of regularization on spectral clustering under the mixed membership stochastic block model

Qing,

Wang

2021

Preprint

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Mixed membership community detection is a challenge problem in network analysis. To estimate the memberships and study the impact of regularized spectral clustering under the mixed membership stochastic block (MMSB) model, this article proposes two efficient spectral clustering approaches based on regularized Laplacian matrix, Simplex Regularized Spectral Clustering (SRSC) and Cone Regularized Spectral Clustering (CRSC). SRSC and CRSC methods are designed based on the ideal simplex structure and the ideal cone structure in the variants of the eigendecomposition of the population regularized Laplacian matrix. We show that these two approaches SRSC and CRSC are asymptotically consistent under mild conditions by providing error bounds for the inferred membership vector of each node under MMSB. Through the theoretical analysis, we give the upper and lower bound for the regularizer τ . By introducing a parametric convergence probability, we can directly see that when τ is large these two methods may still have low error rates but with a smaller probability. Thus we give an empirical optimal choice of τ is O(log(n)) with n the number of nodes to detect sparse networks. The proposed two approaches are successfully applied to synthetic and empirical networks with encouraging results compared with some benchmark methods.

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“…For simplification, a lot of researchers study the networks with assumptions: undirected, unweighted and no-self-loops, such as (Holland et al 1983, Karrer & Newman 2011b, Lancichinetti & Fortunato 2009, Goldenberg & Anna 2010. Some authors consider 'pure' networks in which each node at most belongs to one community/cluster, and in each community the nodes which have similar proprieties or functions are more likely to be linked with each other than random pairs of nodes (Qin & Rohe 2013, Jin 2015, Jing et al 2021. While there are few networks which can be deemed as 'pure' in our real life.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, under DCMM model we extend the symmetric Laplacian inverse matrix (SLIM) method (Jing et al 2021) to mixed membership networks and called the proposed new method as mixed-SLIM. As mentioned in Jing et al (2021), the idea of using the symmetric Laplacian inverse matrix to measure the closeness of nodes comes from the first hitting time in a random walk. Jing et al (2021) combined the SLIM with spectral method based on DCSBM for community detection.…”

Section: Introductionmentioning

confidence: 99%

Estimating mixed-memberships using the Symmetric Laplacian Inverse Matrix

Qing¹,

Wang²

2020

Preprint

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Community detection has been well studied in network analysis, and one popular technique is spectral clustering which is fast and statistically analyzable for detecting clusters for given networks. But the more realistic case of mixed membership community detection remains a challenge. In this paper, we propose a new spectral clustering method Mixed-SLIM for mixed membership community detection. Mixed-SLIM is designed based on the symmetrized Laplacian inverse matrix (SLIM) (Jing et al. 2021) under the degree-corrected mixed membership (DCMM) model. We show that this algorithm and its regularized version Mixed-SLIM τ are asymptotically consistent under mild conditions. Meanwhile, we provide Mixed-SLIM appro and its regularized version Mixed-SLIM τ appro by approximating the SLIM matrix when dealing with large networks in practice. These four Mixed-SLIM methods outperform state-of-art methods in simulations and substantial empirical datasets for both community detection and mixed membership community detection problems.

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Community Detection in Sparse Networks Using the Symmetrized Laplacian Inverse Matrix (SLIM)

Cited by 15 publications

References 7 publications

L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

Impact of regularization on spectral clustering under the mixed membership stochastic block model

Estimating mixed-memberships using the Symmetric Laplacian Inverse Matrix

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