Role of normalization in spectral clustering for stochastic blockmodels

Sarkar, Purnamrita; Bickel, Peter J.

doi:10.1214/14-aos1285

Cited by 96 publications

(75 citation statements)

References 26 publications

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“…(3). As such, our analysis complements existing literature that seeks to understand normalization in the context of spectral methods (Sarkar and Bickel, 2015;von Luxburg, 2007). Moreover, our work together with Rubin- Core-periphery network structure, broadly construed, is demonstrably ubiquitous in realworld networks (Csermely et al, 2013;Holme, 2005;Leskovec et al, 2009).…”

Section: Discussionmentioning

confidence: 68%

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On spectral embedding performance and elucidating network structure in stochastic blockmodel graphs

Cape¹,

Tang

Priebe³

2019

Net Sci

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Statistical inference on graphs often proceeds via spectral methods involving lowdimensional embeddings of matrix-valued graph representations, such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic informationtheoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic block model graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g. homogeneity, affinity, core-periphery, (un)balancedness) via a comprehensive treatment of the two-block stochastic block model and the class of K-block models exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, "Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs." We also provide evidence in support of the claim that "adjacency spectral embedding favors core-periphery network structure." 2010 Mathematics Subject Classification: Primary 62H30; Secondary 62B10.The stochastic block model (SBM) (Holland et al., 1983) is a simple yet ubiquitous network model capable of capturing community structure that has been widely studied via spectral methods in the mathematics, statistics, physics, and engineering communities. Each vertex in an n-vertex K-block SBM graph belongs to one of the K blocks (communities), and the probability of any two vertices sharing an edge depends exclusively on the vertices' block assignments (memberships).This paper provides a detailed comparison of two popular spectral embedding procedures by synthesizing recent advances in random graph limit theory. We undertake an extensive investigation of network structure for stochastic block model graphs by considering sub-models exhibiting various functional relationships, symmetries, and geometric properties within the inherent parameter space consisting of block membership probabilities and block edge probabilities. We also provide a collection of figures depicting relative spectral embedding performance as a function of the SBM parameter space for a range of sub-models exhibiting different forms of network structure, specifically homogeneous community structure, affinity structure, core-periphery structure, and (un)balanced block sizes (see Section 5).The rest of this paper is organized as follows.• Section 2 introduces the formal setting considered in this paper and contextualizes this work with respect to the existing statistical network analysis literature.• Section 3 establishes notation, presents the generalized random dot product graph model of which the stochastic block model is a special case, defines the adjacency and Laplacian spectral embeddings, presents the corresponding spectral embedding limit theorems, and specifies the notion of sparsity considered in this paper.• Section 4 motivates and formulates a measure of large-sample relative ...

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Section: Discussionmentioning

confidence: 68%

“…tion, inference, and community detection, including Fishkind et al (2013); Lei and Rinaldo (2015); McSherry (2001); Rohe et al (2011); Sarkar and Bickel (2015); Sussman et al (2014).…”

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confidence: 99%

On spectral embedding performance and elucidating network structure in stochastic blockmodel graphs

Cape¹,

Tang

Priebe³

2019

Net Sci

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“…As we have discussed, the Laplacian is typically preferred over the adjacency matrix in practice, because the variation in node degrees is reduced by the normalization factor D −1/2 [53]. Figure 2 shows the effect of regularization for the Laplacian of a random network generated from G(n, a n , b n ) with n = 50, a = 5 and b = 0.1.…”

Section: Application To Community Detectionmentioning

confidence: 99%

Concentration of Random Graphs and Application to Community Detection

Levina

Vershynin

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Random matrix theory has played an important role in recent work on statistical network analysis. In this paper, we review recent results on regimes of concentration of random graphs around their expectation, showing that dense graphs concentrate and sparse graphs concentrate after regularization. We also review relevant network models that may be of interest to probabilists considering directions for new random matrix theory developments, and random matrix theory tools that may be of interest to statisticians looking to prove properties of network algorithms. Applications of concentration results to the problem of community detection in networks are discussed in detail.2010 Mathematics Subject Classification. 05C80 (primary) and 05C85, 60B20 (secondary).

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“…Various procedures have been proposed to solve this problem in the last decade or so. These include method of moments (Bickel, Chen, and Levina, 2011), modularity maximization (Newman and Girvan, 2004), semidefinite programming (Abbe, Bandeira, and Hall, 2016;Cai and Li, 2015), spectral clustering (Joseph and Yu, 2016;Lei and Rinaldo, 2015;Qin and Rohe, 2013;Rohe, Chatterjee, and Yu, 2011;Sarkar and Bickel, 2015;Vu, 2018;Proutiere, 2014, 2016), likelihood methods (Amini, Chen, Bickel, and Levina, 2013;Bickel and Chen, 2009;Choi, Wolfe, and Airoldi, 2012;Zhao, Levina, and Zhu, 2012), and spectral embedding (Lyzinski, Sussman, Tang, Athreya, and Priebe, 2014;Sussman, Tang, Fishkind, and Priebe, 2012). Abbe (2018) provides an excellent survey on recent developments on community detection and stochastic block models.…”

Section: Introductionmentioning

confidence: 99%

Strong Consistency of Spectral Clustering for Stochastic Block Models

Wang

Zhang

2020

IEEE Trans. Inform. Theory

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In this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models (SBMs). We show that under some weak conditions on the minimal degree, the number of communities, and the eigenvalues of the probability block matrix, the K-means algorithm applied to the eigenvectors of the graph Laplacian associated with its first few largest eigenvalues can classify all individuals into the true community uniformly correctly almost surely. Extensions to both regularized spectral clustering and degree-corrected SBMs are also considered. We illustrate the performance of different methods on simulated networks.

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Role of normalization in spectral clustering for stochastic blockmodels

Cited by 96 publications

References 26 publications

On spectral embedding performance and elucidating network structure in stochastic blockmodel graphs

On spectral embedding performance and elucidating network structure in stochastic blockmodel graphs

Concentration of Random Graphs and Application to Community Detection

Strong Consistency of Spectral Clustering for Stochastic Block Models

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