Pseudo-likelihood methods for community detection in large sparse networks

Amini, Arash; Chen, Aiyou; Bickel, Peter J.; Levina, Elizaveta

doi:10.1214/13-aos1138

Cited by 333 publications

(389 citation statements)

References 39 publications

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“…Furthermore, we will show thatd 2 12 /d 2 12 P → 1; that is, the two methods do not distinguish between d 2 12 . Interestingly, our results also imply that d 2 11 increases as the graphs become sparser; that is, ρ n decreases.…”

Section: Quality Metricsmentioning

confidence: 82%

Role of normalization in spectral clustering for stochastic blockmodels

Sarkar¹,

Bickel²

2015

Ann. Statist.

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Spectral clustering is a technique that clusters elements using the top few eigenvectors of their (possibly normalized) similarity matrix. The quality of spectral clustering is closely tied to the convergence properties of these principal eigenvectors. This rate of convergence has been shown to be identical for both the normalized and unnormalized variants in recent random matrix theory literature. However, normalization for spectral clustering is commonly believed to be beneficial [Stat. Comput. 17 (2007) 395-416]. Indeed, our experiments show that normalization improves prediction accuracy. In this paper, for the popular stochastic blockmodel, we theoretically show that normalization shrinks the spread of points in a class by a constant fraction under a broad parameter regime. As a byproduct of our work, we also obtain sharp deviation bounds of empirical principal eigenvalues of graphs generated from a stochastic blockmodel.

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Section: Quality Metricsmentioning

confidence: 82%

Role of normalization in spectral clustering for stochastic blockmodels

Sarkar¹,

Bickel²

2015

Ann. Statist.

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“…These networks are much larger than what can easily be analyzed with previous approaches to computing with mixed-membership stochastic blockmodels (10). [Although we note that several efficient methods have recently been developed for blockmodels without overlapping communities (14)(15)(16). ] We analyze a network by setting the number of communities K and running the stochastic inference algorithm.…”

Section: A Study Of Real and Synthetic Networkmentioning

confidence: 99%

“…The first is that many existing community detection algorithms assume that each node belongs to a single community (1,(3)(4)(5)(6)(7)(14)(15)(16). In real-world networks, each node will likely belong to multiple communities and its connections will reflect these multiple memberships (2,(8)(9)(10)(11)(12)(13)17).…”

mentioning

confidence: 99%

“…network analysis | Bayesian statistics | massive data C ommunity detection algorithms (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) analyze networks to find groups of densely connected nodes. These algorithms have become vital to data-driven methods for understanding and exploring network data such as social networks (4), citation networks (18), communication networks (19), and networks induced by scientific observation [e.g., gene regulation networks (20)].…”

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confidence: 99%

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Efficient discovery of overlapping communities in massive networks

Gopalan

Blei

2013

Proc. Natl. Acad. Sci. U.S.A.

220

162

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Detecting overlapping communities is essential to analyzing and exploring natural networks such as social networks, biological networks, and citation networks. However, most existing approaches do not scale to the size of networks that we regularly observe in the real world. In this paper, we develop a scalable approach to community detection that discovers overlapping communities in massive realworld networks. Our approach is based on a Bayesian model of networks that allows nodes to participate in multiple communities, and a corresponding algorithm that naturally interleaves subsampling from the network and updating an estimate of its communities. We demonstrate how we can discover the hidden community structure of several real-world networks, including 3.7 million US patents, 575,000 physics articles from the arXiv preprint server, and 875,000 connected Web pages from the Internet. Furthermore, we demonstrate on large simulated networks that our algorithm accurately discovers the true community structure. This paper opens the door to using sophisticated statistical models to analyze massive networks. Community detection is important for both exploring a network and predicting connections that are not yet observed. For example, by finding the communities in a large citation graph of scientific articles, we can make hypotheses about the fields and subfields that they contain. By finding communities in a large social network, we can more easily make predictions to individual members about who they might be friends with but are not yet connected to.In this paper, we develop an algorithm that discovers communities in modern real-world networks. The challenge is that real-world networks are massive-they can contain hundreds of thousands or even millions of nodes. We will examine a network of scientific articles that contains 575,000 articles, a network of connected Web pages that contains 875,000 pages, and a network of US patents that contains 3,700,000 patents. Most approaches to community detection cannot handle data at this scale.There are two fundamental difficulties to detecting communities in such networks. The first is that many existing community detection algorithms assume that each node belongs to a single community (1,(3)(4)(5)(6)(7)(14)(15)(16). In real-world networks, each node will likely belong to multiple communities and its connections will reflect these multiple memberships (2,(8)(9)(10)(11)(12)(13)17). For example, in a large social network, a member may be connected to coworkers, friends from school, and neighbors. We need algorithms that discover overlapping communities to capture the heterogeneity of each node's connections.The second difficulty is that existing algorithms are too slow. Many community detection algorithms iteratively analyze each pair of nodes, regardless of whether the nodes in the pair are connected in the network (5, 6, 10). Consequently, these algorithms run in time squared in the number of nodes, which makes analyzing massive networks computationally intractable. Other a...

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“…Here we will focus on the following simple way of regularization proposed in [3] and analyzed in [23,18]. Choose τ > 0 and add the same number τ /n to all entries of the adjacency matrix A, thereby replacing it with The following consequence of Theorem 1.1 shows that such regularization indeed forces Laplacian to concentrate.…”

mentioning

confidence: 99%

Concentration and regularization of random graphs

Levina

Vershynin

2017

Random Struct Algorithms

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101

124

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Abstract. This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erdös-Rényi random graphs on n vertices, where edges form independently and possibly with different probabilities pij. Sparse random graphs whose expected degrees are o(log n) fail to concentrate; the obstruction is caused by vertices with abnormally high and low degrees. We show that concentration can be restored if we regularize the degrees of such vertices, and one can do this in various ways. As an example, let us reweight or remove enough edges to make all degrees bounded above by O(d) where d = max npij. Then we show that the resulting adjacency matrix A concentrates with the optimal rate:Similarly, if we make all degrees bounded below by d by adding weight d/n to all edges, then the resulting Laplacian concentrates with the optimal rate:Our approach is based on GrothendieckPietsch factorization, using which we construct a new decomposition of random graphs. We illustrate the concentration results with an application to the community detection problem in the analysis of networks.

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Pseudo-likelihood methods for community detection in large sparse networks

Cited by 333 publications

References 39 publications

Role of normalization in spectral clustering for stochastic blockmodels

Role of normalization in spectral clustering for stochastic blockmodels

Efficient discovery of overlapping communities in massive networks

Concentration and regularization of random graphs

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