Minimax Rates of Community Detection in Stochastic Block Models

Zhang, Anderson Y.; Zhou, Harrison H.

doi:10.48550/arxiv.1507.05313

Cited by 5 publications

(27 citation statements)

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“…For brevity we mention here only the results and proofs that differ from the proof contained in Zhang and Zhou (2015) and refer the reader to the aforementioned paper for a complete description of the techniques involved. We define the homogeneous/symmetric multi layer stochastic blockmodel as the MLSBM with the parameter space Θ M L 1 that has all intra-block connection probabilities equal to each other as well as all inter-block connection probabilities equal to each other for each layer.…”

Section: Proofs Of Minimax and Threshold Resultsmentioning

confidence: 99%

“…a (m) N (Zhang and Zhou 2015). Hence the minimax rate of MLSBM is at most that of the aggregate graph.…”

Section: Minimax Rates and Sharp Thresholdsmentioning

confidence: 97%

“…> 0 for all m. Then Corollary 4.1 of Zhang and Zhou (2015) gives that assuming K = N o(1) , the sharp threshold for the existence of a strongly consistent estimator for the mth layer SBM is…”

Section: Sharp Consistency Thresholdsmentioning

confidence: 99%

“…From now on "aggregate SBM" will refer to this sparse model. We compare this baseline aggregate SBM with the multi-layer models, MLSBM and RMLSBM in terms of minimax rates (Zhang and Zhou 2015;Gao et al 2015) and consistency thresholds (Mossel et al 2014;Abbe and Sandon 2015;Hajek et al 2014) in the next section.…”

Section: Baseline Proceduresmentioning

confidence: 99%

“…for all q, l, m. To align notations and settings with Zhang and Zhou (2015), we slightly modify the growth condition on class sizes of Theorem 4 and 5 as N q ∈ [ N sK , sN K ] with s ≥ 1 and redefine the parameter space of our undirected symmetric MLSBM with no self loops as…”

Section: Minimax Rates and Sharp Thresholdsmentioning

confidence: 99%

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Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel

Paul¹,

Chen²

2016

Electron. J. Statist.

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In recent years there has been an increased interest in statistical analysis of data with multiple types of relations among a set of entities. Such multi-relational data can be represented as multi-layer graphs where the set of vertices represents the entities and multiple types of edges represent the different relations among them. For community detection in multi-layer graphs, we consider two random graph models, the multilayer stochastic blockmodel (MLSBM) and a model with a restricted parameter space, the restricted multi-layer stochastic blockmodel (RMLSBM). We derive consistency results for community assignments of the maximum likelihood estimators (MLEs) in both models where MLSBM is assumed to be the true model, and either the number of nodes or the number of types of edges or both grow. We compare MLEs in the two models with other baseline approaches, such as separate modeling of layers, aggregating the layers and majority voting. RMLSBM is shown to have advantage over MLSBM when either the growth rate of the number of communities is high or the growth rate of the average degree of the component graphs in the multi-graph is low. We also derive minimax rates of error and sharp thresholds for achieving consistency of community detection in both models, which are then used to compare the multi-layer models with a baseline model, the aggregate stochastic block model. The simulation studies and real data applications confirm the superior performance of the multi-layer approaches in comparison to the baseline procedures. 2(1− )N R m I (m)

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Section: Proofs Of Minimax and Threshold Resultsmentioning

confidence: 99%

“…a (m) N (Zhang and Zhou 2015). Hence the minimax rate of MLSBM is at most that of the aggregate graph.…”

Section: Minimax Rates and Sharp Thresholdsmentioning

confidence: 97%

Section: Sharp Consistency Thresholdsmentioning

confidence: 99%

Section: Baseline Proceduresmentioning

confidence: 99%

Section: Minimax Rates and Sharp Thresholdsmentioning

confidence: 99%

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Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel

Paul¹,

Chen²

2016

Electron. J. Statist.

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Local Algorithms for Block Models with Side Information

Mossel

2016

Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

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There has been a recent interest in understanding the power of local algorithms for optimization and inference problems on sparse graphs. Gamarnik and Sudan (2014) showed that local algorithms are weaker than global algorithms for finding large independent sets in sparse random regular graphs thus refuting a conjecture by Hatami, Lovász, and Szegedy (2012). Montanari (2015) showed that local algorithms are suboptimal for finding a community with high connectivity in the sparse Erdős-Rényi random graphs. For the symmetric planted partition problem (also named community detection for the block models) on sparse graphs, a simple observation is that local algorithms cannot have non-trivial performance.In this work we consider the effect of side information on local algorithms for community detection under the binary symmetric stochastic block model. In the block model with side information each of the n vertices is labeled + or − independently and uniformly at random; each pair of vertices is connected independently with probability a/n if both of them have the same label or b/n otherwise. The goal is to estimate the underlying vertex labeling given 1) the graph structure and 2) side information in the form of a vertex labeling positively correlated with the true one. Assuming that the ratio between in and out degree a/b is Θ(1) and the average degree (a + b)/2 = n o(1) , we show that a local algorithm, namely, belief propagation run on the local neighborhoods, maximizes the expected fraction of vertices labeled correctly in the following three regimes:• |a − b| < 2 and all 0 < α < 1/2• (a − b) 2 > C(a + b) for some constant C and all 0 < α < 1/2• For all a, b if the probability that each given vertex label is incorrect is at most α * for some constant α * ∈ (0, 1/2).

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Density Evolution in the Degree-correlated Stochastic Block Model

Mossel,

2015

Preprint

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There is a recent surge of interest in identifying the sharp recovery thresholds for cluster recovery under the stochastic block model. In this paper, we address the more refined question of how many vertices that will be misclassified on average. We consider the binary form of the stochastic block model, where n vertices are partitioned into two clusters with edge probability a/n within the first cluster, c/n within the second cluster, and b/n across clusters. Suppose that 1) . When the cluster sizes are balanced and µ = ν, we show that the minimum fraction of misclassified vertices on average is given by Qand Z is standard normal. Moreover, the minimum misclassified fraction on average is attained by a local algorithm, namely belief propagation, in time linear in the number of edges. Our proof techniques are based on connecting the cluster recovery problem to tree reconstruction problems, and analyzing the density evolution of belief propagation on trees with Gaussian approximations.

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Minimax Rates of Community Detection in Stochastic Block Models

Cited by 5 publications

References 0 publications

Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel

Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel

Local Algorithms for Block Models with Side Information

Density Evolution in the Degree-correlated Stochastic Block Model

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