Density Evolution in the Degree-correlated Stochastic Block Model

Mossel, Elchanan; Xu, Jiaming

doi:10.48550/arxiv.1509.03281

Cited by 3 publications

(8 citation statements)

References 41 publications

(91 reference statements)

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“…Belief propagation, as an algorithm for inference on graphical models, was also applied in community detection by researchers [34,56,55,57,58]. We refer the reader to Yedidia et al [85] for a tutorial introduction to the classical belief propagation method for graphical models such as Bayesian networks and Markov random fields.…”

Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

“…We refer the reader to Yedidia et al [85] for a tutorial introduction to the classical belief propagation method for graphical models such as Bayesian networks and Markov random fields. We now focus on a specific belief propagation algorithm for community detection proposed by Mossel and Xu [57].…”

Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

“…Belief propagation is an alternative approach to assigning community labels approximately but efficiently given the parameters in the SBM. Mossel and Xu [57] proposed a belief propagation algorithm for the SBM with two communities and known (π, P ). When the parameters are unknown, the algorithm can be used as the E step in the EM algorithm.…”

Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

“…Definition 1 (Definition 3.1 in Mossel and Xu [57]). For a node u, denote by (T u , c) the following Poisson two-type branching process tree rooted at u, where c are the labels of the nodes of T u .…”

Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

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A survey on theoretical advances of community detection in networks

Zhao

2017

WIREs Computational Stats

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Real-world networks usually have community structure, that is, nodes are grouped into densely connected communities. Community detection is one of the most popular and best-studied research topics in network science and has attracted attention in many different fields, including computer science, statistics, social sciences, among others. Numerous approaches for community detection have been proposed in literature, from ad hoc algorithms to systematic model-based approaches. The large number of available methods leads to a fundamental question: whether a certain method can provide consistent estimates of community labels. The stochastic blockmodel (SBM) and its variants provide a convenient framework for the study of such problems. This article is a survey on the recent theoretical advances of community detection. The authors review a number of community detection methods and their theoretical properties, including graph cut methods, profile likelihoods, the pseudo-likelihood method, the variational method, belief propagation, spectral clustering, and semidefinite relaxations of the SBM. The authors also briefly discuss other research topics in community detection such as robust community detection, community detection with nodal covariates and model selection, as well as suggest a few possible directions for future research.

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Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

Section: Pseudo-likelihood Variational Methods and Belief Prop-agationmentioning

confidence: 99%

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A survey on theoretical advances of community detection in networks

Zhao

2017

WIREs Computational Stats

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“…It is further assumed that as n → ∞: a, b → ∞ such that a−b √ b = µ, for a fixed positive constant µ and that the average degree (a+b) 2 = n o (1) . The latter condition is crucial in our analysis, by enabling the approximation of the neighborhood of a given node in the graph by a tree [20], [33].…”

Section: System Modelsmentioning

confidence: 99%

EXIT Analysis for Community Detection

Saad¹,

Nosratinia²

2019

Preprint

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This paper employs the extrinsic information transfer (EXIT) method, a technique imported from the analysis of the iterative decoding of error control codes, to study the performance of belief propagation in community detection in the presence of side information. We consider both the detection of a single (hidden) community, as well as the problem of identifying two symmetric communities. For single community detection, this paper demonstrates the suitability of EXIT to predict the asymptotic phase transition for weak recovery. More importantly, EXIT analysis is leveraged to produce useful insights such as the performance of belief propagation near the threshold. For two symmetric communities, the asymptotic residual error for belief propagation is calculated under finite-alphabet side information, generalizing a previous result with noisy labels. EXIT analysis is used to illuminate the effect of side information on community detection, its relative importance depending on the correlation of the graphical information with node labels, as well as the effect of side information on residual errors.

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Recovering Asymmetric Communities in the Stochastic Block Model

Caltagirone

Lelarge

Leo

2018

IEEE Trans. Netw. Sci. Eng.

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We consider the sparse stochastic block model in the case where the degrees are uninformative. The case where the two communities have approximately the same size has been extensively studied and we concentrate here on the community detection problem in the case of unbalanced communities. In this setting, spectral algorithms based on the non-backtracking matrix are known to solve the community detection problem (i.e. do strictly better than a random guess) when the signal is sufficiently large namely above the so-called Kesten Stigum threshold. In this regime and when the average degree tends to infinity, we show that if the community of a vanishing fraction of the vertices is revealed, then a local algorithm (belief propagation) is optimal down to Kesten Stigum threshold and we quantify explicitly its performance. Below the Kesten Stigum threshold, we show that, in the large degree limit, there is a second threshold called the spinodal curve below which, the community detection problem is not solvable. The spinodal curve is equal to the Kesten Stigum threshold when the fraction of vertices in the smallest community is above p * = 1 2 − 1 2 √ 3 , so that the Kesten Stigum threshold is the threshold for solvability of the community detection in this case. However when the smallest community is smaller than p * , the spinodal curve only provides a lower bound on the threshold for solvability. In the regime below the Kesten Stigum bound and above the spinodal curve, we also characterize the performance of best local algorithms as a function of the fraction of revealed vertices. Our proof relies on a careful analysis of the associated reconstruction problem on trees which might be of independent interest. In particular, we show that the spinodal curve corresponds to the reconstruction threshold on the tree.

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Density Evolution in the Degree-correlated Stochastic Block Model

Cited by 3 publications

References 41 publications

A survey on theoretical advances of community detection in networks

A survey on theoretical advances of community detection in networks

EXIT Analysis for Community Detection

Recovering Asymmetric Communities in the Stochastic Block Model

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