Global and Local Information in Clustering Labeled Block Models

Kanade, Varun; Mossel, Elchanan; Schramm, Tselil

doi:10.1109/tit.2016.2516564

Cited by 22 publications

(27 citation statements)

References 33 publications

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“…Similar results in the case where (2) does not hold have been proved in [8]. In the large degree d regime, our result improves Proposition 3 in [9] which deals only with the case p = 0.5 and λ larger than a large constant C. The fact that local algorithms are very efficient as soon as q > 0 (even optimal in the limit q → 0) leads to linear time algorithms for community detection (when some labels are revealed). Indeed from a practical perspective, we believe that our analysis carries over to the labeled stochastic block model [10], [11].…”

Section: Reconstructability Above the Kesten-stigum Boundsupporting

confidence: 89%

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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Section: Reconstructability Above the Kesten-stigum Boundsupporting

confidence: 89%

Recovering Asymmetric Communities in the Stochastic Block Model

Caltagirone

Lelarge

Leo

2018

IEEE Trans. Netw. Sci. Eng.

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“…A local algorithm is an algorithm that bases the estimate of the spin of a vertex i only on the neighborhood of vertex i of radius t. In general, local algorithms are not able to obtain a success probability (2) larger than zero in the stochastic block model [9], so that an estimator based on a local algorithm does not satisfy (4). However, when a vanishing fraction of vertices reveals their labels, a local algorithm is able to achieve the maximum possible success probability (2) when the parameters of the stochastic block model are above the Kesten Stigum threshold [9].…”

Section: Local Algorithmsmentioning

confidence: 99%

Efficient Inference in Stochastic Block Models With Vertex Labels

Stegehuis¹,

Massoulié²

2020

IEEE Trans. Netw. Sci. Eng.

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We study the stochastic block model with two communities where vertices contain side information in the form of a vertex label. These vertex labels may have arbitrary label distributions, depending on the community memberships. We analyze a linearized version of the popular belief propagation algorithm. We show that this algorithm achieves the highest accuracy possible whenever a certain function of the network parameters has a unique fixed point. Whenever this function has multiple fixed points, the belief propagation algorithm may not perform optimally. We show that increasing the information in the vertex labels may reduce the number of fixed points and hence lead to optimality of belief propagation.

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“…In sparse SBM, only weak recovery is expected, where the goal is to predict labels which have non-trivial correlation with true labels. It was shown in [13], [14] that observing labels of a vanishingly small fraction of nodes randomly does not affect the weak recovery asymptotically but can lead to efficient local algorithms (in which the label prediction at each node depends only on its local neighborhood), whenever recovery is possible. The effect of sampling on exact recovery in SBM with logdegree, i.e., with p = a ln n/n and q = b ln n/n has not been studied.…”

Section: Introductionmentioning

confidence: 99%

Active learning for community detection in stochastic block models

Gadde

Gad

Avestimehr

et al. 2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Abstract-The stochastic block model (SBM) is an important generative model for random graphs in network science and machine learning, useful for benchmarking community detection (or clustering) algorithms. The symmetric SBM generates a graph with 2n nodes which cluster into two equally sized communities. Nodes connect with probability p within a community and q across different communities. We consider the case of p = a ln(n)/n and q = b ln(n)/n. In this case, it was recently shown that recovering the community membership (or label) of every node with high probability (w.h.p.) using only the graph is possible if and only if the Chernoff-Hellinger (CH) divergenceIn this work, we study if, and by how much, community detection below the clustering threshold (i.e. D(a, b) < 1) is possible by querying the labels of a limited number of chosen nodes (i.e., active learning). Our main result is to show that, under certain conditions, sampling the labels of a vanishingly small fraction of nodes (a number sub-linear in n) is sufficient for exact community detection even when D(a, b) < 1. Furthermore, we provide an efficient learning algorithm which recovers the community memberships of all nodes w.h.p. as long as the number of sampled points meets the sufficient condition. We also show that recovery is not possible if the number of observed labels is less than n 1−D(a,b) . The validity of our results is demonstrated through numerical experiments.

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Global and Local Information in Clustering Labeled Block Models

Cited by 22 publications

References 33 publications

Recovering Asymmetric Communities in the Stochastic Block Model

Recovering Asymmetric Communities in the Stochastic Block Model

Efficient Inference in Stochastic Block Models With Vertex Labels

Active learning for community detection in stochastic block models

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