Density Evolution in the Degree-correlated Stochastic Block Model

Elchanan Mossel, Jiaming Xu
29th Annual Conference on Learning Theory, PMLR 49:1319-1356, 2016.

Abstract

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 as n \to ∞, a= b+ μ\sqrt b , c=b+ ν\sqrt b for two fixed constants μ, ν, and b \to ∞with b=n^o(1). When the cluster sizes are balanced and μ≠ν, we show that the minimum fraction of misclassified vertices on average is given by Q(\sqrtv^*), where Q(x) is the Q-function for standard normal, v^* is the unique fixed point of v= \frac(μ-ν)^216 + \frac (μ+ν)^2 16 \mathbbE[ \tanh(v+ \sqrtv Z)], and 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v49-mossel16, title = {Density Evolution in the Degree-correlated Stochastic Block Model}, author = {Mossel, Elchanan and Xu, Jiaming}, booktitle = {29th Annual Conference on Learning Theory}, pages = {1319--1356}, year = {2016}, editor = {Feldman, Vitaly and Rakhlin, Alexander and Shamir, Ohad}, volume = {49}, series = {Proceedings of Machine Learning Research}, address = {Columbia University, New York, New York, USA}, month = {23--26 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v49/mossel16.pdf}, url = {https://proceedings.mlr.press/v49/mossel16.html}, abstract = {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 as n \to ∞, a= b+ μ\sqrt b , c=b+ ν\sqrt b for two fixed constants μ, ν, and b \to ∞with b=n^o(1). When the cluster sizes are balanced and μ≠ν, we show that the minimum fraction of misclassified vertices on average is given by Q(\sqrtv^*), where Q(x) is the Q-function for standard normal, v^* is the unique fixed point of v= \frac(μ-ν)^216 + \frac (μ+ν)^2 16 \mathbbE[ \tanh(v+ \sqrtv Z)], and 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. } }
Endnote
%0 Conference Paper %T Density Evolution in the Degree-correlated Stochastic Block Model %A Elchanan Mossel %A Jiaming Xu %B 29th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2016 %E Vitaly Feldman %E Alexander Rakhlin %E Ohad Shamir %F pmlr-v49-mossel16 %I PMLR %P 1319--1356 %U https://proceedings.mlr.press/v49/mossel16.html %V 49 %X 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 as n \to ∞, a= b+ μ\sqrt b , c=b+ ν\sqrt b for two fixed constants μ, ν, and b \to ∞with b=n^o(1). When the cluster sizes are balanced and μ≠ν, we show that the minimum fraction of misclassified vertices on average is given by Q(\sqrtv^*), where Q(x) is the Q-function for standard normal, v^* is the unique fixed point of v= \frac(μ-ν)^216 + \frac (μ+ν)^2 16 \mathbbE[ \tanh(v+ \sqrtv Z)], and 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.
RIS
TY - CPAPER TI - Density Evolution in the Degree-correlated Stochastic Block Model AU - Elchanan Mossel AU - Jiaming Xu BT - 29th Annual Conference on Learning Theory DA - 2016/06/06 ED - Vitaly Feldman ED - Alexander Rakhlin ED - Ohad Shamir ID - pmlr-v49-mossel16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 49 SP - 1319 EP - 1356 L1 - http://proceedings.mlr.press/v49/mossel16.pdf UR - https://proceedings.mlr.press/v49/mossel16.html AB - 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 as n \to ∞, a= b+ μ\sqrt b , c=b+ ν\sqrt b for two fixed constants μ, ν, and b \to ∞with b=n^o(1). When the cluster sizes are balanced and μ≠ν, we show that the minimum fraction of misclassified vertices on average is given by Q(\sqrtv^*), where Q(x) is the Q-function for standard normal, v^* is the unique fixed point of v= \frac(μ-ν)^216 + \frac (μ+ν)^2 16 \mathbbE[ \tanh(v+ \sqrtv Z)], and 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. ER -
APA
Mossel, E. & Xu, J.. (2016). Density Evolution in the Degree-correlated Stochastic Block Model. 29th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 49:1319-1356 Available from https://proceedings.mlr.press/v49/mossel16.html.

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