NDLI: Reconstruction and estimation in the planted partition model

Content Provider	Springer Nature Link
Author	Neeman, Joe Sly, Allan Mossel, Elchanan
Copyright Year	2014
Abstract	The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $$n$$ nodes with two equal-sized clusters, with an between-class edge probability of $$q$$ and a within-class edge probability of $$p$$ . Although most of the literature on this model has focused on the case of increasing degrees (ie. $$pn, qn \rightarrow \infty $$ as $$n \rightarrow \infty $$ ), the sparse case $$p, q = O(1/n)$$ is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $$p = a/n$$ and $$q = b/n$$ , then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if $$(a - b)^2 > 2(a + b)$$ , and impossible if $$(a - b)^2 < 2(a + b)$$ . By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $$(a - b)^2 > C (a + b)$$ for some sufficiently large $$C$$ . We prove half of their prediction, showing that it is indeed impossible to cluster if $$(a - b)^2 < 2(a + b)$$ . Furthermore we show that it is impossible even to estimate the model parameters from the graph when $$(a - b)^2 < 2(a + b)$$ ; on the other hand, we provide a simple and efficient algorithm for estimating $$a$$ and $$b$$ when $$(a - b)^2 > 2(a + b)$$ . Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.
Ending Page	461
Page Count	31
Starting Page	431
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	3
Volume Number	162
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2014-07-20
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Random graphs Mathematical and Computational Biology Statistics for Business/Economics/Mathematical Finance/Insurance Network models, stochastic Theoretical, Mathematical and Computational Physics Operations Research/Decision Theory Applications of branching processes Social networks Probability Theory and Stochastic Processes Quantitative Finance
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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