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2018 Contiguity and non-reconstruction results for planted partition models: the dense case
Debapratim Banerjee
Electron. J. Probab. 23: 1-28 (2018). DOI: 10.1214/17-EJP128

Abstract

We consider the two block stochastic block model on $n$ nodes with asymptotically equal cluster sizes. The connection probabilities within and between cluster are denoted by $p_n:=\frac{a_n} {n}$ and $q_n:=\frac{b_n} {n}$ respectively. Mossel et al. [27] considered the case when $a_n=a$ and $b_n=b$ are fixed. They proved the probability models of the stochastic block model and that of Erdős–Rényi graph with same average degree are mutually contiguous whenever $(a-b)^2<2(a+b)$ and are asymptotically singular whenever $(a-b)^2>2(a+b)$. Mossel et al. [27] also proved that when $(a-b)^2<2(a+b)$ no algorithm is able to find an estimate of the labeling of the nodes which is positively correlated with the true labeling. It is natural to ask what happens when $a_n$ and $b_n$ both grow to infinity. In this paper we consider the case when $a_{n} \to \infty $, $\frac{a_n} {n} \to p \in [0,1)$ and $(a_n-b_n)^2= \Theta (a_n+b_n)$. Observe that in this case $\frac{b_n} {n} \to p$ also. We show that here the models are mutually contiguous if asymptotically $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$ and they are asymptotically singular if asymptotically $(a_n-b_n)^2 > 2(1-p)(a_n+b_n)$. Further we also prove it is impossible find an estimate of the labeling of the nodes which is positively correlated with the true labeling whenever $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$ asymptotically. The results of this paper justify the negative part of a conjecture made in Decelle et al. (2011) [17] for dense graphs.

Citation

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Debapratim Banerjee. "Contiguity and non-reconstruction results for planted partition models: the dense case." Electron. J. Probab. 23 1 - 28, 2018. https://doi.org/10.1214/17-EJP128

Information

Received: 22 November 2016; Accepted: 27 November 2017; Published: 2018
First available in Project Euclid: 23 February 2018

zbMATH: 1387.05230
MathSciNet: MR3771755
Digital Object Identifier: 10.1214/17-EJP128

Subjects:
Primary: 05C80

Keywords: Community detection , linear statistics , phase transition , planted partition model , random network , Stochastic block model , threshold

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