## Electronic Journal of Probability

### Contiguity and non-reconstruction results for planted partition models: the dense case

Debapratim Banerjee

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 18, 28 pp.

Dates
Accepted: 27 November 2017
First available in Project Euclid: 23 February 2018

https://projecteuclid.org/euclid.ejp/1519354947

Digital Object Identifier
doi:10.1214/17-EJP128

Mathematical Reviews number (MathSciNet)
MR3771755

Zentralblatt MATH identifier
1387.05230

Subjects

#### Citation

Banerjee, Debapratim. Contiguity and non-reconstruction results for planted partition models: the dense case. Electron. J. Probab. 23 (2018), paper no. 18, 28 pp. doi:10.1214/17-EJP128. https://projecteuclid.org/euclid.ejp/1519354947

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