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

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.(2012) 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 Erd{\"o}s-R{\'e}nyi 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.(2012) 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. We prove that their results extend to the case when $a_n=o(n)$ and $b_n=o(n)$. We also consider the case when $\frac{a_n}{n} \to p \in (0,1)$ 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 $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$ and they are asymptotically singular if $(a_n-b_n)^2 > 2(1-p)(a_n+b_n)$. 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 $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$. The results of this paper justify the negative part of a conjecture made in Decelle et al.(2011) for dense graphs.