On the Log Partition Function of Ising Model on Stochastic Block Model

Abstract

A sparse stochastic block model (SBM) with two communities is defined by the community probability $π_0,π_1$, and the connection probability between communities $a,b\in\{0,1\}$, namely $q_{ab} = \frac{α_{ab}}{n}$. When $q_{ab}$ is constant in $a,b$, the random graph is simply the Erdős-Rény random graph. We evaluate the log partition function of the Ising model on sparse SBM with two communities. As an application, we give consistent parameter estimation of the sparse SBM with two communities in a special case. More specifically, let $d_0,d_1$ be the average degree of the two communities, i.e., $d_0\overset{def}{=}π_0α_{00}+π_1α_{01},d_1\overset{def}{=}π_0α_{10}+π_1α_{11}$. We focus on the regime $d_0=d_1$ (the regime $d_0\ne d_1$ is trivial). In this regime, there exists $d,λ$ and $r\geq 0$ with $π_0=\frac{1}{1+r}, π_1=\frac{r}{1+r}$, $α_{00}=d(1+rλ), α_{01}=α_{10} = d(1-λ), α_{11} = d(1+\fracλ{r})$. We give a consistent estimator of $r$ when $λ<0$. The estimator of $λ$ given by \citep{mossel2015reconstruction} is valid in the general situation. We also provide a random clustering algorithm which does not require knowledge of parameters and which is positively correlated with the true community label when $λ<0$.