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

A sparse stochastic block model (SBM) with two communities is defined by the community probability $\pi_0,\pi_1$, and the connection probability between communities $a,b\in\{0,1\}$, namely $q_{ab} = \frac{\alpha_{ab}}{n}$. When $q_{ab}$ is constant in $a,b$, the random graph is simply the Erd\H{o}s-R\'{e}ny 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., $d_0\overset{def}{=}\pi_0\alpha_{00}+\pi_1\alpha_{01},d_1\overset{def}{=}\pi_0\alpha_{10}+\pi_1\alpha_{11}$. We focus on the regime $d_0=d_1$ (the regime $d_0\ne d_1$ is trivial). In this regime, there exists $d,\lambda$ and $r\geq 0$ with $\pi_0=\frac{1}{1+r}, \pi_1=\frac{r}{1+r}$, $\alpha_{00}=d(1+r\lambda), \alpha_{01}=\alpha_{10} = d(1-\lambda), \alpha_{11} = d(1+\frac{\lambda}{r})$. We give a consistent estimator of $r$ when $\lambda<0$. The estimator of $\lambda$ 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 $\lambda<0$.