Hitting times in the stochastic block model

Given a large connected graph $G=(V,E)$, and two vertices $w,\neq v$, let $T_{w,v}$ be the first hitting time to $v$ starting from $w$ for the simple random walk on $G$. We prove a general theorem that guarantees, under some assumptions on $G$, to approximate $\mathbb E[T_{w,v}]$ up to $o(1)$ terms. As a corollary, we derive explicit formulas for the stochastic block model with two communities and connectivity parameters $p$ and $q$, and show that the average hitting times, for fixed $v$ and as $w$ varies, concentrates around four possible values. The proof is purely probabilistic and uses a coupling argument.