Hitting times in the stochastic block model

Given a large connected graph $G=(V,E)$, and two vertices w,≠ v, let Tw,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 E[Tw,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.