(The) local law of centered generalized stochastic block models in the sparse regime

Abstract

We consider the centered generalized stochastic block model, especially in sparse regime. We obtain a tighter bound for $T_k$, the row average of the block model's Green function by cumulant expansion. By using our new bound, we prove the local law of cgSBM in sparse regime. Precisely, we prove that when $\phi>1/8$, $1/N$ of the sum of all entries of the Green function is close to the semicircular distribution $m_{sc}$.