Detectability of macroscopic structures in directed asymmetric stochastic block model

We study the problem of identifying macroscopic structures in networks, characterizing the impact of introducing link directions on the detectability phase transition. To this end, building on the stochastic block model, we construct a class of nontrivially detectable directed networks. We find closed-form solutions by using the belief propagation method, showing how the transition line depends on the assortativity and the asymmetry of the network. Finally, we numerically identify the existence of a hard phase for detection close to the transition point.