Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees

In this paper, we rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborova (2011) regarding the block model, in particular in the case of q = 3 and q = 4 communities. We prove that for q = 3 and q = 4 there is no computational-statistical gap if the average degree is above some constant by showing that it is information theoretically impossible to detect below the Kesten-Stigum bound. We proceed by showing that for the broadcast process on Galton-Watson trees, reconstruction is impossible for q = 3 and q = 4 if the average degree is suficiently large. This improves on the result of Sly (2009), who proved similar results for d-regular trees when q = 3. Our analysis of the critical case q = 4 provides a detailed picture showing that the tightness of the Kesten-Stigum bound in the antiferromagnetic regime depends on the average degree of the tree. We also prove that for q >= 5, the Kesten-Stigum bound is not sharp. Our results prove conjectures of Decelle, Krzakala, Moore, Zdeborova (2011), Moore (2017), Abbe and Sandon (2018) and RicciTersenghi, Semerjian, and Zdeborova (2019). Our proofs are based on a new general coupling of the tree and graph processes and on a refined analysis of the broadcast process on the tree.