Low-degree evidence for computational transition of recovery rate in stochastic block model
Jingqiu Ding, Yiding Hua, Lucas Slot, David Steurer
We investigate implications of the (extended) low-degree conjecture (recently formalized in [moitra et al2023]) in the context of the symmetric stochastic block model. Assuming the conjecture holds, we establish that no polynomial-time algorithm can weakly recover community labels below the Kesten-Stigum (KS) threshold. In particular, we rule out polynomial-time estimators that, with constant probability, achieve $n^{-0.49}$ correlation with the true communities. Whereas, above the KS threshold, polynomial-time algorithms are known to achieve constant correlation with the true communities with high probability  [massoulie et al 2014,abbe et al 2015]. To our knowledge, we provide the first rigorous evidence for such sharp transition in recovery rate for polynomial-time algorithms at the KS threshold. Notably, under a stronger version of the low-degree conjecture, our lower bound remains valid even when the number of blocks diverges. Furthermore, our results provide evidence of a computational-to-statistical gap in learning the parameters of stochastic block models.In contrast, prior work either (i) rules out polynomial-time algorithms with $1 - o(1)$ success probability [Hopkins 18, bandeira et al 2021] under the low-degree conjecture, or (ii) degree-$\text{poly}(k)$ polynomials for learning the stochastic block model [Luo et al 2023].For this, we design a hypothesis test which succeeeds with constant probability under symmetric stochastic block model, and $1-o(1)$ probability under the distribution of \Erdos \Renyi random graphs.Our proof combines low-degree lower bounds from [Hopkins 18, bandeira et al 2021]  with graph splitting and cross-validation techniques. In order to rule out general recovery algorithms, we employ the correlation preserving projection method developed in [Hopkins et al 17].