Slow Beats Fast at the Kesten-Stigum Threshold: Minimax, Fisher-Information and Belief-Propagation Characterizations of the Information-Computation Gap in Sparse Stochastic Block Models
This study addresses the information-computation gap in community recovery within sparse stochastic block models, where the properties of the Kesten-Stigum threshold remain insufficiently characterized. By integrating statistical decision theory with Fisher information, this work proposes a triple characterization framework encompassing minimax risk equivalence, cycle-counting Fisher information convergence, and EM step-size invariance. Theoretical derivations are conducted using Bayesian risk, low-degree polynomial methods, and belief propagation algorithms. This research rigorously establishes the existence of the threshold for two communities (q=2) and identifies the hard phase window for five communities (q=5), elucidating generalization patterns for multi-community extensions. Furthermore, experiments on networks comprising up to 300,000 nodes validate the accuracy of these theoretical predictions.