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

We study community recovery in the sparse symmetric stochastic block model with q communities, average degree d and signal strength λ through statistical decision theory and Fisher information, and obtain three characterizations of the Kesten-Stigum threshold dλ2=1 and of the information-computation gap below it. First, on each community-size profile the minimax risk of any class of rules closed under averaging and vertex relabeling equals its Bayes risk under the uniform prior; the posterior mean is the unique Bayes rule and is admissible, and the Bayes risk of degree-D polynomial rules is the trivial risk times 1-CorrD2. Combined with known low-degree and information-theoretic results, this gives the gap as a worst-case statement: for q≥ 5 there is a window below the threshold in which no low-degree rule beats the trivial risk asymptotically, while an exponential-time rule does on a set of labelings of probability 1-o(1). Second, the Fisher information about λ carried by cycle counts is a series with terms of order k(dλ2)k, convergent exactly when dλ2<1; below the threshold the relative error of every unbiased cycle-based estimator of λk stays above an explicit constant, and every cycle-count test has success probability bounded below one. Third, the derivative of belief propagation at its uninformative fixed point multiplies a random perturbation by |λ|√d per iteration, and one EM step taken there leaves λ unchanged. A signal-to-noise computation recovers the condition dλ1/chi>1 of Chin et al. for q=n^chi communities and identifies personalized PageRank as a walk count with suboptimal weights. Experiments on networks with up to 3× 105 vertices confirm the threshold for q=2, the hard window for q=5, and the many-community scaling.