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

Abstract

We study community recovery in the sparse symmetric stochastic block model with $q$ communities, average degree $d$ and signal strength $\lambda$ through statistical decision theory and Fisher information, and obtain three characterizations of the Kesten-Stigum threshold $d\lambda^2=1$ 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 $1-\mathrm{Corr}_D^2$. Combined with known low-degree and information-theoretic results, this gives the gap as a worst-case statement: for $q\ge 5$ 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 $\lambda$ carried by cycle counts is a series with terms of order $k(d\lambda^2)^k$, convergent exactly when $d\lambda^2<1$; below the threshold the relative error of every unbiased cycle-based estimator of $\lambda^k$ 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 $|\lambda|\sqrt{d}$ per iteration, and one EM step taken there leaves $\lambda$ unchanged. A signal-to-noise computation recovers the condition $d\lambda^{1/\chi}>1$ 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 $3\times 10^5$ vertices confirm the threshold for $q=2$, the hard window for $q=5$, and the many-community scaling.