The Value of Depth in Message Passing on Sparse Graphs: A Kesten-Stigum Dichotomy

How deep does a graph neural network need to be on a sparse graph? We study its purest statistical form: node classification on the sparse contextual stochastic block model (CSBM) with average degree =O(1), whose local weak limit is a broadcast-labelled Poisson Galton-Watson tree. Prior work derived a message-passing classifier h that aggregates from each vertex at distance k the attenuated evidence 2 artanh(gamma k t(X v)), with gamma the edge signal and t a bounded likelihood-ratio transform of the feature. We prove that the value of depth is governed by a single number, the Kesten-Stigum ratio kappa= gamma squared. Below the threshold (kappa<1), the error sequence is Cauchy at a geometric rate, E-E(') C kappa(+1)/3 for all '>, so all layers beyond depth O((1/epsilon)) change the error by less than epsilon; conversely, under mild regularity each sufficiently deep layer still flips the decision with probability at least c kappa/2, the empirically sharp exponent. Above the threshold (kappa>1), depth is geometrically productive: E is driven to a branching-process floor of order at most 1/(kappa-1) at any geometric rate kappa-s, s<1 (this bound has content only for kappa>17). No local classifier of any depth beats the universal floor e- (-zeta) set by isolated roots (zeta the feature signal-to-noise ratio), while the first layer provably helps by an explicit total-variation amount. Simulations with an exact belief-propagation baseline on the same trees show that the pairwise rule's error curve is mildly non-monotone in, so an optimal finite depth exists (an exact instance is certified in the appendix), while BP saturates strictly faster, at an effective per-layer ratio below kappa that we identify.