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 Delta=O(1), whose local weak limit is a broadcast-labelled Poisson Galton-Watson tree. Prior work derived a message-passing classifier hℓ that aggregates from each vertex at distance k≤ℓ the attenuated evidence 2operatornameartanh(γk t(Xv)), with γ 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=γ2Delta. Below the threshold (kappa<1), the error sequence is Cauchy at a geometric rate, |E(ℓ)-E(ℓ')|≤ Ckappa(ℓ+1)/3 for all ℓ'>ℓ, so all layers beyond depth O(log(1/ε)) change the error by less than ε; conversely, under mild regularity each sufficiently deep layer still flips the decision with probability at least ckappaℓ/2, the empirically sharp exponent. Above the threshold (kappa>1), depth is geometrically productive: E(ℓ) is driven to a branching-process floor of order at most 1/(kappa-1) at any geometric rate kappa-sℓ, s<1 (this bound has content only for kappa>17). No local classifier of any depth beats the universal floor e-DeltaPhi(-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 ℓ, 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.