{"paper":{"title":"The Value of Depth in Message Passing on Sparse Graphs: A Kesten-Stigum Dichotomy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.PR","stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Aseem Raj Baranwal","submitted_at":"2026-07-18T07:11:45Z","abstract_excerpt":"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 $h_\\ell$ that aggregates from each vertex at distance $k\\le\\ell$ the attenuated evidence $2\\operatorname{artanh}(\\gamma^k t(X_v))$, with $\\gamma$ the edge signal and $t$ a bounded likelihood-ratio transform of the feature. We prove that the value of depth is gov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16676","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.16676/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}