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Specifically, consider the symmetric stochastic block model with $q$ groups, average degree $d$, and connection probabilities $c_\\text{in}/n$ and $c_\\text{out}/n$ for within-group and between-group edges respectively; let $\\lambda = (c_\\text{in}-c_\\text{out})/(qd)$. We show that, when $q$ is large, and $\\lambda = O(1/q)$, the critical value of $d$ at which community detection becomes possible---in physical terms, the condensation threshold---is \\[ d_\\text{c} = \\Theta\\!\\l"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1607.01760","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-07-06T19:42:06Z","cross_cats_sorted":["cond-mat.stat-mech","cs.CC","cs.IT","cs.SI","math.IT"],"title_canon_sha256":"d6126ed9aa988d677b07790a601401039c1d635904281706bbb765cdeac27b3d","abstract_canon_sha256":"6583d7b46db2840ff5a6d385227653dcb2a0bc88e84a1c27f8121319c0d95bff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:25.553910Z","signature_b64":"Bloktf27iUX6Fys5bFi84wysNUivplicp3xL6BNCSLNyPpk1XRo46K4h5f0/Wg4Udo47JotY0lTXrTaDMDLXDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"310eed39e83bd2ec82c189f5f3794ba16cbf6bcfb2bd2ff11603f69fdd02ef40","last_reissued_at":"2026-05-18T01:11:25.553156Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:25.553156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Information-theoretic thresholds for community detection in sparse networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","cs.CC","cs.IT","cs.SI","math.IT"],"primary_cat":"math.PR","authors_text":"Cristopher Moore, Jess Banks, Joe Neeman, Praneeth Netrapalli","submitted_at":"2016-07-06T19:42:06Z","abstract_excerpt":"We give upper and lower bounds on the information-theoretic threshold for community detection in the stochastic block model. 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