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We assume that for each node $1 \\leq i \\leq n$, there is a probability mass function $\\pi_i$ defined over $\\{1, 2, \\ldots, K\\}$ such that \\[ \\pi_i(k) = \\mbox{the weight of node $i$ on community $k$}, \\qquad 1 \\leq k \\leq K. \\] The goal is to estimate $\\{\\pi_i, 1 \\leq i \\leq n\\}$ (i.e., membership estimation).\n  We model the network with the {\\it degree-corrected mixed membership (DCMM)} model \\cite{Mixed-SCORE}. Since for many natural networks, the degrees have an approxi"},"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":"1709.05603","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2017-09-17T03:23:43Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"fbc3ed1a6c2c9bbd21cdc5775108c279aa089e829cae7b7e0274a485473317f5","abstract_canon_sha256":"21ed270b70e13c3b0d1255ab0ece9c2e9da06bcbcc6bda1551a20cf2f2841c67"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:00.160277Z","signature_b64":"34gB6QVG/QL6E3GXe9mRVzxcpFFydfEZbvwjPiKhXmtfu72tQnHAoZ9QkPJwUm7W6Y/83o+/yIwKgdXYTwsCAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ada81e702988d3890d677734545f6ec1e45d523a83b6847341fac5ec43f0413a","last_reissued_at":"2026-05-18T00:35:00.159638Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:00.159638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Sharp Lower Bound for Mixed-membership Estimation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Jiashun Jin, Zheng Tracy Ke","submitted_at":"2017-09-17T03:23:43Z","abstract_excerpt":"Consider an undirected network with $n$ nodes and $K$ perceivable communities, where some nodes may have mixed memberships. 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