{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HKIXXO5ZBLN2E5VWYSS3IVWVN2","short_pith_number":"pith:HKIXXO5Z","canonical_record":{"source":{"id":"1602.01016","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SI","submitted_at":"2016-02-02T17:23:59Z","cross_cats_sorted":["cs.DS","physics.soc-ph"],"title_canon_sha256":"6db27e048874ede462468b01b71e4d6b71e6b1e3a4b0cd963bf3ba01fe97561b","abstract_canon_sha256":"e9db40a921898ec856f96fa762f3e7e3b5cd0ec5986421eae3fb84e585af3f80"},"schema_version":"1.0"},"canonical_sha256":"3a917bbbb90adba276b6c4a5b456d56e8005d15d9fd9165ca29c742ec7e3241c","source":{"kind":"arxiv","id":"1602.01016","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.01016","created_at":"2026-05-18T01:21:23Z"},{"alias_kind":"arxiv_version","alias_value":"1602.01016v1","created_at":"2026-05-18T01:21:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.01016","created_at":"2026-05-18T01:21:23Z"},{"alias_kind":"pith_short_12","alias_value":"HKIXXO5ZBLN2","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HKIXXO5ZBLN2E5VW","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HKIXXO5Z","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HKIXXO5ZBLN2E5VWYSS3IVWVN2","target":"record","payload":{"canonical_record":{"source":{"id":"1602.01016","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SI","submitted_at":"2016-02-02T17:23:59Z","cross_cats_sorted":["cs.DS","physics.soc-ph"],"title_canon_sha256":"6db27e048874ede462468b01b71e4d6b71e6b1e3a4b0cd963bf3ba01fe97561b","abstract_canon_sha256":"e9db40a921898ec856f96fa762f3e7e3b5cd0ec5986421eae3fb84e585af3f80"},"schema_version":"1.0"},"canonical_sha256":"3a917bbbb90adba276b6c4a5b456d56e8005d15d9fd9165ca29c742ec7e3241c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:23.982234Z","signature_b64":"D7S7R/zHbGC4mBffISEHBOTPP4fS12h8bmdaiMAKBniLH6UmSM3ITwgpkZGPH6oGIepH+uK+Rwqyf1PEQb5oAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a917bbbb90adba276b6c4a5b456d56e8005d15d9fd9165ca29c742ec7e3241c","last_reissued_at":"2026-05-18T01:21:23.981610Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:23.981610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.01016","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:21:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rf73KvPx1Rc9PNGTCrzfTutmq+dG5Q2iPgrj2ddhpxfPFa6y+9z1s5k79vRq0i6hH2nm23bwOam520aLn3jyAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:05:06.954532Z"},"content_sha256":"79a3dcabe1d5a189fd905f6b9815a2d2ef16a9d14c66fb9e6d78e0f5197b61ab","schema_version":"1.0","event_id":"sha256:79a3dcabe1d5a189fd905f6b9815a2d2ef16a9d14c66fb9e6d78e0f5197b61ab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HKIXXO5ZBLN2E5VWYSS3IVWVN2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Network Clustering via Maximizing Modularity: Approximation Algorithms and Theoretical Limits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","physics.soc-ph"],"primary_cat":"cs.SI","authors_text":"My T. Thai, Thang N. Dinh, Xiang Li","submitted_at":"2016-02-02T17:23:59Z","abstract_excerpt":"Many social networks and complex systems are found to be naturally divided into clusters of densely connected nodes, known as community structure (CS). Finding CS is one of fundamental yet challenging topics in network science. One of the most popular classes of methods for this problem is to maximize Newman's modularity. However, there is a little understood on how well we can approximate the maximum modularity as well as the implications of finding community structure with provable guarantees. In this paper, we settle definitely the approximability of modularity clustering, proving that appr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01016","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:21:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NsroSR4uA30O8W5ZZRVSun9SlapkyAHsoOJqQLcSKEGb8kkscmtX8KxzovAYpiPyyQ1E5xIzakgyZdjZkdc8Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:05:06.955154Z"},"content_sha256":"3c95dc5b6cd7789f0f6a01a9d3f2039b3fe2175f72d28ecbaca769ff6bf9cec8","schema_version":"1.0","event_id":"sha256:3c95dc5b6cd7789f0f6a01a9d3f2039b3fe2175f72d28ecbaca769ff6bf9cec8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HKIXXO5ZBLN2E5VWYSS3IVWVN2/bundle.json","state_url":"https://pith.science/pith/HKIXXO5ZBLN2E5VWYSS3IVWVN2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HKIXXO5ZBLN2E5VWYSS3IVWVN2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T13:05:06Z","links":{"resolver":"https://pith.science/pith/HKIXXO5ZBLN2E5VWYSS3IVWVN2","bundle":"https://pith.science/pith/HKIXXO5ZBLN2E5VWYSS3IVWVN2/bundle.json","state":"https://pith.science/pith/HKIXXO5ZBLN2E5VWYSS3IVWVN2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HKIXXO5ZBLN2E5VWYSS3IVWVN2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HKIXXO5ZBLN2E5VWYSS3IVWVN2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e9db40a921898ec856f96fa762f3e7e3b5cd0ec5986421eae3fb84e585af3f80","cross_cats_sorted":["cs.DS","physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SI","submitted_at":"2016-02-02T17:23:59Z","title_canon_sha256":"6db27e048874ede462468b01b71e4d6b71e6b1e3a4b0cd963bf3ba01fe97561b"},"schema_version":"1.0","source":{"id":"1602.01016","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.01016","created_at":"2026-05-18T01:21:23Z"},{"alias_kind":"arxiv_version","alias_value":"1602.01016v1","created_at":"2026-05-18T01:21:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.01016","created_at":"2026-05-18T01:21:23Z"},{"alias_kind":"pith_short_12","alias_value":"HKIXXO5ZBLN2","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HKIXXO5ZBLN2E5VW","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HKIXXO5Z","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:3c95dc5b6cd7789f0f6a01a9d3f2039b3fe2175f72d28ecbaca769ff6bf9cec8","target":"graph","created_at":"2026-05-18T01:21:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Many social networks and complex systems are found to be naturally divided into clusters of densely connected nodes, known as community structure (CS). Finding CS is one of fundamental yet challenging topics in network science. One of the most popular classes of methods for this problem is to maximize Newman's modularity. However, there is a little understood on how well we can approximate the maximum modularity as well as the implications of finding community structure with provable guarantees. In this paper, we settle definitely the approximability of modularity clustering, proving that appr","authors_text":"My T. Thai, Thang N. Dinh, Xiang Li","cross_cats":["cs.DS","physics.soc-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SI","submitted_at":"2016-02-02T17:23:59Z","title":"Network Clustering via Maximizing Modularity: Approximation Algorithms and Theoretical Limits"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01016","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:79a3dcabe1d5a189fd905f6b9815a2d2ef16a9d14c66fb9e6d78e0f5197b61ab","target":"record","created_at":"2026-05-18T01:21:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e9db40a921898ec856f96fa762f3e7e3b5cd0ec5986421eae3fb84e585af3f80","cross_cats_sorted":["cs.DS","physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SI","submitted_at":"2016-02-02T17:23:59Z","title_canon_sha256":"6db27e048874ede462468b01b71e4d6b71e6b1e3a4b0cd963bf3ba01fe97561b"},"schema_version":"1.0","source":{"id":"1602.01016","kind":"arxiv","version":1}},"canonical_sha256":"3a917bbbb90adba276b6c4a5b456d56e8005d15d9fd9165ca29c742ec7e3241c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a917bbbb90adba276b6c4a5b456d56e8005d15d9fd9165ca29c742ec7e3241c","first_computed_at":"2026-05-18T01:21:23.981610Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:23.981610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D7S7R/zHbGC4mBffISEHBOTPP4fS12h8bmdaiMAKBniLH6UmSM3ITwgpkZGPH6oGIepH+uK+Rwqyf1PEQb5oAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:23.982234Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.01016","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:79a3dcabe1d5a189fd905f6b9815a2d2ef16a9d14c66fb9e6d78e0f5197b61ab","sha256:3c95dc5b6cd7789f0f6a01a9d3f2039b3fe2175f72d28ecbaca769ff6bf9cec8"],"state_sha256":"6a4111cd0bd038e898770c36cf0e748371fbdb3b8d1c05581862d2a45cf6d197"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PumAZDwVyMo3JH7PY0K67v5O93tNfJYM5NSmVjWqQarHhHKz/coceAIYQfnKqtjLiTDq3pCBb/ykxvXdOZyOCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:05:06.957561Z","bundle_sha256":"9afec377da7166c3ca869b562b0fdc4103d77ebac7ce24be7f00d7ef6007654e"}}