{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:S5N5OI3HH5PKHTK3F6D6NMHLGT","short_pith_number":"pith:S5N5OI3H","canonical_record":{"source":{"id":"1207.2834","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-12T03:21:02Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"ef638cf7b4741e2e462fcdecb69418b48f1d1a1a66a817630e8e426ac0ee8a3d","abstract_canon_sha256":"b3350e3ba9c8e5e6b84dac9b4406e0c3ede7a52cc7ac498578ecc206e71c92c4"},"schema_version":"1.0"},"canonical_sha256":"975bd723673f5ea3cd5b2f87e6b0eb34d4b3ca2c6c2b152796b656ce2976343a","source":{"kind":"arxiv","id":"1207.2834","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2834","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2834v4","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2834","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"pith_short_12","alias_value":"S5N5OI3HH5PK","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"S5N5OI3HH5PKHTK3","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"S5N5OI3H","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:S5N5OI3HH5PKHTK3F6D6NMHLGT","target":"record","payload":{"canonical_record":{"source":{"id":"1207.2834","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-12T03:21:02Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"ef638cf7b4741e2e462fcdecb69418b48f1d1a1a66a817630e8e426ac0ee8a3d","abstract_canon_sha256":"b3350e3ba9c8e5e6b84dac9b4406e0c3ede7a52cc7ac498578ecc206e71c92c4"},"schema_version":"1.0"},"canonical_sha256":"975bd723673f5ea3cd5b2f87e6b0eb34d4b3ca2c6c2b152796b656ce2976343a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:59.400451Z","signature_b64":"LvaF4vH7iLCsHw435cmf7+X5srJiWRUvpKnjoSMz34/gexjvlc+mccQLZQehxKSGglng4G+AMzotczRrLcqmDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"975bd723673f5ea3cd5b2f87e6b0eb34d4b3ca2c6c2b152796b656ce2976343a","last_reissued_at":"2026-05-18T03:25:59.399927Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:59.399927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.2834","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:25:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"68/PDrBTGVa3vkJXyJFp78hjF6wDYzvEcYRHDAuennJT49+P1jL7apPAt1BxF+Jsn6r0k/mpXJ3DR63XPbuJDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:02:55.365671Z"},"content_sha256":"1d95e68cf5286cc013c62040a36c94f3ab22912d27e415a8396f973f4fcef22d","schema_version":"1.0","event_id":"sha256:1d95e68cf5286cc013c62040a36c94f3ab22912d27e415a8396f973f4fcef22d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:S5N5OI3HH5PKHTK3F6D6NMHLGT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homologies of path complexes and digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Alexander Grigor'yan, Shing-Tung Yau, Yong Lin, Yuri Muranov","submitted_at":"2012-07-12T03:21:02Z","abstract_excerpt":"In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the path homology and cohomology of a digraph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2834","kind":"arxiv","version":4},"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":"2026-05-18T03:25:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nEzh8UzXrApZ+9ywbmIgdA8YtlGvgKIQujSH1nC/dKutB7ml/PcAzqI2u4/CGCsDOJeIu0q6Rz0cXuEzreORDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:02:55.366310Z"},"content_sha256":"7ae3dc38285c4bfcf224026f135a1d24df1c767fa01acb41482c138498312d87","schema_version":"1.0","event_id":"sha256:7ae3dc38285c4bfcf224026f135a1d24df1c767fa01acb41482c138498312d87"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S5N5OI3HH5PKHTK3F6D6NMHLGT/bundle.json","state_url":"https://pith.science/pith/S5N5OI3HH5PKHTK3F6D6NMHLGT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S5N5OI3HH5PKHTK3F6D6NMHLGT/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":"2026-05-30T13:02:55Z","links":{"resolver":"https://pith.science/pith/S5N5OI3HH5PKHTK3F6D6NMHLGT","bundle":"https://pith.science/pith/S5N5OI3HH5PKHTK3F6D6NMHLGT/bundle.json","state":"https://pith.science/pith/S5N5OI3HH5PKHTK3F6D6NMHLGT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S5N5OI3HH5PKHTK3F6D6NMHLGT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:S5N5OI3HH5PKHTK3F6D6NMHLGT","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":"b3350e3ba9c8e5e6b84dac9b4406e0c3ede7a52cc7ac498578ecc206e71c92c4","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-12T03:21:02Z","title_canon_sha256":"ef638cf7b4741e2e462fcdecb69418b48f1d1a1a66a817630e8e426ac0ee8a3d"},"schema_version":"1.0","source":{"id":"1207.2834","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2834","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2834v4","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2834","created_at":"2026-05-18T03:25:59Z"},{"alias_kind":"pith_short_12","alias_value":"S5N5OI3HH5PK","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"S5N5OI3HH5PKHTK3","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"S5N5OI3H","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:7ae3dc38285c4bfcf224026f135a1d24df1c767fa01acb41482c138498312d87","target":"graph","created_at":"2026-05-18T03:25:59Z","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":"In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the path homology and cohomology of a digraph.","authors_text":"Alexander Grigor'yan, Shing-Tung Yau, Yong Lin, Yuri Muranov","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-12T03:21:02Z","title":"Homologies of path complexes and digraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2834","kind":"arxiv","version":4},"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":"sha256:1d95e68cf5286cc013c62040a36c94f3ab22912d27e415a8396f973f4fcef22d","target":"record","created_at":"2026-05-18T03:25:59Z","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":"b3350e3ba9c8e5e6b84dac9b4406e0c3ede7a52cc7ac498578ecc206e71c92c4","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-12T03:21:02Z","title_canon_sha256":"ef638cf7b4741e2e462fcdecb69418b48f1d1a1a66a817630e8e426ac0ee8a3d"},"schema_version":"1.0","source":{"id":"1207.2834","kind":"arxiv","version":4}},"canonical_sha256":"975bd723673f5ea3cd5b2f87e6b0eb34d4b3ca2c6c2b152796b656ce2976343a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"975bd723673f5ea3cd5b2f87e6b0eb34d4b3ca2c6c2b152796b656ce2976343a","first_computed_at":"2026-05-18T03:25:59.399927Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:59.399927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LvaF4vH7iLCsHw435cmf7+X5srJiWRUvpKnjoSMz34/gexjvlc+mccQLZQehxKSGglng4G+AMzotczRrLcqmDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:59.400451Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.2834","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d95e68cf5286cc013c62040a36c94f3ab22912d27e415a8396f973f4fcef22d","sha256:7ae3dc38285c4bfcf224026f135a1d24df1c767fa01acb41482c138498312d87"],"state_sha256":"31e5cf6f45d3d812167bd1fef589cf2ac80746429f766aaa70fd593994494a0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DZjyWjkefRe8Iz1/ZhQAr6vnEzcCDb/2c26t19ExpZsOQaafRqaItxhgqNVedfvUI/a3xtajz7RqHHscrohQAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:02:55.372624Z","bundle_sha256":"4c8538e053381121b8f680fce7b6dafc51d69bdbb06cb6e085c6c11958a59f76"}}