{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:GMWQGPYBCNLESSD67CD2VISVMV","short_pith_number":"pith:GMWQGPYB","canonical_record":{"source":{"id":"1610.00890","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-10-04T08:03:30Z","cross_cats_sorted":[],"title_canon_sha256":"64b9d7379278f52e2faad04e592704973a039e245bb6abee72f79d6646e2d304","abstract_canon_sha256":"f8acb69d317635ae1aba8fe97e62a20e8db146d502b909352bef76db3577ad08"},"schema_version":"1.0"},"canonical_sha256":"332d033f01135649487ef887aaa25565743aafdc9e58da47200246f219505c88","source":{"kind":"arxiv","id":"1610.00890","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.00890","created_at":"2026-05-18T00:21:05Z"},{"alias_kind":"arxiv_version","alias_value":"1610.00890v5","created_at":"2026-05-18T00:21:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00890","created_at":"2026-05-18T00:21:05Z"},{"alias_kind":"pith_short_12","alias_value":"GMWQGPYBCNLE","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GMWQGPYBCNLESSD6","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GMWQGPYB","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:GMWQGPYBCNLESSD67CD2VISVMV","target":"record","payload":{"canonical_record":{"source":{"id":"1610.00890","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-10-04T08:03:30Z","cross_cats_sorted":[],"title_canon_sha256":"64b9d7379278f52e2faad04e592704973a039e245bb6abee72f79d6646e2d304","abstract_canon_sha256":"f8acb69d317635ae1aba8fe97e62a20e8db146d502b909352bef76db3577ad08"},"schema_version":"1.0"},"canonical_sha256":"332d033f01135649487ef887aaa25565743aafdc9e58da47200246f219505c88","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:05.172359Z","signature_b64":"vVAu0niRxAscdA0DN2nbwJ0v0iAjUrNMnpuNN9Oo1f5yTyTu82ZVO2rYkvz9K6JUBL7vpUbjZiMAdndk4mY8DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"332d033f01135649487ef887aaa25565743aafdc9e58da47200246f219505c88","last_reissued_at":"2026-05-18T00:21:05.171473Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:05.171473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.00890","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:21:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VLN9tP0D296co348ea1/0g7Pr7VqfWjVrcIwlCRiFKLUF9KHF2qAAfgR7B1JojKG92eMedcC906HalejuqeXBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:05:49.249657Z"},"content_sha256":"2b10040e82f8a00a5dff68991e08b1c03e10923a47fca28cdcf8b69da5a86fcd","schema_version":"1.0","event_id":"sha256:2b10040e82f8a00a5dff68991e08b1c03e10923a47fca28cdcf8b69da5a86fcd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:GMWQGPYBCNLESSD67CD2VISVMV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Embedded Homology of Hypergraphs and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Jie Wu, Jingyan Li, Shiquan Ren, Stephane Bressan","submitted_at":"2016-10-04T08:03:30Z","abstract_excerpt":"Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this paper, generalising the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalisation of the Mayer-Vietoris sequence for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for the embedded homology of hypergraphs. Moreover, as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00890","kind":"arxiv","version":5},"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":"2026-05-18T00:21:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ridfAiwoObfNjRCcihk+F6u0Cu5mkEGLITXdXc1jFv9MSJh+Hq9URe3IbSInZ2R6Otx+rx2cIJWu9gxjxZtoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:05:49.250011Z"},"content_sha256":"80adf9bf93283bc99516db631a83265824ee640da616f1a6d09c8f01fffa786b","schema_version":"1.0","event_id":"sha256:80adf9bf93283bc99516db631a83265824ee640da616f1a6d09c8f01fffa786b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GMWQGPYBCNLESSD67CD2VISVMV/bundle.json","state_url":"https://pith.science/pith/GMWQGPYBCNLESSD67CD2VISVMV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GMWQGPYBCNLESSD67CD2VISVMV/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":"2026-06-05T03:05:49Z","links":{"resolver":"https://pith.science/pith/GMWQGPYBCNLESSD67CD2VISVMV","bundle":"https://pith.science/pith/GMWQGPYBCNLESSD67CD2VISVMV/bundle.json","state":"https://pith.science/pith/GMWQGPYBCNLESSD67CD2VISVMV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GMWQGPYBCNLESSD67CD2VISVMV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:GMWQGPYBCNLESSD67CD2VISVMV","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":"f8acb69d317635ae1aba8fe97e62a20e8db146d502b909352bef76db3577ad08","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-10-04T08:03:30Z","title_canon_sha256":"64b9d7379278f52e2faad04e592704973a039e245bb6abee72f79d6646e2d304"},"schema_version":"1.0","source":{"id":"1610.00890","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.00890","created_at":"2026-05-18T00:21:05Z"},{"alias_kind":"arxiv_version","alias_value":"1610.00890v5","created_at":"2026-05-18T00:21:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00890","created_at":"2026-05-18T00:21:05Z"},{"alias_kind":"pith_short_12","alias_value":"GMWQGPYBCNLE","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GMWQGPYBCNLESSD6","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GMWQGPYB","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:80adf9bf93283bc99516db631a83265824ee640da616f1a6d09c8f01fffa786b","target":"graph","created_at":"2026-05-18T00:21:05Z","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":"Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this paper, generalising the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalisation of the Mayer-Vietoris sequence for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for the embedded homology of hypergraphs. Moreover, as","authors_text":"Jie Wu, Jingyan Li, Shiquan Ren, Stephane Bressan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-10-04T08:03:30Z","title":"The Embedded Homology of Hypergraphs and Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00890","kind":"arxiv","version":5},"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":"sha256:2b10040e82f8a00a5dff68991e08b1c03e10923a47fca28cdcf8b69da5a86fcd","target":"record","created_at":"2026-05-18T00:21:05Z","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":"f8acb69d317635ae1aba8fe97e62a20e8db146d502b909352bef76db3577ad08","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-10-04T08:03:30Z","title_canon_sha256":"64b9d7379278f52e2faad04e592704973a039e245bb6abee72f79d6646e2d304"},"schema_version":"1.0","source":{"id":"1610.00890","kind":"arxiv","version":5}},"canonical_sha256":"332d033f01135649487ef887aaa25565743aafdc9e58da47200246f219505c88","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"332d033f01135649487ef887aaa25565743aafdc9e58da47200246f219505c88","first_computed_at":"2026-05-18T00:21:05.171473Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:05.171473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vVAu0niRxAscdA0DN2nbwJ0v0iAjUrNMnpuNN9Oo1f5yTyTu82ZVO2rYkvz9K6JUBL7vpUbjZiMAdndk4mY8DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:05.172359Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.00890","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b10040e82f8a00a5dff68991e08b1c03e10923a47fca28cdcf8b69da5a86fcd","sha256:80adf9bf93283bc99516db631a83265824ee640da616f1a6d09c8f01fffa786b"],"state_sha256":"fe079f927680aed355660eb2b2e4e1a6f7d6303573ac1609edff21d5b65f4878"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CQC/zGKD5Eu7Hr0jeZaB28Vnh2nGgjnkp5b12ETUir3uYYqik0MFGOwP2/gXl42MkSOEJM9eHBZsGLvbu4/LAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T03:05:49.252493Z","bundle_sha256":"f47b10aa07efe562398f4ba749f3e292ac3bf5d3efe2429fa45217f40c35b9b0"}}