{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:B72N3FWVWBV3IH3OP6QHTJW6UM","short_pith_number":"pith:B72N3FWV","schema_version":"1.0","canonical_sha256":"0ff4dd96d5b06bb41f6e7fa079a6dea3377a8a9971a4b864f386cf46f09e7b97","source":{"kind":"arxiv","id":"1906.04861","version":2},"attestation_state":"computed","paper":{"title":"Homological Connectivity in Random \\v{C}ech Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.PR","authors_text":"Omer Bobrowski","submitted_at":"2019-06-11T23:32:06Z","abstract_excerpt":"We study the homology of random \\v{C}ech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology \"stabilizes\" and becomes isomorphic to that of the underlying topological space. Our results form a comprehensive high-dimensional analogue of well-known phenomena related to connectivity in the Erd\\H{o}s-R\\'enyi graph and random geometric graphs. We first prove that there is a sharp phase transition describing homological connectivity. Next, we analyze the behavior of the complex in the "},"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":"1906.04861","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-06-11T23:32:06Z","cross_cats_sorted":["math.AT","math.CO"],"title_canon_sha256":"abeb12e8e0a646c6563155f35cc6cf48c2df9137a35b4bbeb6a1ee21de17b1d7","abstract_canon_sha256":"74646c71f9e3f9dbc0305ff9c0fc2b8c08a785a7dffc4cbf0cc473f0f7008788"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:25.686994Z","signature_b64":"mpGpYKPPAmDZtNh60ZAIDgb/IG0bHRkWB9nn83o7v+xxA7Qr7yhgpHHsGnHaT9ZR1mEYtQDDFA2xsdgaYCj7Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ff4dd96d5b06bb41f6e7fa079a6dea3377a8a9971a4b864f386cf46f09e7b97","last_reissued_at":"2026-05-17T23:43:25.685720Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:25.685720Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homological Connectivity in Random \\v{C}ech Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.PR","authors_text":"Omer Bobrowski","submitted_at":"2019-06-11T23:32:06Z","abstract_excerpt":"We study the homology of random \\v{C}ech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology \"stabilizes\" and becomes isomorphic to that of the underlying topological space. Our results form a comprehensive high-dimensional analogue of well-known phenomena related to connectivity in the Erd\\H{o}s-R\\'enyi graph and random geometric graphs. We first prove that there is a sharp phase transition describing homological connectivity. Next, we analyze the behavior of the complex in the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.04861","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1906.04861","created_at":"2026-05-17T23:43:25.685815+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.04861v2","created_at":"2026-05-17T23:43:25.685815+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.04861","created_at":"2026-05-17T23:43:25.685815+00:00"},{"alias_kind":"pith_short_12","alias_value":"B72N3FWVWBV3","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"B72N3FWVWBV3IH3O","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"B72N3FWV","created_at":"2026-05-18T12:33:12.712433+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM","json":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM.json","graph_json":"https://pith.science/api/pith-number/B72N3FWVWBV3IH3OP6QHTJW6UM/graph.json","events_json":"https://pith.science/api/pith-number/B72N3FWVWBV3IH3OP6QHTJW6UM/events.json","paper":"https://pith.science/paper/B72N3FWV"},"agent_actions":{"view_html":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM","download_json":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM.json","view_paper":"https://pith.science/paper/B72N3FWV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.04861&json=true","fetch_graph":"https://pith.science/api/pith-number/B72N3FWVWBV3IH3OP6QHTJW6UM/graph.json","fetch_events":"https://pith.science/api/pith-number/B72N3FWVWBV3IH3OP6QHTJW6UM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM/action/storage_attestation","attest_author":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM/action/author_attestation","sign_citation":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM/action/citation_signature","submit_replication":"https://pith.science/pith/B72N3FWVWBV3IH3OP6QHTJW6UM/action/replication_record"}},"created_at":"2026-05-17T23:43:25.685815+00:00","updated_at":"2026-05-17T23:43:25.685815+00:00"}