{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AEMEDIZK56ACWY6BR5L6QUHBLA","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":"7e644fa85fbf80cbc718cf828484a79ee1ee00a49292d52e4757c1d11a80abb6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-02-22T18:23:52Z","title_canon_sha256":"188d9975a7bb4b8aad205fd9c7eb2928817b6155e9d2e55a1f9e334cfe730561"},"schema_version":"1.0","source":{"id":"1802.08224","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.08224","created_at":"2026-05-18T00:22:45Z"},{"alias_kind":"arxiv_version","alias_value":"1802.08224v1","created_at":"2026-05-18T00:22:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.08224","created_at":"2026-05-18T00:22:45Z"},{"alias_kind":"pith_short_12","alias_value":"AEMEDIZK56AC","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AEMEDIZK56ACWY6B","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AEMEDIZK","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:7b6b4236b92a73b6ea69aaddf910a2d33c377351c97552b6a1a8bd718e4f4c9d","target":"graph","created_at":"2026-05-18T00:22:45Z","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":"We study combinatorial connectivity for two models of random geometric complexes. These two models - \\v{C}ech and Vietoris-Rips complexes - are built on a homogeneous Poisson point process of intensity $n$ on a $d$-dimensional torus using balls of radius $r_n$. In the former, the $k$-simplices/faces are formed by subsets of $(k+1)$ Poisson points such that the balls of radius $r_n$ centred at these points have a mutual interesection and in the latter, we require only a pairwise intersection of the balls. Given a (simplicial) complex (i.e., a collection of $k$-simplices for all $k \\geq 1$), we ","authors_text":"D. Yogeshwaran, Srikanth K. Iyer","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-02-22T18:23:52Z","title":"Thresholds for vanishing of `Isolated' faces in random \\v{C}ech and Vietoris-Rips complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.08224","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":"sha256:11b5294ef917d0e86efd95c24d900d60a99c101eb87010e6fe4b26fc79484322","target":"record","created_at":"2026-05-18T00:22:45Z","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":"7e644fa85fbf80cbc718cf828484a79ee1ee00a49292d52e4757c1d11a80abb6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-02-22T18:23:52Z","title_canon_sha256":"188d9975a7bb4b8aad205fd9c7eb2928817b6155e9d2e55a1f9e334cfe730561"},"schema_version":"1.0","source":{"id":"1802.08224","kind":"arxiv","version":1}},"canonical_sha256":"011841a32aef802b63c18f57e850e158170d57540bc5e54ad3bd81db8ed3440f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"011841a32aef802b63c18f57e850e158170d57540bc5e54ad3bd81db8ed3440f","first_computed_at":"2026-05-18T00:22:45.243168Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:45.243168Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HtdeDn29fJ+gr76Zcyr9XHvfk14nqrYNf28kVq1JDgZ6ghT0Qidu2YEK3xsdQ9PU/Jyt7hdnxErl3iYmhvPQCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:45.243730Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.08224","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:11b5294ef917d0e86efd95c24d900d60a99c101eb87010e6fe4b26fc79484322","sha256:7b6b4236b92a73b6ea69aaddf910a2d33c377351c97552b6a1a8bd718e4f4c9d"],"state_sha256":"933e6d74c602a47e14f47c41d1167ebbe9eaa9215523be15613ab541025cda74"}