{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:67ZGSTLLFODE76WBDNACCZM4U4","short_pith_number":"pith:67ZGSTLL","canonical_record":{"source":{"id":"1501.07835","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T16:48:38Z","cross_cats_sorted":[],"title_canon_sha256":"df3d7ae8151d9cf4b4a7aa0d7d45a8e63a838831e5f528a859c6bf9d4cf4d337","abstract_canon_sha256":"36f382ae00533e9031bb48c3d6cb122217d3885115608a17f6248289769e0200"},"schema_version":"1.0"},"canonical_sha256":"f7f2694d6b2b864ffac11b4021659ca73b5bc9d85590f3e7d848d5fd81310375","source":{"kind":"arxiv","id":"1501.07835","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07835","created_at":"2026-05-18T02:28:14Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07835v1","created_at":"2026-05-18T02:28:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07835","created_at":"2026-05-18T02:28:14Z"},{"alias_kind":"pith_short_12","alias_value":"67ZGSTLLFODE","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"67ZGSTLLFODE76WB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"67ZGSTLL","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:67ZGSTLLFODE76WBDNACCZM4U4","target":"record","payload":{"canonical_record":{"source":{"id":"1501.07835","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T16:48:38Z","cross_cats_sorted":[],"title_canon_sha256":"df3d7ae8151d9cf4b4a7aa0d7d45a8e63a838831e5f528a859c6bf9d4cf4d337","abstract_canon_sha256":"36f382ae00533e9031bb48c3d6cb122217d3885115608a17f6248289769e0200"},"schema_version":"1.0"},"canonical_sha256":"f7f2694d6b2b864ffac11b4021659ca73b5bc9d85590f3e7d848d5fd81310375","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:14.028952Z","signature_b64":"X+ArClsMl+LOdDmApxjo4tIvzht8FewDSe9zRMhAE8Udh0cJsH4QZNDpdEoxdwDkoMUSmOJQZsD/h3wSKOpABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7f2694d6b2b864ffac11b4021659ca73b5bc9d85590f3e7d848d5fd81310375","last_reissued_at":"2026-05-18T02:28:14.027995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:14.027995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.07835","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":"2026-05-18T02:28:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mvt2SxIdCQq7CeIgG/ZfAOZXsy7nfEeMihNtAUbCnWpqG8EkL0YZ9BdHYxZ1TgD5S2hK5Y4dNoM/AGQNU40SAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:12:23.074877Z"},"content_sha256":"a3761ba428a703c2f4d1934b931868ffd12d37c4df62e6a6c5eb2aaf94bf2491","schema_version":"1.0","event_id":"sha256:a3761ba428a703c2f4d1934b931868ffd12d37c4df62e6a6c5eb2aaf94bf2491"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:67ZGSTLLFODE76WBDNACCZM4U4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The size of the giant component in random hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christoph Koch, Mihyun Kang, Oliver Cooley","submitted_at":"2015-01-30T16:48:38Z","abstract_excerpt":"The phase transition in the size of the giant component in random graphs is one of the most well-studied phenomena in random graph theory. For hypergraphs, there are many possible generalisations of the notion of a component, and for all but the simplest example, the phase transition phenomenon was first proved by Cooley, Kang and Person. In this paper we build on this and determine the asymptotic size of the unique giant component."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07835","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":"2026-05-18T02:28:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YG3LyDx5A6KFXnu8BniTDFVcvX9iw334sYpVSRGYKz1yX6MzZy3dFx3HGMOeNAWwSxpluExRM9ZVXo16pY4ZAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:12:23.075533Z"},"content_sha256":"9a0cea3227959ebfc5667d4f16f1e3c4757ab0f1f0d35c812340c9b9684d11ee","schema_version":"1.0","event_id":"sha256:9a0cea3227959ebfc5667d4f16f1e3c4757ab0f1f0d35c812340c9b9684d11ee"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/67ZGSTLLFODE76WBDNACCZM4U4/bundle.json","state_url":"https://pith.science/pith/67ZGSTLLFODE76WBDNACCZM4U4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/67ZGSTLLFODE76WBDNACCZM4U4/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":"2026-05-31T18:12:23Z","links":{"resolver":"https://pith.science/pith/67ZGSTLLFODE76WBDNACCZM4U4","bundle":"https://pith.science/pith/67ZGSTLLFODE76WBDNACCZM4U4/bundle.json","state":"https://pith.science/pith/67ZGSTLLFODE76WBDNACCZM4U4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/67ZGSTLLFODE76WBDNACCZM4U4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:67ZGSTLLFODE76WBDNACCZM4U4","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":"36f382ae00533e9031bb48c3d6cb122217d3885115608a17f6248289769e0200","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T16:48:38Z","title_canon_sha256":"df3d7ae8151d9cf4b4a7aa0d7d45a8e63a838831e5f528a859c6bf9d4cf4d337"},"schema_version":"1.0","source":{"id":"1501.07835","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07835","created_at":"2026-05-18T02:28:14Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07835v1","created_at":"2026-05-18T02:28:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07835","created_at":"2026-05-18T02:28:14Z"},{"alias_kind":"pith_short_12","alias_value":"67ZGSTLLFODE","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"67ZGSTLLFODE76WB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"67ZGSTLL","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:9a0cea3227959ebfc5667d4f16f1e3c4757ab0f1f0d35c812340c9b9684d11ee","target":"graph","created_at":"2026-05-18T02:28:14Z","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":"The phase transition in the size of the giant component in random graphs is one of the most well-studied phenomena in random graph theory. For hypergraphs, there are many possible generalisations of the notion of a component, and for all but the simplest example, the phase transition phenomenon was first proved by Cooley, Kang and Person. In this paper we build on this and determine the asymptotic size of the unique giant component.","authors_text":"Christoph Koch, Mihyun Kang, Oliver Cooley","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T16:48:38Z","title":"The size of the giant component in random hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07835","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:a3761ba428a703c2f4d1934b931868ffd12d37c4df62e6a6c5eb2aaf94bf2491","target":"record","created_at":"2026-05-18T02:28:14Z","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":"36f382ae00533e9031bb48c3d6cb122217d3885115608a17f6248289769e0200","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T16:48:38Z","title_canon_sha256":"df3d7ae8151d9cf4b4a7aa0d7d45a8e63a838831e5f528a859c6bf9d4cf4d337"},"schema_version":"1.0","source":{"id":"1501.07835","kind":"arxiv","version":1}},"canonical_sha256":"f7f2694d6b2b864ffac11b4021659ca73b5bc9d85590f3e7d848d5fd81310375","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f7f2694d6b2b864ffac11b4021659ca73b5bc9d85590f3e7d848d5fd81310375","first_computed_at":"2026-05-18T02:28:14.027995Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:14.027995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X+ArClsMl+LOdDmApxjo4tIvzht8FewDSe9zRMhAE8Udh0cJsH4QZNDpdEoxdwDkoMUSmOJQZsD/h3wSKOpABQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:14.028952Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.07835","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a3761ba428a703c2f4d1934b931868ffd12d37c4df62e6a6c5eb2aaf94bf2491","sha256:9a0cea3227959ebfc5667d4f16f1e3c4757ab0f1f0d35c812340c9b9684d11ee"],"state_sha256":"e16880c4bd6db6fbde100476d09feb40e32660ab0c3b05c1cf5ca9eea5c8d953"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uk1bYM05cE7y1pcE63DOF2r5ulgZycXEMEmamC/6aT43dxYl/fK1msZkpCTSon4McA6w/32O93qHhvd41eIoDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T18:12:23.079111Z","bundle_sha256":"d61128ce3135f92a2cf825d5220eca74d88a8b41d7957277b9a1e833897e2676"}}