{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:TFRBYATZCMW5ZJY7O6V4WU5OSB","short_pith_number":"pith:TFRBYATZ","canonical_record":{"source":{"id":"1503.01831","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-03-06T01:47:26Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"4e61335397ed2c8ec380c9bbef531ac411dc22a6ca00df3de63eef6fc8598b6a","abstract_canon_sha256":"b483c55b8b399188d960b55adaaad95f1df43d440bd4caa2a8c88ddd8f6f73ec"},"schema_version":"1.0"},"canonical_sha256":"99621c0279132ddca71f77abcb53ae904c6edf8ca38f6eae677789a74c0bb8a8","source":{"kind":"arxiv","id":"1503.01831","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01831","created_at":"2026-05-18T01:58:26Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01831v2","created_at":"2026-05-18T01:58:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01831","created_at":"2026-05-18T01:58:26Z"},{"alias_kind":"pith_short_12","alias_value":"TFRBYATZCMW5","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TFRBYATZCMW5ZJY7","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TFRBYATZ","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:TFRBYATZCMW5ZJY7O6V4WU5OSB","target":"record","payload":{"canonical_record":{"source":{"id":"1503.01831","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-03-06T01:47:26Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"4e61335397ed2c8ec380c9bbef531ac411dc22a6ca00df3de63eef6fc8598b6a","abstract_canon_sha256":"b483c55b8b399188d960b55adaaad95f1df43d440bd4caa2a8c88ddd8f6f73ec"},"schema_version":"1.0"},"canonical_sha256":"99621c0279132ddca71f77abcb53ae904c6edf8ca38f6eae677789a74c0bb8a8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:26.816682Z","signature_b64":"BOjuuvWH7UzEOAz6tVwnZWSe3TCDxoE8DwhSJAFKzr0EkbT6k+FO7GOrjFLBDN8J6AodBKB3MNTaVUTEIEJSBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99621c0279132ddca71f77abcb53ae904c6edf8ca38f6eae677789a74c0bb8a8","last_reissued_at":"2026-05-18T01:58:26.814703Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:26.814703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.01831","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:58:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vwluh55AfD4+UkBW6lLnbziOVX52YwibSl4/Jrf07PjWMc8Mw8//LnribPzf/TwlqjeSybBvyeoEbeXrJ1FtBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T19:39:41.192972Z"},"content_sha256":"c54b8c0b81a348392dd5d4418dd8763c452b3ebe1d3238e7f5e80a7b0f2bb6f8","schema_version":"1.0","event_id":"sha256:c54b8c0b81a348392dd5d4418dd8763c452b3ebe1d3238e7f5e80a7b0f2bb6f8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:TFRBYATZCMW5ZJY7O6V4WU5OSB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Random Simplicial Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.AT","authors_text":"Christopher F. Fowler","submitted_at":"2015-03-06T01:47:26Z","abstract_excerpt":"We consider a multi-parameter model for randomly constructing simplicial complexes. This model interpolates between random clique complexes and Linial-Meshulam random $k$-dimensional complexes, two models that have been extensively studied. While these models asymptotically exhibit nontrivial cohomology in only one or two dimensions, we show that in this generalized setting nontrivial cohomology can occur in several dimensions simultaneously. We establish upper and lower thresholds for the appearance of nontrivial cohomology in a particular dimension, and in some instances characterize the beh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01831","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:58:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Cg6Mablkaz0vYP4bYv4gS01LUNZVKPupoveen4Y/ogtoRsBsHMbRnddblXHr5vlrE1sexw/j2dBOlP51CU1FCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T19:39:41.193349Z"},"content_sha256":"d045015c8f6145672e135bcf017c7d51e4655d39a8038730637227b1780b96b6","schema_version":"1.0","event_id":"sha256:d045015c8f6145672e135bcf017c7d51e4655d39a8038730637227b1780b96b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TFRBYATZCMW5ZJY7O6V4WU5OSB/bundle.json","state_url":"https://pith.science/pith/TFRBYATZCMW5ZJY7O6V4WU5OSB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TFRBYATZCMW5ZJY7O6V4WU5OSB/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":"2026-06-30T19:39:41Z","links":{"resolver":"https://pith.science/pith/TFRBYATZCMW5ZJY7O6V4WU5OSB","bundle":"https://pith.science/pith/TFRBYATZCMW5ZJY7O6V4WU5OSB/bundle.json","state":"https://pith.science/pith/TFRBYATZCMW5ZJY7O6V4WU5OSB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TFRBYATZCMW5ZJY7O6V4WU5OSB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TFRBYATZCMW5ZJY7O6V4WU5OSB","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":"b483c55b8b399188d960b55adaaad95f1df43d440bd4caa2a8c88ddd8f6f73ec","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-03-06T01:47:26Z","title_canon_sha256":"4e61335397ed2c8ec380c9bbef531ac411dc22a6ca00df3de63eef6fc8598b6a"},"schema_version":"1.0","source":{"id":"1503.01831","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01831","created_at":"2026-05-18T01:58:26Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01831v2","created_at":"2026-05-18T01:58:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01831","created_at":"2026-05-18T01:58:26Z"},{"alias_kind":"pith_short_12","alias_value":"TFRBYATZCMW5","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TFRBYATZCMW5ZJY7","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TFRBYATZ","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:d045015c8f6145672e135bcf017c7d51e4655d39a8038730637227b1780b96b6","target":"graph","created_at":"2026-05-18T01:58:26Z","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 consider a multi-parameter model for randomly constructing simplicial complexes. This model interpolates between random clique complexes and Linial-Meshulam random $k$-dimensional complexes, two models that have been extensively studied. While these models asymptotically exhibit nontrivial cohomology in only one or two dimensions, we show that in this generalized setting nontrivial cohomology can occur in several dimensions simultaneously. We establish upper and lower thresholds for the appearance of nontrivial cohomology in a particular dimension, and in some instances characterize the beh","authors_text":"Christopher F. Fowler","cross_cats":["math.CO","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-03-06T01:47:26Z","title":"Generalized Random Simplicial Complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01831","kind":"arxiv","version":2},"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:c54b8c0b81a348392dd5d4418dd8763c452b3ebe1d3238e7f5e80a7b0f2bb6f8","target":"record","created_at":"2026-05-18T01:58:26Z","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":"b483c55b8b399188d960b55adaaad95f1df43d440bd4caa2a8c88ddd8f6f73ec","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-03-06T01:47:26Z","title_canon_sha256":"4e61335397ed2c8ec380c9bbef531ac411dc22a6ca00df3de63eef6fc8598b6a"},"schema_version":"1.0","source":{"id":"1503.01831","kind":"arxiv","version":2}},"canonical_sha256":"99621c0279132ddca71f77abcb53ae904c6edf8ca38f6eae677789a74c0bb8a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99621c0279132ddca71f77abcb53ae904c6edf8ca38f6eae677789a74c0bb8a8","first_computed_at":"2026-05-18T01:58:26.814703Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:58:26.814703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BOjuuvWH7UzEOAz6tVwnZWSe3TCDxoE8DwhSJAFKzr0EkbT6k+FO7GOrjFLBDN8J6AodBKB3MNTaVUTEIEJSBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:58:26.816682Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.01831","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c54b8c0b81a348392dd5d4418dd8763c452b3ebe1d3238e7f5e80a7b0f2bb6f8","sha256:d045015c8f6145672e135bcf017c7d51e4655d39a8038730637227b1780b96b6"],"state_sha256":"23f629132ce31734c6a215bce65f087a8fccc32019a1cdbae9e34ce9fa6ac882"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6YB1UrQzB4mPQUA7dPgdvKFqQelIfLtguKdulmmz/3bBSxRrPfX7KzkdqS19V70jdDSGqaMgAJGR/d+bBxVBCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T19:39:41.196474Z","bundle_sha256":"05d94adb66c37468af15e75afcc371af7c4e7ed53e533eccb9146694c94d8d20"}}