{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:WYRWG5Q67UAZVRROQXI2QNDUTV","short_pith_number":"pith:WYRWG5Q6","canonical_record":{"source":{"id":"0910.0499","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-02T23:23:13Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"b847d954eb81ef75182f791f5359cceadb049590224ba34ab547730d7964668d","abstract_canon_sha256":"e5afd2c8a481e8b16ccb6cdb04304b61b8dca91a8ec69ba10e4cf19f1562318d"},"schema_version":"1.0"},"canonical_sha256":"b62363761efd019ac62e85d1a834749d5d7e0590b9ca9b0dfb50de3fff8e2c58","source":{"kind":"arxiv","id":"0910.0499","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.0499","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"arxiv_version","alias_value":"0910.0499v1","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.0499","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"pith_short_12","alias_value":"WYRWG5Q67UAZ","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"pith_short_16","alias_value":"WYRWG5Q67UAZVRRO","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"pith_short_8","alias_value":"WYRWG5Q6","created_at":"2026-07-04T15:53:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:WYRWG5Q67UAZVRROQXI2QNDUTV","target":"record","payload":{"canonical_record":{"source":{"id":"0910.0499","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-02T23:23:13Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"b847d954eb81ef75182f791f5359cceadb049590224ba34ab547730d7964668d","abstract_canon_sha256":"e5afd2c8a481e8b16ccb6cdb04304b61b8dca91a8ec69ba10e4cf19f1562318d"},"schema_version":"1.0"},"canonical_sha256":"b62363761efd019ac62e85d1a834749d5d7e0590b9ca9b0dfb50de3fff8e2c58","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:53:38.357463Z","signature_b64":"Z0GBOITK4QGIBo49KtPKXfA4YguZPdGu5KZEIq4KCLyHvREZ8gct+sPTqB8DC3X3Y5xevRCg+x0Yrm9kehvcDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b62363761efd019ac62e85d1a834749d5d7e0590b9ca9b0dfb50de3fff8e2c58","last_reissued_at":"2026-07-04T15:53:38.356921Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:53:38.356921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0910.0499","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":"2026-07-04T15:53:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uHQK9sF0pEjcij268y+IlWrOLYuvexWLz9WThtYQPmSaBt7wT0ArWlgRfNZfoOElUzRI8WCWxJykk0yPDqZZBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T20:58:59.499855Z"},"content_sha256":"343dce3117429d4c0bd4e23a39aac3cfaf2bd90b6e98be1964006f423df69a40","schema_version":"1.0","event_id":"sha256:343dce3117429d4c0bd4e23a39aac3cfaf2bd90b6e98be1964006f423df69a40"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:WYRWG5Q67UAZVRROQXI2QNDUTV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A zero-one law for the existence of triangles in random key graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Armand M. Makowski, Osman Yagan","submitted_at":"2009-10-02T23:23:13Z","abstract_excerpt":"Random key graphs are random graphs induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. For this class of random graphs we show the existence of a zero-one law for the appearance of triangles, and identify the corresponding critical scaling. This is done by applying the method of first and second moments to the number of triangles in the graph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.0499","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0910.0499/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T15:53:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"//F8K0WAktvzeRosKC2LnKZK57gX0FgLlVSn+fAmGX4PSw7ebtEs/JyVUOYAxksyVG/qrMCCX9iK/acOdpY1BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T20:58:59.500215Z"},"content_sha256":"081e596c98af962bb975c43e92cf176a078fa0f805df5a4f27c0c9f5394cb22e","schema_version":"1.0","event_id":"sha256:081e596c98af962bb975c43e92cf176a078fa0f805df5a4f27c0c9f5394cb22e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WYRWG5Q67UAZVRROQXI2QNDUTV/bundle.json","state_url":"https://pith.science/pith/WYRWG5Q67UAZVRROQXI2QNDUTV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WYRWG5Q67UAZVRROQXI2QNDUTV/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":"2026-07-04T20:58:59Z","links":{"resolver":"https://pith.science/pith/WYRWG5Q67UAZVRROQXI2QNDUTV","bundle":"https://pith.science/pith/WYRWG5Q67UAZVRROQXI2QNDUTV/bundle.json","state":"https://pith.science/pith/WYRWG5Q67UAZVRROQXI2QNDUTV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WYRWG5Q67UAZVRROQXI2QNDUTV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:WYRWG5Q67UAZVRROQXI2QNDUTV","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":"e5afd2c8a481e8b16ccb6cdb04304b61b8dca91a8ec69ba10e4cf19f1562318d","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-02T23:23:13Z","title_canon_sha256":"b847d954eb81ef75182f791f5359cceadb049590224ba34ab547730d7964668d"},"schema_version":"1.0","source":{"id":"0910.0499","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.0499","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"arxiv_version","alias_value":"0910.0499v1","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.0499","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"pith_short_12","alias_value":"WYRWG5Q67UAZ","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"pith_short_16","alias_value":"WYRWG5Q67UAZVRRO","created_at":"2026-07-04T15:53:38Z"},{"alias_kind":"pith_short_8","alias_value":"WYRWG5Q6","created_at":"2026-07-04T15:53:38Z"}],"graph_snapshots":[{"event_id":"sha256:081e596c98af962bb975c43e92cf176a078fa0f805df5a4f27c0c9f5394cb22e","target":"graph","created_at":"2026-07-04T15:53:38Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0910.0499/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Random key graphs are random graphs induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. For this class of random graphs we show the existence of a zero-one law for the appearance of triangles, and identify the corresponding critical scaling. This is done by applying the method of first and second moments to the number of triangles in the graph.","authors_text":"Armand M. Makowski, Osman Yagan","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-02T23:23:13Z","title":"A zero-one law for the existence of triangles in random key graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.0499","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":"sha256:343dce3117429d4c0bd4e23a39aac3cfaf2bd90b6e98be1964006f423df69a40","target":"record","created_at":"2026-07-04T15:53:38Z","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":"e5afd2c8a481e8b16ccb6cdb04304b61b8dca91a8ec69ba10e4cf19f1562318d","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-02T23:23:13Z","title_canon_sha256":"b847d954eb81ef75182f791f5359cceadb049590224ba34ab547730d7964668d"},"schema_version":"1.0","source":{"id":"0910.0499","kind":"arxiv","version":1}},"canonical_sha256":"b62363761efd019ac62e85d1a834749d5d7e0590b9ca9b0dfb50de3fff8e2c58","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b62363761efd019ac62e85d1a834749d5d7e0590b9ca9b0dfb50de3fff8e2c58","first_computed_at":"2026-07-04T15:53:38.356921Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:53:38.356921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z0GBOITK4QGIBo49KtPKXfA4YguZPdGu5KZEIq4KCLyHvREZ8gct+sPTqB8DC3X3Y5xevRCg+x0Yrm9kehvcDw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:53:38.357463Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.0499","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:343dce3117429d4c0bd4e23a39aac3cfaf2bd90b6e98be1964006f423df69a40","sha256:081e596c98af962bb975c43e92cf176a078fa0f805df5a4f27c0c9f5394cb22e"],"state_sha256":"fed177bab00bc32e8e812810df8d4b62a734cc52610ae6fc64ff6107d715ba5f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pRPq+va270d3/54FL/3kUdugELCDA9SZO6e9Wg6knAudT10CgVveMEBhJwBEbEGM1C02x8NSGJl9KcOLSGNNDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T20:58:59.502124Z","bundle_sha256":"6189f7a723fd9f0cf50852b014066d113ecb9aa082352cc88c7cb3b16547ce6e"}}