{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:EONMQ36XKHXVWPFOI5S3VFSMM2","short_pith_number":"pith:EONMQ36X","canonical_record":{"source":{"id":"1308.5493","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-26T06:20:13Z","cross_cats_sorted":[],"title_canon_sha256":"c38554f6f27cf7aa869734165d6b494ffdd85c3472959f1797a4ab1848d419f8","abstract_canon_sha256":"b4ee3ea540343bbf19fc5198abe7603f88c1ad8655c37886a4b7cfbaacd5a8de"},"schema_version":"1.0"},"canonical_sha256":"239ac86fd751ef5b3cae4765ba964c669f98e81871f7f54ab3db44bdeb168980","source":{"kind":"arxiv","id":"1308.5493","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5493","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5493v3","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5493","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"pith_short_12","alias_value":"EONMQ36XKHXV","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EONMQ36XKHXVWPFO","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EONMQ36X","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:EONMQ36XKHXVWPFOI5S3VFSMM2","target":"record","payload":{"canonical_record":{"source":{"id":"1308.5493","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-26T06:20:13Z","cross_cats_sorted":[],"title_canon_sha256":"c38554f6f27cf7aa869734165d6b494ffdd85c3472959f1797a4ab1848d419f8","abstract_canon_sha256":"b4ee3ea540343bbf19fc5198abe7603f88c1ad8655c37886a4b7cfbaacd5a8de"},"schema_version":"1.0"},"canonical_sha256":"239ac86fd751ef5b3cae4765ba964c669f98e81871f7f54ab3db44bdeb168980","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:13.135932Z","signature_b64":"b+8DH0ChQtmJtyDRg94E4gA8qSrqP20uNquEK+8GWZNJDWjzpEfbf5+Rtkzd28fYPo6Ax0tnjhKQSkCL9H6JCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"239ac86fd751ef5b3cae4765ba964c669f98e81871f7f54ab3db44bdeb168980","last_reissued_at":"2026-05-18T02:42:13.135478Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:13.135478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.5493","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:42:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/fexamX1lYms05MiUoouGb60CqU9l6h7dkB9ggyZ1ZYc3cRXTK5jAgic4hV3G9h+DAiVFDYgEuAAk6+XWbKUDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:09:28.990473Z"},"content_sha256":"9c9ac8e88a34f1b03a35bec48592da7ac405712b8196c04746d6b61d7078d525","schema_version":"1.0","event_id":"sha256:9c9ac8e88a34f1b03a35bec48592da7ac405712b8196c04746d6b61d7078d525"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:EONMQ36XKHXVWPFOI5S3VFSMM2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Law of large numbers for the SIR epidemic on a random graph with given degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Malwina Luczak, Peter Windridge, Svante Janson","submitted_at":"2013-08-26T06:20:13Z","abstract_excerpt":"We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate.\n  Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak occurs with probability bounded away from zero. Our main result is that, conditional on a large outbreak, the evo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5493","kind":"arxiv","version":3},"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":"2026-05-18T02:42:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9ptSFcamo9ow5NiPNPE4B2Ns8otz7fCCbtaEQ+VvgsqGEvpY+WdSRGGN5GVqvNMk00LfTfW7ptc7KWG8l5DcDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:09:28.991173Z"},"content_sha256":"b61d713e648265df30b1bcdf521e49a4e2d6001ef5d0db4934349c6aaa292e5a","schema_version":"1.0","event_id":"sha256:b61d713e648265df30b1bcdf521e49a4e2d6001ef5d0db4934349c6aaa292e5a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EONMQ36XKHXVWPFOI5S3VFSMM2/bundle.json","state_url":"https://pith.science/pith/EONMQ36XKHXVWPFOI5S3VFSMM2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EONMQ36XKHXVWPFOI5S3VFSMM2/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":"2026-05-25T20:09:28Z","links":{"resolver":"https://pith.science/pith/EONMQ36XKHXVWPFOI5S3VFSMM2","bundle":"https://pith.science/pith/EONMQ36XKHXVWPFOI5S3VFSMM2/bundle.json","state":"https://pith.science/pith/EONMQ36XKHXVWPFOI5S3VFSMM2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EONMQ36XKHXVWPFOI5S3VFSMM2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EONMQ36XKHXVWPFOI5S3VFSMM2","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":"b4ee3ea540343bbf19fc5198abe7603f88c1ad8655c37886a4b7cfbaacd5a8de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-26T06:20:13Z","title_canon_sha256":"c38554f6f27cf7aa869734165d6b494ffdd85c3472959f1797a4ab1848d419f8"},"schema_version":"1.0","source":{"id":"1308.5493","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5493","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5493v3","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5493","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"pith_short_12","alias_value":"EONMQ36XKHXV","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EONMQ36XKHXVWPFO","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EONMQ36X","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:b61d713e648265df30b1bcdf521e49a4e2d6001ef5d0db4934349c6aaa292e5a","target":"graph","created_at":"2026-05-18T02:42:13Z","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 the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate.\n  Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak occurs with probability bounded away from zero. Our main result is that, conditional on a large outbreak, the evo","authors_text":"Malwina Luczak, Peter Windridge, Svante Janson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-26T06:20:13Z","title":"Law of large numbers for the SIR epidemic on a random graph with given degrees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5493","kind":"arxiv","version":3},"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":"sha256:9c9ac8e88a34f1b03a35bec48592da7ac405712b8196c04746d6b61d7078d525","target":"record","created_at":"2026-05-18T02:42:13Z","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":"b4ee3ea540343bbf19fc5198abe7603f88c1ad8655c37886a4b7cfbaacd5a8de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-26T06:20:13Z","title_canon_sha256":"c38554f6f27cf7aa869734165d6b494ffdd85c3472959f1797a4ab1848d419f8"},"schema_version":"1.0","source":{"id":"1308.5493","kind":"arxiv","version":3}},"canonical_sha256":"239ac86fd751ef5b3cae4765ba964c669f98e81871f7f54ab3db44bdeb168980","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"239ac86fd751ef5b3cae4765ba964c669f98e81871f7f54ab3db44bdeb168980","first_computed_at":"2026-05-18T02:42:13.135478Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:13.135478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b+8DH0ChQtmJtyDRg94E4gA8qSrqP20uNquEK+8GWZNJDWjzpEfbf5+Rtkzd28fYPo6Ax0tnjhKQSkCL9H6JCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:13.135932Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.5493","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9c9ac8e88a34f1b03a35bec48592da7ac405712b8196c04746d6b61d7078d525","sha256:b61d713e648265df30b1bcdf521e49a4e2d6001ef5d0db4934349c6aaa292e5a"],"state_sha256":"0ba8803363c4e67c0e89b8447cc5be0ed16fdc732d630ea9e60af5d5968bf107"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"29X1vGnn4bqZZoxUXVdZjseLB2Zv0W504/XacNKpIKWMe2T/4Hde765LasOVRN8GT68qxg/rJaRCTf71+hxjBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T20:09:28.994767Z","bundle_sha256":"6aa00df95420edd98dcd5f2d33c6beb99e97a17414b899ff29101151c3de8eca"}}