{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:3GBJ33K4VQSVMUA2D3R7Y6HVB2","short_pith_number":"pith:3GBJ33K4","canonical_record":{"source":{"id":"1212.5396","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-12-21T11:11:45Z","cross_cats_sorted":["physics.soc-ph"],"title_canon_sha256":"ccd1bd12e2ff45062a8d00e1b5ec3c9e8224442becfab10ff80cd5e687350170","abstract_canon_sha256":"ab3cd1670f21b07dc166e7c3950c860e4568df73baa5fa122c937da0a865b18d"},"schema_version":"1.0"},"canonical_sha256":"d9829ded5cac2556501a1ee3fc78f50e8dd27d6bad834d76435272c56873ec15","source":{"kind":"arxiv","id":"1212.5396","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.5396","created_at":"2026-05-18T01:52:29Z"},{"alias_kind":"arxiv_version","alias_value":"1212.5396v3","created_at":"2026-05-18T01:52:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5396","created_at":"2026-05-18T01:52:29Z"},{"alias_kind":"pith_short_12","alias_value":"3GBJ33K4VQSV","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3GBJ33K4VQSVMUA2","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3GBJ33K4","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:3GBJ33K4VQSVMUA2D3R7Y6HVB2","target":"record","payload":{"canonical_record":{"source":{"id":"1212.5396","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-12-21T11:11:45Z","cross_cats_sorted":["physics.soc-ph"],"title_canon_sha256":"ccd1bd12e2ff45062a8d00e1b5ec3c9e8224442becfab10ff80cd5e687350170","abstract_canon_sha256":"ab3cd1670f21b07dc166e7c3950c860e4568df73baa5fa122c937da0a865b18d"},"schema_version":"1.0"},"canonical_sha256":"d9829ded5cac2556501a1ee3fc78f50e8dd27d6bad834d76435272c56873ec15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:52:29.312966Z","signature_b64":"8GIMLqac+L38/F8gw6DxNzMzLCb93ETk2/aD9n5HV5/H/7LWTAB4RjKajzL1cO7EJb5sp3YuW8AlLk+6NnwtCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9829ded5cac2556501a1ee3fc78f50e8dd27d6bad834d76435272c56873ec15","last_reissued_at":"2026-05-18T01:52:29.312521Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:52:29.312521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.5396","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":"2026-05-18T01:52:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AYxd5QBEvjnMXGkyizrI6YdJYrxXSUCSPC0mu2J9VFY94gOgHwqGAYXBGxGCTUcMWw+dsHXOQ3Gff94rCwzZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T22:52:42.635883Z"},"content_sha256":"62374abfc867556e672d63f6450fe37f9f3e7264a04e9f233f9fdd8e08f03f5d","schema_version":"1.0","event_id":"sha256:62374abfc867556e672d63f6450fe37f9f3e7264a04e9f233f9fdd8e08f03f5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:3GBJ33K4VQSVMUA2D3R7Y6HVB2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"SIR epidemics with long range infection in one dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["physics.soc-ph"],"primary_cat":"cond-mat.stat-mech","authors_text":"Peter Grassberger","submitted_at":"2012-12-21T11:11:45Z","abstract_excerpt":"We study epidemic processes with immunization on very large 1-dimensional lattices, where at least some of the infections are non-local, with rates decaying as power laws p(x) ~ x^{-sigma-1} for large distances x. When starting with a single infected site, the cluster of infected sites stays always bounded if $\\sigma >1$ (and dies with probability 1, of its size is allowed to fluctuate down to zero), but the process can lead to an infinite epidemic for sigma <1. For sigma <0 the behavior is essentially of mean field type, but for 0 < sigma <= 1 the behavior is non-trivial, both for the critica"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5396","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":"2026-05-18T01:52:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ff/2Idfv2+sJVWP24URmTOBza3ZiwFdSuPNb8caYME5atnqi66O/2k8CA0oQONn0bA/oNEGLjRh1tE05a+IqBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T22:52:42.636483Z"},"content_sha256":"b45ad0ab4ec47fec38ee17169896d3e513caad2e3e28fd54e42e8cba0bfbdffe","schema_version":"1.0","event_id":"sha256:b45ad0ab4ec47fec38ee17169896d3e513caad2e3e28fd54e42e8cba0bfbdffe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3GBJ33K4VQSVMUA2D3R7Y6HVB2/bundle.json","state_url":"https://pith.science/pith/3GBJ33K4VQSVMUA2D3R7Y6HVB2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3GBJ33K4VQSVMUA2D3R7Y6HVB2/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":"2026-05-26T22:52:42Z","links":{"resolver":"https://pith.science/pith/3GBJ33K4VQSVMUA2D3R7Y6HVB2","bundle":"https://pith.science/pith/3GBJ33K4VQSVMUA2D3R7Y6HVB2/bundle.json","state":"https://pith.science/pith/3GBJ33K4VQSVMUA2D3R7Y6HVB2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3GBJ33K4VQSVMUA2D3R7Y6HVB2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3GBJ33K4VQSVMUA2D3R7Y6HVB2","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":"ab3cd1670f21b07dc166e7c3950c860e4568df73baa5fa122c937da0a865b18d","cross_cats_sorted":["physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-12-21T11:11:45Z","title_canon_sha256":"ccd1bd12e2ff45062a8d00e1b5ec3c9e8224442becfab10ff80cd5e687350170"},"schema_version":"1.0","source":{"id":"1212.5396","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.5396","created_at":"2026-05-18T01:52:29Z"},{"alias_kind":"arxiv_version","alias_value":"1212.5396v3","created_at":"2026-05-18T01:52:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5396","created_at":"2026-05-18T01:52:29Z"},{"alias_kind":"pith_short_12","alias_value":"3GBJ33K4VQSV","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3GBJ33K4VQSVMUA2","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3GBJ33K4","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:b45ad0ab4ec47fec38ee17169896d3e513caad2e3e28fd54e42e8cba0bfbdffe","target":"graph","created_at":"2026-05-18T01:52:29Z","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 epidemic processes with immunization on very large 1-dimensional lattices, where at least some of the infections are non-local, with rates decaying as power laws p(x) ~ x^{-sigma-1} for large distances x. When starting with a single infected site, the cluster of infected sites stays always bounded if $\\sigma >1$ (and dies with probability 1, of its size is allowed to fluctuate down to zero), but the process can lead to an infinite epidemic for sigma <1. For sigma <0 the behavior is essentially of mean field type, but for 0 < sigma <= 1 the behavior is non-trivial, both for the critica","authors_text":"Peter Grassberger","cross_cats":["physics.soc-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-12-21T11:11:45Z","title":"SIR epidemics with long range infection in one dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5396","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":"sha256:62374abfc867556e672d63f6450fe37f9f3e7264a04e9f233f9fdd8e08f03f5d","target":"record","created_at":"2026-05-18T01:52:29Z","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":"ab3cd1670f21b07dc166e7c3950c860e4568df73baa5fa122c937da0a865b18d","cross_cats_sorted":["physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-12-21T11:11:45Z","title_canon_sha256":"ccd1bd12e2ff45062a8d00e1b5ec3c9e8224442becfab10ff80cd5e687350170"},"schema_version":"1.0","source":{"id":"1212.5396","kind":"arxiv","version":3}},"canonical_sha256":"d9829ded5cac2556501a1ee3fc78f50e8dd27d6bad834d76435272c56873ec15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d9829ded5cac2556501a1ee3fc78f50e8dd27d6bad834d76435272c56873ec15","first_computed_at":"2026-05-18T01:52:29.312521Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:52:29.312521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8GIMLqac+L38/F8gw6DxNzMzLCb93ETk2/aD9n5HV5/H/7LWTAB4RjKajzL1cO7EJb5sp3YuW8AlLk+6NnwtCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:52:29.312966Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.5396","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62374abfc867556e672d63f6450fe37f9f3e7264a04e9f233f9fdd8e08f03f5d","sha256:b45ad0ab4ec47fec38ee17169896d3e513caad2e3e28fd54e42e8cba0bfbdffe"],"state_sha256":"bbb914400f90d8d50a6b1878108b09d9f5a297c512811f2b79ad531d1d6438d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D1bkjAncvUstas7B6VV7D5sld3BMgSjYaZvSmmniM/QYoYWNJcGehrQNzzEgyiQ7EnK7C8Fs3nBVPV1fUL/UDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T22:52:42.639693Z","bundle_sha256":"3e1301a2c1b72da280ad3f23b8b3feb58002b2906d885fa86dc3a04800e35191"}}