{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JJJV4JKKYDEUWYRQONCWAZTMQK","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":"04d63283f758ececdfb13f47fbd63f284da297c88664f3d7406b45e64365a475","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-05T10:30:33Z","title_canon_sha256":"7932219f1ab8f17e44ecfbe4e1efa022fdf6a2bf6f51990e22b9b3438b8c0c41"},"schema_version":"1.0","source":{"id":"1502.01490","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01490","created_at":"2026-05-18T01:34:24Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01490v2","created_at":"2026-05-18T01:34:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01490","created_at":"2026-05-18T01:34:24Z"},{"alias_kind":"pith_short_12","alias_value":"JJJV4JKKYDEU","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JJJV4JKKYDEUWYRQ","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JJJV4JKK","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:839db246599ffbd58e7e9ba8edc8ceb215c4e526779c18923a5ac694e75f8091","target":"graph","created_at":"2026-05-18T01:34:24Z","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":"Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with probability $p$ between any pair of vertices. Bootstrap percolation on a random graph is a process of spread of \"activation\" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r \\geq 2$ active neighbours become active as well. We study the size of the final act","authors_text":"Tatyana Turova, Thomas Vallier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-05T10:30:33Z","title":"Bootstrap percolation on a graph with random and local connections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01490","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":"sha256:321a0d6e157e634e39f10acdebb153e48643b55f51fc1e8c79609cf1ab7be850","target":"record","created_at":"2026-05-18T01:34:24Z","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":"04d63283f758ececdfb13f47fbd63f284da297c88664f3d7406b45e64365a475","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-05T10:30:33Z","title_canon_sha256":"7932219f1ab8f17e44ecfbe4e1efa022fdf6a2bf6f51990e22b9b3438b8c0c41"},"schema_version":"1.0","source":{"id":"1502.01490","kind":"arxiv","version":2}},"canonical_sha256":"4a535e254ac0c94b6230734560666c82bce90f4bd1dadaf697085c67641775c1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a535e254ac0c94b6230734560666c82bce90f4bd1dadaf697085c67641775c1","first_computed_at":"2026-05-18T01:34:24.695545Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:24.695545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YFTd/jzCm7s4UQC7udoot7sXVJLCDijAy98ENXtTTIe10pl8EBEmGsu4JhbjA9viFNTAbmTlBIftl+UELCm9CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:24.696067Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.01490","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:321a0d6e157e634e39f10acdebb153e48643b55f51fc1e8c79609cf1ab7be850","sha256:839db246599ffbd58e7e9ba8edc8ceb215c4e526779c18923a5ac694e75f8091"],"state_sha256":"8bdb79ca2113ebfdf73d1e4c2001b25eadf5c11030087c8405905b917ad48d07"}