{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:JULLR56FAMSHX35YLAKBCNHMAW","short_pith_number":"pith:JULLR56F","canonical_record":{"source":{"id":"1107.1410","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-07T14:48:24Z","cross_cats_sorted":[],"title_canon_sha256":"0831d8aa717b2d69a342dc186ed1d01e0055b24353abc1ced5013fd472358897","abstract_canon_sha256":"8d2fba6ed7c54d85ca4b5ac24d1f3fd5b882ea95e2cefd4c7a2d430e60cd9967"},"schema_version":"1.0"},"canonical_sha256":"4d16b8f7c503247befb858141134ec05837dbcc2c3f7ca8aa8651ff8e0ba08a7","source":{"kind":"arxiv","id":"1107.1410","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.1410","created_at":"2026-05-18T04:01:26Z"},{"alias_kind":"arxiv_version","alias_value":"1107.1410v2","created_at":"2026-05-18T04:01:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1410","created_at":"2026-05-18T04:01:26Z"},{"alias_kind":"pith_short_12","alias_value":"JULLR56FAMSH","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JULLR56FAMSHX35Y","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JULLR56F","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:JULLR56FAMSHX35YLAKBCNHMAW","target":"record","payload":{"canonical_record":{"source":{"id":"1107.1410","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-07T14:48:24Z","cross_cats_sorted":[],"title_canon_sha256":"0831d8aa717b2d69a342dc186ed1d01e0055b24353abc1ced5013fd472358897","abstract_canon_sha256":"8d2fba6ed7c54d85ca4b5ac24d1f3fd5b882ea95e2cefd4c7a2d430e60cd9967"},"schema_version":"1.0"},"canonical_sha256":"4d16b8f7c503247befb858141134ec05837dbcc2c3f7ca8aa8651ff8e0ba08a7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:26.198665Z","signature_b64":"UvrxrmPlRVk526rsfoJUvgI5CaUSbN+ZE92COJVNGgROfyCRfZvJ5rbk442ISs3urxgkOQXrTqFYeE0AwVtcAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d16b8f7c503247befb858141134ec05837dbcc2c3f7ca8aa8651ff8e0ba08a7","last_reissued_at":"2026-05-18T04:01:26.197723Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:26.197723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.1410","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":"2026-05-18T04:01:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BFSMzCf90Urjzn9keGeUUMO2+xOxBLLmRwWBHoPRqpZi4uPWC4x4DFHvAA1Mk+Czj5U2uCTfUCBIeM5MpbyIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:11:32.798641Z"},"content_sha256":"87669c52cae1e812d1a00838b2aff82a258d97edbfd932b010149fc4b1d61d52","schema_version":"1.0","event_id":"sha256:87669c52cae1e812d1a00838b2aff82a258d97edbfd932b010149fc4b1d61d52"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:JULLR56FAMSHX35YLAKBCNHMAW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear algebra and bootstrap percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"B\\'ela Bollob\\'as, J\\'ozsef Balogh, Oliver Riordan, Robert Morris","submitted_at":"2011-07-07T14:48:24Z","abstract_excerpt":"In $\\HH$-bootstrap percolation, a set $A \\subset V(\\HH)$ of initially 'infected' vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph $\\HH$. A particular case of this is the $H$-bootstrap process, in which $\\HH$ encodes copies of $H$ in a graph $G$. We find the minimum size of a set $A$ that leads to complete infection when $G$ and $H$ are powers of complete graphs and $\\HH$ encodes induced copies of $H$ in $G$. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1410","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":"2026-05-18T04:01:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eDpcH4lv/wT5BCROdE6WnRsunVOdQ8r4l626DNaNptvheqvfDBY2TqWRAUYY4GPuXzdywutdg8jzbqTuNzTXCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:11:32.798986Z"},"content_sha256":"54a94f47cc679458b5e3595d75a2983c346c094ed94cfd56d921169bceb98ede","schema_version":"1.0","event_id":"sha256:54a94f47cc679458b5e3595d75a2983c346c094ed94cfd56d921169bceb98ede"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JULLR56FAMSHX35YLAKBCNHMAW/bundle.json","state_url":"https://pith.science/pith/JULLR56FAMSHX35YLAKBCNHMAW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JULLR56FAMSHX35YLAKBCNHMAW/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":"2026-06-02T23:11:32Z","links":{"resolver":"https://pith.science/pith/JULLR56FAMSHX35YLAKBCNHMAW","bundle":"https://pith.science/pith/JULLR56FAMSHX35YLAKBCNHMAW/bundle.json","state":"https://pith.science/pith/JULLR56FAMSHX35YLAKBCNHMAW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JULLR56FAMSHX35YLAKBCNHMAW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JULLR56FAMSHX35YLAKBCNHMAW","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":"8d2fba6ed7c54d85ca4b5ac24d1f3fd5b882ea95e2cefd4c7a2d430e60cd9967","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-07T14:48:24Z","title_canon_sha256":"0831d8aa717b2d69a342dc186ed1d01e0055b24353abc1ced5013fd472358897"},"schema_version":"1.0","source":{"id":"1107.1410","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.1410","created_at":"2026-05-18T04:01:26Z"},{"alias_kind":"arxiv_version","alias_value":"1107.1410v2","created_at":"2026-05-18T04:01:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1410","created_at":"2026-05-18T04:01:26Z"},{"alias_kind":"pith_short_12","alias_value":"JULLR56FAMSH","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JULLR56FAMSHX35Y","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JULLR56F","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:54a94f47cc679458b5e3595d75a2983c346c094ed94cfd56d921169bceb98ede","target":"graph","created_at":"2026-05-18T04:01:26Z","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":"In $\\HH$-bootstrap percolation, a set $A \\subset V(\\HH)$ of initially 'infected' vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph $\\HH$. A particular case of this is the $H$-bootstrap process, in which $\\HH$ encodes copies of $H$ in a graph $G$. We find the minimum size of a set $A$ that leads to complete infection when $G$ and $H$ are powers of complete graphs and $\\HH$ encodes induced copies of $H$ in $G$. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated ","authors_text":"B\\'ela Bollob\\'as, J\\'ozsef Balogh, Oliver Riordan, Robert Morris","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-07T14:48:24Z","title":"Linear algebra and bootstrap percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1410","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":"sha256:87669c52cae1e812d1a00838b2aff82a258d97edbfd932b010149fc4b1d61d52","target":"record","created_at":"2026-05-18T04:01:26Z","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":"8d2fba6ed7c54d85ca4b5ac24d1f3fd5b882ea95e2cefd4c7a2d430e60cd9967","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-07T14:48:24Z","title_canon_sha256":"0831d8aa717b2d69a342dc186ed1d01e0055b24353abc1ced5013fd472358897"},"schema_version":"1.0","source":{"id":"1107.1410","kind":"arxiv","version":2}},"canonical_sha256":"4d16b8f7c503247befb858141134ec05837dbcc2c3f7ca8aa8651ff8e0ba08a7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d16b8f7c503247befb858141134ec05837dbcc2c3f7ca8aa8651ff8e0ba08a7","first_computed_at":"2026-05-18T04:01:26.197723Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:01:26.197723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UvrxrmPlRVk526rsfoJUvgI5CaUSbN+ZE92COJVNGgROfyCRfZvJ5rbk442ISs3urxgkOQXrTqFYeE0AwVtcAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:01:26.198665Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.1410","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87669c52cae1e812d1a00838b2aff82a258d97edbfd932b010149fc4b1d61d52","sha256:54a94f47cc679458b5e3595d75a2983c346c094ed94cfd56d921169bceb98ede"],"state_sha256":"66d05af2ae6c852500ea2d63e875ce0fd99ff69f1f82fa8ebe252cb5daeee0b4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kdunYlbkgsJyFh0iMfURsVNK/I4IZbHvuIE2XK9cfITJbYanqSRn9GFDIhQfbtgJKLviVHdajZeDOKPSbWbcDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T23:11:32.801115Z","bundle_sha256":"d65e6684ffc6aaef5120755dab99c2ea471736fe36cd9216ae06061a81fecc35"}}