{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:BH4KNMP3X3ANLTKRFOFIUJTMEP","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":"69531f7364eb1001c474e99f7fb123e96529ebd96163e02b06cb0afaeccf2fb2","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-12-20T14:08:43Z","title_canon_sha256":"b85c14bdb23204e6019bafc32a2d50c24ab248d57d90037e1ebe298e6477f6f1"},"schema_version":"1.0","source":{"id":"1012.4336","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.4336","created_at":"2026-05-18T04:19:03Z"},{"alias_kind":"arxiv_version","alias_value":"1012.4336v1","created_at":"2026-05-18T04:19:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.4336","created_at":"2026-05-18T04:19:03Z"},{"alias_kind":"pith_short_12","alias_value":"BH4KNMP3X3AN","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"BH4KNMP3X3ANLTKR","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"BH4KNMP3","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:1c4e6afc0edb63027f8b937dc9b975c6bd7c057df90bf826baf17b2e3561b8e3","target":"graph","created_at":"2026-05-18T04:19:03Z","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 introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is pruned from the network. The heterogeneous-$k$-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds $k_i$ are $1$ with probability $f$ or $k \\geq 3$ with probability $(1-f)$, the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolati","authors_text":"A. V. Goltsev, G. J. Baxter, J. F. F. Mendes, S. N. Dorogovtsev","cross_cats":["math-ph","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-12-20T14:08:43Z","title":"Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4336","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:a7ceb3a5e59e68f83dfd2c91a31aeaef97363073b674033b6f7e77cfabbdb4da","target":"record","created_at":"2026-05-18T04:19:03Z","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":"69531f7364eb1001c474e99f7fb123e96529ebd96163e02b06cb0afaeccf2fb2","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-12-20T14:08:43Z","title_canon_sha256":"b85c14bdb23204e6019bafc32a2d50c24ab248d57d90037e1ebe298e6477f6f1"},"schema_version":"1.0","source":{"id":"1012.4336","kind":"arxiv","version":1}},"canonical_sha256":"09f8a6b1fbbec0d5cd512b8a8a266c23eddd7ddbc01db386983402c2d9630b6f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09f8a6b1fbbec0d5cd512b8a8a266c23eddd7ddbc01db386983402c2d9630b6f","first_computed_at":"2026-05-18T04:19:03.234094Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:19:03.234094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"txsKNCTUUaFHCUCQgrSqJxGjA/8Sdla6c5k29RntSNXOwmLa/R7MWsdvjF3yquBQrI85PWmUcGeJs/Wjv51GBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:19:03.234595Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.4336","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a7ceb3a5e59e68f83dfd2c91a31aeaef97363073b674033b6f7e77cfabbdb4da","sha256:1c4e6afc0edb63027f8b937dc9b975c6bd7c057df90bf826baf17b2e3561b8e3"],"state_sha256":"8f194b468dd0f155af03008f70088a8167aa9ab6ef3823fe0d39de5a9b9b3a22"}