{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4HNTFCFFP36GQHAHDVASABJMNC","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":"c6c017a09b0f229fcbcee38957b6f7a5c8baf1841ae2e9153c4c05e37320df5a","cross_cats_sorted":["cond-mat.dis-nn","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-05-15T05:41:03Z","title_canon_sha256":"a8a3416e1c81a1f92167bc2098b70a0633e314a5b7de1d532d6fea0a57c91e3c"},"schema_version":"1.0","source":{"id":"1505.03963","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.03963","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"arxiv_version","alias_value":"1505.03963v5","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03963","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"pith_short_12","alias_value":"4HNTFCFFP36G","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4HNTFCFFP36GQHAH","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4HNTFCFF","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:36eadead076e8f7f633a04b2c2964333b10d7555c193e39a67b2c50440e9e42e","target":"graph","created_at":"2026-05-18T01:02:13Z","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 analyze site percolation on directed and undirected graphs with site-dependent open-site probabilities. We construct upper bounds on cluster susceptibilities, vertex connectivity functions, and the expected number of simple open cycles through a chosen arc; separate bounds are given on finite and infinite (di)graphs. These produce lower bounds for percolation and uniqueness transitions in infinite (di)graphs, and for the formation of a giant component in finite (di)graphs. The bounds are formulated in terms of appropriately weighted adjacency and non-backtracking (Hashimoto) matrices. It tu","authors_text":"Kathleen E. Hamilton, Leonid P. Pryadko","cross_cats":["cond-mat.dis-nn","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-05-15T05:41:03Z","title":"Algebraic bounds for heterogeneous site percolation on directed and undirected graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03963","kind":"arxiv","version":5},"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":"sha256:512eed37b542cd46ef8d119236285574450a851ed8242f0f7243caf42d7d6edf","target":"record","created_at":"2026-05-18T01:02:13Z","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":"c6c017a09b0f229fcbcee38957b6f7a5c8baf1841ae2e9153c4c05e37320df5a","cross_cats_sorted":["cond-mat.dis-nn","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-05-15T05:41:03Z","title_canon_sha256":"a8a3416e1c81a1f92167bc2098b70a0633e314a5b7de1d532d6fea0a57c91e3c"},"schema_version":"1.0","source":{"id":"1505.03963","kind":"arxiv","version":5}},"canonical_sha256":"e1db3288a57efc681c071d4120052c688addeb8132f867d60c36dcd6beb5efdd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1db3288a57efc681c071d4120052c688addeb8132f867d60c36dcd6beb5efdd","first_computed_at":"2026-05-18T01:02:13.934341Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:13.934341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8TyFOOIPSd8oQ22EnMBDoPPgwwXslClNsIE355MZsiLFEiWJoeoC2rJqazMui8RB/CBoCZjGxhjo7XihMnrrDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:13.935052Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.03963","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:512eed37b542cd46ef8d119236285574450a851ed8242f0f7243caf42d7d6edf","sha256:36eadead076e8f7f633a04b2c2964333b10d7555c193e39a67b2c50440e9e42e"],"state_sha256":"491c4082f04ac008a84c31488081391a7fbfbafa41efcf1932350f6366d0d93d"}