{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:QI2Z64H23SERLYV42CA2GYI6IJ","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":"00b5311994ecb4c7df4ce2020b30c6d0be31471a296a7fe668532b46afb5a071","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-06T18:40:24Z","title_canon_sha256":"b30eeb4f4c10ca76f5a58bde8d5b0a33190af4287e449bd34fbe3497c64579c0"},"schema_version":"1.0","source":{"id":"0902.1156","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.1156","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"arxiv_version","alias_value":"0902.1156v2","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.1156","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"pith_short_12","alias_value":"QI2Z64H23SER","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"QI2Z64H23SERLYV4","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"QI2Z64H2","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:8bcfe251a89cf29075827b0d900ccf918f7a95329dc25943362a4b46b19258d4","target":"graph","created_at":"2026-05-18T03:49:09Z","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":"The spread of a connected graph G was introduced by Alon, Boppana and Spencer (1998) and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph; in particular for random regular graphs G(n,d), for Erd\\H{o}s-R\\'enyi random graphs G_{n,p} in the supercritical range p>1/n, and for a 'small world' model. For supercritical G_{n,p}, we show that if p=c/n with c>1 fixed then with high probability the spread of th","authors_text":"Colin McDiarmid, Louigi Addario-Berry, Svante Janson","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-06T18:40:24Z","title":"On the spread of random graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.1156","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":"sha256:81632fffc202a2148faba1c607788a88290d24e1136f6bb91c738d5dc700975a","target":"record","created_at":"2026-05-18T03:49:09Z","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":"00b5311994ecb4c7df4ce2020b30c6d0be31471a296a7fe668532b46afb5a071","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-02-06T18:40:24Z","title_canon_sha256":"b30eeb4f4c10ca76f5a58bde8d5b0a33190af4287e449bd34fbe3497c64579c0"},"schema_version":"1.0","source":{"id":"0902.1156","kind":"arxiv","version":2}},"canonical_sha256":"82359f70fadc8915e2bcd081a3611e426dd3d458685450703f00881eaeb19e13","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82359f70fadc8915e2bcd081a3611e426dd3d458685450703f00881eaeb19e13","first_computed_at":"2026-05-18T03:49:09.464137Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:09.464137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bD0GBINMnNKvUn/ZAkvYjh6sweHIxSqEGIrEivLWoMemq+LVF+xk/xONGOHxQl38yIoWNH9MRw+s0+O9xWTMDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:09.464700Z","signed_message":"canonical_sha256_bytes"},"source_id":"0902.1156","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:81632fffc202a2148faba1c607788a88290d24e1136f6bb91c738d5dc700975a","sha256:8bcfe251a89cf29075827b0d900ccf918f7a95329dc25943362a4b46b19258d4"],"state_sha256":"7820d71a831c1cf327618a3fab523723ed6bffbfa08fc128ecd74e5491f0d24a"}