{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FRTPKD4CLYGU2PVUIGJPQU7EZB","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":"d3cbb43d8f1adf70b45063abdbac72f80fb674401d9805d53d14f0df434229ef","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-14T20:14:16Z","title_canon_sha256":"7afbada766f3b1614ebdeaf48c57801bfee48515a74e13f06035a5505fc133dd"},"schema_version":"1.0","source":{"id":"1210.3830","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.3830","created_at":"2026-05-18T02:22:30Z"},{"alias_kind":"arxiv_version","alias_value":"1210.3830v2","created_at":"2026-05-18T02:22:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.3830","created_at":"2026-05-18T02:22:30Z"},{"alias_kind":"pith_short_12","alias_value":"FRTPKD4CLYGU","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FRTPKD4CLYGU2PVU","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FRTPKD4C","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:71c99852d03775315b1befd1544a3acbd838ee4d8bd44414c23943000b441e67","target":"graph","created_at":"2026-05-18T02:22:30Z","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 define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $\\tau>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distributi","authors_text":"Emmanuel Jacob, Peter M\\\"orters","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-14T20:14:16Z","title":"Spatial preferential attachment networks: Power laws and clustering coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3830","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:b52bddb7968b57920a7da8e081dee493b774d32055230061f0263a8848b1e7d4","target":"record","created_at":"2026-05-18T02:22:30Z","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":"d3cbb43d8f1adf70b45063abdbac72f80fb674401d9805d53d14f0df434229ef","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-14T20:14:16Z","title_canon_sha256":"7afbada766f3b1614ebdeaf48c57801bfee48515a74e13f06035a5505fc133dd"},"schema_version":"1.0","source":{"id":"1210.3830","kind":"arxiv","version":2}},"canonical_sha256":"2c66f50f825e0d4d3eb44192f853e4c842d9ccdb52c59548e6e4e47be1e6598f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c66f50f825e0d4d3eb44192f853e4c842d9ccdb52c59548e6e4e47be1e6598f","first_computed_at":"2026-05-18T02:22:30.261765Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:22:30.261765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U3PqIF09fJ5A9GqCydxJ5tp7Edi7fZYqTbuM2BJLOfglMUvmtU5FXUJndW3NbRo9Hg8NjSYynmR5YTLPMUgQBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:22:30.262387Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.3830","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b52bddb7968b57920a7da8e081dee493b774d32055230061f0263a8848b1e7d4","sha256:71c99852d03775315b1befd1544a3acbd838ee4d8bd44414c23943000b441e67"],"state_sha256":"c2977bd455e1a8e19f04092b58bf25755eaa70e7af4c129c3f561ea0fec47ddc"}