{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:63WIWXC2ZFUWBUBCVNGOAWUNCJ","short_pith_number":"pith:63WIWXC2","schema_version":"1.0","canonical_sha256":"f6ec8b5c5ac96960d022ab4ce05a8d12410c395441e51cd7aef0dc6bdee5dfb8","source":{"kind":"arxiv","id":"2607.17597","version":1},"attestation_state":"computed","paper":{"title":"Spatial Dependence in Directed Preferential-Attachment Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"stat.ME","authors_text":"Tiandong Wang, Zihan Li","submitted_at":"2026-07-20T06:30:19Z","abstract_excerpt":"Spatially embedded directed networks, such as airline networks, often exhibit simultaneous high activity at nearby nodes. Preferential attachment (PA) explains hub dominance. We extend it to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields. Under sublinear PA, out-degree proportions converge to explicit normalized powered weights, whereas self-loop exclusion yields a coupled in-degree limit. We derive a strictly concave inverse that recovers the in-weights from terminal degree proportions. For ordered "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.17597","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ME","submitted_at":"2026-07-20T06:30:19Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"72cda41afe4750cb6303a8ec890890d87c2d8c43d31b0d692b064724e7b51a38","abstract_canon_sha256":"36a5a91aba7735291eddbcc8e649993ae365f19385009c28c294a8895eb80f60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T01:21:54.361340Z","signature_b64":"H8pd1OI10TtsbNwi7nUxx8OY91bRW1+0mBN8Z43NH3ptCgvaEV2YPJdwUiw45JMtYXUHw1NPmcJEQB6liwqHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6ec8b5c5ac96960d022ab4ce05a8d12410c395441e51cd7aef0dc6bdee5dfb8","last_reissued_at":"2026-07-21T01:21:54.360514Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T01:21:54.360514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spatial Dependence in Directed Preferential-Attachment Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"stat.ME","authors_text":"Tiandong Wang, Zihan Li","submitted_at":"2026-07-20T06:30:19Z","abstract_excerpt":"Spatially embedded directed networks, such as airline networks, often exhibit simultaneous high activity at nearby nodes. Preferential attachment (PA) explains hub dominance. We extend it to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields. Under sublinear PA, out-degree proportions converge to explicit normalized powered weights, whereas self-loop exclusion yields a coupled in-degree limit. We derive a strictly concave inverse that recovers the in-weights from terminal degree proportions. For ordered "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17597","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.17597/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.17597","created_at":"2026-07-21T01:21:54.360940+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.17597v1","created_at":"2026-07-21T01:21:54.360940+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.17597","created_at":"2026-07-21T01:21:54.360940+00:00"},{"alias_kind":"pith_short_12","alias_value":"63WIWXC2ZFUW","created_at":"2026-07-21T01:21:54.360940+00:00"},{"alias_kind":"pith_short_16","alias_value":"63WIWXC2ZFUWBUBC","created_at":"2026-07-21T01:21:54.360940+00:00"},{"alias_kind":"pith_short_8","alias_value":"63WIWXC2","created_at":"2026-07-21T01:21:54.360940+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ","json":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ.json","graph_json":"https://pith.science/api/pith-number/63WIWXC2ZFUWBUBCVNGOAWUNCJ/graph.json","events_json":"https://pith.science/api/pith-number/63WIWXC2ZFUWBUBCVNGOAWUNCJ/events.json","paper":"https://pith.science/paper/63WIWXC2"},"agent_actions":{"view_html":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ","download_json":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ.json","view_paper":"https://pith.science/paper/63WIWXC2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.17597&json=true","fetch_graph":"https://pith.science/api/pith-number/63WIWXC2ZFUWBUBCVNGOAWUNCJ/graph.json","fetch_events":"https://pith.science/api/pith-number/63WIWXC2ZFUWBUBCVNGOAWUNCJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ/action/storage_attestation","attest_author":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ/action/author_attestation","sign_citation":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ/action/citation_signature","submit_replication":"https://pith.science/pith/63WIWXC2ZFUWBUBCVNGOAWUNCJ/action/replication_record"}},"created_at":"2026-07-21T01:21:54.360940+00:00","updated_at":"2026-07-21T01:21:54.360940+00:00"}