{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:DUL5NOCZCQIQWJTRQB6TJ4AM3I","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":"5bb2d82588c73399351816ed1c8068c53d7786dd510364282c2336d1576532c9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.PR","submitted_at":"2015-04-02T17:10:20Z","title_canon_sha256":"99c63a98e2522fc55c697aed0f0cd541fe2244fd4f5ff4f54d80bfcf3fb7350b"},"schema_version":"1.0","source":{"id":"1504.00618","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.00618","created_at":"2026-05-18T02:19:27Z"},{"alias_kind":"arxiv_version","alias_value":"1504.00618v2","created_at":"2026-05-18T02:19:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00618","created_at":"2026-05-18T02:19:27Z"},{"alias_kind":"pith_short_12","alias_value":"DUL5NOCZCQIQ","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DUL5NOCZCQIQWJTR","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DUL5NOCZ","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:982805b252e42a4f2d2dcd381976f56d6ea000ea076acc033475097b1237bd45","target":"graph","created_at":"2026-05-18T02:19:27Z","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":"A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $\\tau$ of the degree distribution and the rate $\\delta d$ at which the connection probability decreases with the distance of tw","authors_text":"Emmanuel Jacob, Peter Morters","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.PR","submitted_at":"2015-04-02T17:10:20Z","title":"Robustness of scale-free spatial networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00618","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":"sha256:57f0690ec588e9cdb49617ef74e0947533b44011714efd88377a899ca1e2562f","target":"record","created_at":"2026-05-18T02:19:27Z","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":"5bb2d82588c73399351816ed1c8068c53d7786dd510364282c2336d1576532c9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.PR","submitted_at":"2015-04-02T17:10:20Z","title_canon_sha256":"99c63a98e2522fc55c697aed0f0cd541fe2244fd4f5ff4f54d80bfcf3fb7350b"},"schema_version":"1.0","source":{"id":"1504.00618","kind":"arxiv","version":2}},"canonical_sha256":"1d17d6b85914110b2671807d34f00cda06dddc73641cfa2d0020d3d10ef8641b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d17d6b85914110b2671807d34f00cda06dddc73641cfa2d0020d3d10ef8641b","first_computed_at":"2026-05-18T02:19:27.415843Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:27.415843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nV3Sr2cXEo7bjvaTeHpp0QfoXLTQYmPtq5LDc1UYBM70KFJz7yJypXeveOe2vVQiaCDXnUBNvFUTB4+5/QwoAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:27.416548Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.00618","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57f0690ec588e9cdb49617ef74e0947533b44011714efd88377a899ca1e2562f","sha256:982805b252e42a4f2d2dcd381976f56d6ea000ea076acc033475097b1237bd45"],"state_sha256":"53bdc1cef99f26fbe15d18d1fd11654efac358e83305986ae426ef3d37d7a7ce"}