{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7T25Z5UNPRHYOFLOOZ4BQMVBTI","short_pith_number":"pith:7T25Z5UN","canonical_record":{"source":{"id":"1109.4311","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T14:51:30Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"b0114e6edbaabd9a51d2dc30306d2cb80599ae18d814f996d8c5b87036ef5448","abstract_canon_sha256":"e3239b0a2b1f70eeab86b6a584c21d29c7bf5749ea729a556f7a1f268bc93392"},"schema_version":"1.0"},"canonical_sha256":"fcf5dcf68d7c4f87156e76781832a19a0948e956c3f02998674ca039d32e1786","source":{"kind":"arxiv","id":"1109.4311","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4311","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4311v1","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4311","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"pith_short_12","alias_value":"7T25Z5UNPRHY","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7T25Z5UNPRHYOFLO","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7T25Z5UN","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7T25Z5UNPRHYOFLOOZ4BQMVBTI","target":"record","payload":{"canonical_record":{"source":{"id":"1109.4311","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T14:51:30Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"b0114e6edbaabd9a51d2dc30306d2cb80599ae18d814f996d8c5b87036ef5448","abstract_canon_sha256":"e3239b0a2b1f70eeab86b6a584c21d29c7bf5749ea729a556f7a1f268bc93392"},"schema_version":"1.0"},"canonical_sha256":"fcf5dcf68d7c4f87156e76781832a19a0948e956c3f02998674ca039d32e1786","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:43.588131Z","signature_b64":"CJbRhRIws7qinoGBmAhG7aMxabDldvcZjtQbfRYJ1FUg4IX//aoEFXTN5++wmKI3xvJM4XAkYeKh/UK68DlcCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcf5dcf68d7c4f87156e76781832a19a0948e956c3f02998674ca039d32e1786","last_reissued_at":"2026-05-18T04:12:43.587441Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:43.587441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.4311","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:12:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TMBuGzJ62iBQYjNTIW4b8c/60BRjldkrSPNidVLu9pibVTryF5pP4CDUoAm7EbTjQJrrPSgJiOifeDuQhfkEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T11:10:41.442441Z"},"content_sha256":"609a264cf4076c557bf3951a87a0abfbcc48fa0017faa900859b9f398ac2ded9","schema_version":"1.0","event_id":"sha256:609a264cf4076c557bf3951a87a0abfbcc48fa0017faa900859b9f398ac2ded9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7T25Z5UNPRHYOFLOOZ4BQMVBTI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Mixing Time of Geographical Threshold Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.PR","authors_text":"Andrew Beveridge, Milan Bradonji\\'c","submitted_at":"2011-09-20T14:51:30Z","abstract_excerpt":"We study the mixing time of random graphs in the $d$-dimensional toric unit cube $[0,1]^d$ generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights, drawn from some distribution. The connectivity threshold for GTGs is comparable to that of RGGs, essentially corresponding to a connectivity radius of $r=(\\log n/n)^{1/d}$. However, the degree distributions at "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4311","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:12:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FMsu0X7jCsIO3vwXCzOyaXgW4C8PJS4P99MI10yup+EFrpEMw5TzCmwNyJ3i2MPC7fdEFRTMG+Vka+FHld5ODQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T11:10:41.442779Z"},"content_sha256":"e88b3e35b75572ad2513f7777a565ee5482c1d8a8241b5bb0ab5319937d88847","schema_version":"1.0","event_id":"sha256:e88b3e35b75572ad2513f7777a565ee5482c1d8a8241b5bb0ab5319937d88847"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7T25Z5UNPRHYOFLOOZ4BQMVBTI/bundle.json","state_url":"https://pith.science/pith/7T25Z5UNPRHYOFLOOZ4BQMVBTI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7T25Z5UNPRHYOFLOOZ4BQMVBTI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T11:10:41Z","links":{"resolver":"https://pith.science/pith/7T25Z5UNPRHYOFLOOZ4BQMVBTI","bundle":"https://pith.science/pith/7T25Z5UNPRHYOFLOOZ4BQMVBTI/bundle.json","state":"https://pith.science/pith/7T25Z5UNPRHYOFLOOZ4BQMVBTI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7T25Z5UNPRHYOFLOOZ4BQMVBTI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7T25Z5UNPRHYOFLOOZ4BQMVBTI","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":"e3239b0a2b1f70eeab86b6a584c21d29c7bf5749ea729a556f7a1f268bc93392","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T14:51:30Z","title_canon_sha256":"b0114e6edbaabd9a51d2dc30306d2cb80599ae18d814f996d8c5b87036ef5448"},"schema_version":"1.0","source":{"id":"1109.4311","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4311","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4311v1","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4311","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"pith_short_12","alias_value":"7T25Z5UNPRHY","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7T25Z5UNPRHYOFLO","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7T25Z5UN","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:e88b3e35b75572ad2513f7777a565ee5482c1d8a8241b5bb0ab5319937d88847","target":"graph","created_at":"2026-05-18T04:12:43Z","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 study the mixing time of random graphs in the $d$-dimensional toric unit cube $[0,1]^d$ generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights, drawn from some distribution. The connectivity threshold for GTGs is comparable to that of RGGs, essentially corresponding to a connectivity radius of $r=(\\log n/n)^{1/d}$. However, the degree distributions at ","authors_text":"Andrew Beveridge, Milan Bradonji\\'c","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T14:51:30Z","title":"On the Mixing Time of Geographical Threshold Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4311","kind":"arxiv","version":1},"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":"sha256:609a264cf4076c557bf3951a87a0abfbcc48fa0017faa900859b9f398ac2ded9","target":"record","created_at":"2026-05-18T04:12:43Z","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":"e3239b0a2b1f70eeab86b6a584c21d29c7bf5749ea729a556f7a1f268bc93392","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T14:51:30Z","title_canon_sha256":"b0114e6edbaabd9a51d2dc30306d2cb80599ae18d814f996d8c5b87036ef5448"},"schema_version":"1.0","source":{"id":"1109.4311","kind":"arxiv","version":1}},"canonical_sha256":"fcf5dcf68d7c4f87156e76781832a19a0948e956c3f02998674ca039d32e1786","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fcf5dcf68d7c4f87156e76781832a19a0948e956c3f02998674ca039d32e1786","first_computed_at":"2026-05-18T04:12:43.587441Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:43.587441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CJbRhRIws7qinoGBmAhG7aMxabDldvcZjtQbfRYJ1FUg4IX//aoEFXTN5++wmKI3xvJM4XAkYeKh/UK68DlcCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:43.588131Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.4311","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:609a264cf4076c557bf3951a87a0abfbcc48fa0017faa900859b9f398ac2ded9","sha256:e88b3e35b75572ad2513f7777a565ee5482c1d8a8241b5bb0ab5319937d88847"],"state_sha256":"e901f8869e13d145785c9bc467c671929d7c61637045a2684e01f6ba9520acfe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"45EeUjv9J0LhX7g8TRyZCTNT6iT59FJwOk5QJIcU3CzSSNREMdtauFj/bg9aBm8vO4lymCCn5So+zZbdhXEYCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T11:10:41.444618Z","bundle_sha256":"70b35608f5583e98ab72197490b605a97899ca859ad6137c4bbc77001da2edfe"}}