{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:LIFHYIYYEP3VQXH47KCAV6OZSQ","short_pith_number":"pith:LIFHYIYY","canonical_record":{"source":{"id":"1801.10131","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-30T18:29:49Z","cross_cats_sorted":[],"title_canon_sha256":"4870beaeb08c2445b807215b72556cef5bfe360194f94d1eaa1ef2877ac85fab","abstract_canon_sha256":"8c0e4e03c1a6ad7f3ba321e600ce445d8f40707833ce5a94a809238f2d6a885e"},"schema_version":"1.0"},"canonical_sha256":"5a0a7c231823f7585cfcfa840af9d9941ffbadb208d2e434294b71bd186ec5d8","source":{"kind":"arxiv","id":"1801.10131","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.10131","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"arxiv_version","alias_value":"1801.10131v1","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.10131","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"pith_short_12","alias_value":"LIFHYIYYEP3V","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"LIFHYIYYEP3VQXH4","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"LIFHYIYY","created_at":"2026-05-18T12:32:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:LIFHYIYYEP3VQXH47KCAV6OZSQ","target":"record","payload":{"canonical_record":{"source":{"id":"1801.10131","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-30T18:29:49Z","cross_cats_sorted":[],"title_canon_sha256":"4870beaeb08c2445b807215b72556cef5bfe360194f94d1eaa1ef2877ac85fab","abstract_canon_sha256":"8c0e4e03c1a6ad7f3ba321e600ce445d8f40707833ce5a94a809238f2d6a885e"},"schema_version":"1.0"},"canonical_sha256":"5a0a7c231823f7585cfcfa840af9d9941ffbadb208d2e434294b71bd186ec5d8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:45.798729Z","signature_b64":"508y6ADPcjzyuNwvBzm/6L9/WDCnoGOyDDrWhYtE4f/554Qbn52XcFG2OEYFODYWtiRzH0359wyfkw9NqN2EBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a0a7c231823f7585cfcfa840af9d9941ffbadb208d2e434294b71bd186ec5d8","last_reissued_at":"2026-05-18T00:24:45.798099Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:45.798099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.10131","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":"2026-05-18T00:24:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8zXgLywV0JrfNRDm/1i2Ivfp3kEOFr+a2Up+OpUHepvL2P/Px1u+O+iYR9j+NanJiD7oEbxrrSt/vEdX3pjMBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:06:51.239136Z"},"content_sha256":"636b112ee6d88e48b2879ca6bab91ea4a7b77cc1de62fd1cef8bcddf6bc1858b","schema_version":"1.0","event_id":"sha256:636b112ee6d88e48b2879ca6bab91ea4a7b77cc1de62fd1cef8bcddf6bc1858b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:LIFHYIYYEP3VQXH47KCAV6OZSQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Long scale Ollivier-Ricci curvature of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Cushing, Supanat Kamtue","submitted_at":"2018-01-30T18:29:49Z","abstract_excerpt":"We study the long scale Ollivier-Ricci curvature of graphs as a function of the chosen idleness. As in the previous work on the short scale, we show that this idleness function is concave and piecewise linear with at most $3$ linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long scale curvature inside the Cartesian product of two regular graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.10131","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":"2026-05-18T00:24:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PVjBVU5zJyJKN1sSnZ4OZqfcwgWPrU5jS3ULYCOq4xpVLohy9CzNONyzEdQz/2bElIaaAiQgjM4BCVOfZfgCDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:06:51.239503Z"},"content_sha256":"f3b23219e47e64dc6acfef0c525de695e098608576f9b5d6ef10fa252bb937b2","schema_version":"1.0","event_id":"sha256:f3b23219e47e64dc6acfef0c525de695e098608576f9b5d6ef10fa252bb937b2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LIFHYIYYEP3VQXH47KCAV6OZSQ/bundle.json","state_url":"https://pith.science/pith/LIFHYIYYEP3VQXH47KCAV6OZSQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LIFHYIYYEP3VQXH47KCAV6OZSQ/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":"2026-06-01T19:06:51Z","links":{"resolver":"https://pith.science/pith/LIFHYIYYEP3VQXH47KCAV6OZSQ","bundle":"https://pith.science/pith/LIFHYIYYEP3VQXH47KCAV6OZSQ/bundle.json","state":"https://pith.science/pith/LIFHYIYYEP3VQXH47KCAV6OZSQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LIFHYIYYEP3VQXH47KCAV6OZSQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:LIFHYIYYEP3VQXH47KCAV6OZSQ","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":"8c0e4e03c1a6ad7f3ba321e600ce445d8f40707833ce5a94a809238f2d6a885e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-30T18:29:49Z","title_canon_sha256":"4870beaeb08c2445b807215b72556cef5bfe360194f94d1eaa1ef2877ac85fab"},"schema_version":"1.0","source":{"id":"1801.10131","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.10131","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"arxiv_version","alias_value":"1801.10131v1","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.10131","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"pith_short_12","alias_value":"LIFHYIYYEP3V","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"LIFHYIYYEP3VQXH4","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"LIFHYIYY","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:f3b23219e47e64dc6acfef0c525de695e098608576f9b5d6ef10fa252bb937b2","target":"graph","created_at":"2026-05-18T00:24:45Z","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 long scale Ollivier-Ricci curvature of graphs as a function of the chosen idleness. As in the previous work on the short scale, we show that this idleness function is concave and piecewise linear with at most $3$ linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long scale curvature inside the Cartesian product of two regular graphs.","authors_text":"David Cushing, Supanat Kamtue","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-30T18:29:49Z","title":"Long scale Ollivier-Ricci curvature of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.10131","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":"sha256:636b112ee6d88e48b2879ca6bab91ea4a7b77cc1de62fd1cef8bcddf6bc1858b","target":"record","created_at":"2026-05-18T00:24:45Z","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":"8c0e4e03c1a6ad7f3ba321e600ce445d8f40707833ce5a94a809238f2d6a885e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-30T18:29:49Z","title_canon_sha256":"4870beaeb08c2445b807215b72556cef5bfe360194f94d1eaa1ef2877ac85fab"},"schema_version":"1.0","source":{"id":"1801.10131","kind":"arxiv","version":1}},"canonical_sha256":"5a0a7c231823f7585cfcfa840af9d9941ffbadb208d2e434294b71bd186ec5d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5a0a7c231823f7585cfcfa840af9d9941ffbadb208d2e434294b71bd186ec5d8","first_computed_at":"2026-05-18T00:24:45.798099Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:45.798099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"508y6ADPcjzyuNwvBzm/6L9/WDCnoGOyDDrWhYtE4f/554Qbn52XcFG2OEYFODYWtiRzH0359wyfkw9NqN2EBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:45.798729Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.10131","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:636b112ee6d88e48b2879ca6bab91ea4a7b77cc1de62fd1cef8bcddf6bc1858b","sha256:f3b23219e47e64dc6acfef0c525de695e098608576f9b5d6ef10fa252bb937b2"],"state_sha256":"00e5010dbb07e7ab6e75b3e14752143eafad46d6090bdc1effb89016cc8a3197"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zdBVPnXn1NQoSB8pymQwDWHeMtoKuBoDamXbq4odVlQL06l43BLEU57kfYaU7ZfWdZI1GyVUVJ4TTN3ZcbiSBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T19:06:51.241489Z","bundle_sha256":"42579303c3c817d003a6f1540c484e831c55def2cd2d5bf149ef98c82314f44c"}}