{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BKDO2JR5J64CAPWL7UNRBAEEJE","short_pith_number":"pith:BKDO2JR5","canonical_record":{"source":{"id":"1111.2687","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-11T09:12:15Z","cross_cats_sorted":["math.DG","math.FA","math.PR"],"title_canon_sha256":"222327f1bc79cef9b789dfc3f49d55927c998f48acd7cdebaaf376ebd6b71a3a","abstract_canon_sha256":"23c1db2b38c127507827d0d5073bf19dbeb23a0cac4996a1fa2c7840ebc82eab"},"schema_version":"1.0"},"canonical_sha256":"0a86ed263d4fb8203ecbfd1b108084490bddbe6e6deb5862492f4552f30d5411","source":{"kind":"arxiv","id":"1111.2687","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2687","created_at":"2026-05-18T01:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2687v2","created_at":"2026-05-18T01:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2687","created_at":"2026-05-18T01:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"BKDO2JR5J64C","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BKDO2JR5J64CAPWL","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BKDO2JR5","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BKDO2JR5J64CAPWL7UNRBAEEJE","target":"record","payload":{"canonical_record":{"source":{"id":"1111.2687","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-11T09:12:15Z","cross_cats_sorted":["math.DG","math.FA","math.PR"],"title_canon_sha256":"222327f1bc79cef9b789dfc3f49d55927c998f48acd7cdebaaf376ebd6b71a3a","abstract_canon_sha256":"23c1db2b38c127507827d0d5073bf19dbeb23a0cac4996a1fa2c7840ebc82eab"},"schema_version":"1.0"},"canonical_sha256":"0a86ed263d4fb8203ecbfd1b108084490bddbe6e6deb5862492f4552f30d5411","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:40.307121Z","signature_b64":"qpr8TA4PmEPQhlGfJixrFmRAro0puOr069ofqTUs1+uFZeAuuRR6aj5Yqj9ZsAmSFWmKh28h8W2/w+XLv0/wDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a86ed263d4fb8203ecbfd1b108084490bddbe6e6deb5862492f4552f30d5411","last_reissued_at":"2026-05-18T01:59:40.306524Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:40.306524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.2687","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E84AjEr4r4QEAZV55xvGRkm0nUBTal4kYTN4z6IYFexjYL6I07ACAoC0dLAT/WQx1TqGYUf6oEKJSWl86WPLCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T22:49:24.509485Z"},"content_sha256":"830392d11ca4dfdd0d5b8b5dd1786281406e78b7acd565f50e99c8a8631cead6","schema_version":"1.0","event_id":"sha256:830392d11ca4dfdd0d5b8b5dd1786281406e78b7acd565f50e99c8a8631cead6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BKDO2JR5J64CAPWL7UNRBAEEJE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ricci curvature of finite Markov chains via convexity of the entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Jan Maas, Matthias Erbar","submitted_at":"2011-11-11T09:12:15Z","abstract_excerpt":"We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy.\n  Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2687","kind":"arxiv","version":2},"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":"2026-05-18T01:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JVUxMwdoS/5uoZX0EjWnietI2e+vbuCvPXV7W+R+Vh79k/cpZNY+JGPXNErOTFucNRBgRPdtBz44otMOmKssAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T22:49:24.509853Z"},"content_sha256":"257febe34d429c079313935ddd29bc27ca474d045f907315fd5df5d4853b12a4","schema_version":"1.0","event_id":"sha256:257febe34d429c079313935ddd29bc27ca474d045f907315fd5df5d4853b12a4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BKDO2JR5J64CAPWL7UNRBAEEJE/bundle.json","state_url":"https://pith.science/pith/BKDO2JR5J64CAPWL7UNRBAEEJE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BKDO2JR5J64CAPWL7UNRBAEEJE/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":"2026-05-29T22:49:24Z","links":{"resolver":"https://pith.science/pith/BKDO2JR5J64CAPWL7UNRBAEEJE","bundle":"https://pith.science/pith/BKDO2JR5J64CAPWL7UNRBAEEJE/bundle.json","state":"https://pith.science/pith/BKDO2JR5J64CAPWL7UNRBAEEJE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BKDO2JR5J64CAPWL7UNRBAEEJE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BKDO2JR5J64CAPWL7UNRBAEEJE","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":"23c1db2b38c127507827d0d5073bf19dbeb23a0cac4996a1fa2c7840ebc82eab","cross_cats_sorted":["math.DG","math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-11T09:12:15Z","title_canon_sha256":"222327f1bc79cef9b789dfc3f49d55927c998f48acd7cdebaaf376ebd6b71a3a"},"schema_version":"1.0","source":{"id":"1111.2687","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2687","created_at":"2026-05-18T01:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2687v2","created_at":"2026-05-18T01:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2687","created_at":"2026-05-18T01:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"BKDO2JR5J64C","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BKDO2JR5J64CAPWL","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BKDO2JR5","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:257febe34d429c079313935ddd29bc27ca474d045f907315fd5df5d4853b12a4","target":"graph","created_at":"2026-05-18T01:59:40Z","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 a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy.\n  Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ri","authors_text":"Jan Maas, Matthias Erbar","cross_cats":["math.DG","math.FA","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-11T09:12:15Z","title":"Ricci curvature of finite Markov chains via convexity of the entropy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2687","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":"sha256:830392d11ca4dfdd0d5b8b5dd1786281406e78b7acd565f50e99c8a8631cead6","target":"record","created_at":"2026-05-18T01:59:40Z","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":"23c1db2b38c127507827d0d5073bf19dbeb23a0cac4996a1fa2c7840ebc82eab","cross_cats_sorted":["math.DG","math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-11T09:12:15Z","title_canon_sha256":"222327f1bc79cef9b789dfc3f49d55927c998f48acd7cdebaaf376ebd6b71a3a"},"schema_version":"1.0","source":{"id":"1111.2687","kind":"arxiv","version":2}},"canonical_sha256":"0a86ed263d4fb8203ecbfd1b108084490bddbe6e6deb5862492f4552f30d5411","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a86ed263d4fb8203ecbfd1b108084490bddbe6e6deb5862492f4552f30d5411","first_computed_at":"2026-05-18T01:59:40.306524Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:59:40.306524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qpr8TA4PmEPQhlGfJixrFmRAro0puOr069ofqTUs1+uFZeAuuRR6aj5Yqj9ZsAmSFWmKh28h8W2/w+XLv0/wDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:59:40.307121Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.2687","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:830392d11ca4dfdd0d5b8b5dd1786281406e78b7acd565f50e99c8a8631cead6","sha256:257febe34d429c079313935ddd29bc27ca474d045f907315fd5df5d4853b12a4"],"state_sha256":"76c50fb03bb6299c98b061d76a1275c9b083c69bac7c24b2eb160e4566ff359a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cRZmS5rUmCw67rcCQ9d0KkeYjQKuWxjkjbOZPx6VNpMdN/5823x+ZXQR30MofNJss3MX9u1VFC4S4UqgNAGcAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T22:49:24.511921Z","bundle_sha256":"c8b2a35e7d43eed2c5acf00d0c5f1afe9fbe550a28e710cb56eda3ad8796da28"}}