{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:3OWGAYWAO62D5EGKUY754QG4RM","short_pith_number":"pith:3OWGAYWA","canonical_record":{"source":{"id":"1607.04453","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-15T10:52:22Z","cross_cats_sorted":[],"title_canon_sha256":"805ad2ba81d4289d5a608c050859f7e774a1269f4aac7253ceaed7850ff7ccc4","abstract_canon_sha256":"b718a70b2454cfd8e03b6689d47dd082ee420679c819513cc134c7302996fc94"},"schema_version":"1.0"},"canonical_sha256":"dbac6062c077b43e90caa63fde40dc8b27d907b39effd58c5275b8f0106a6701","source":{"kind":"arxiv","id":"1607.04453","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.04453","created_at":"2026-05-18T01:11:01Z"},{"alias_kind":"arxiv_version","alias_value":"1607.04453v1","created_at":"2026-05-18T01:11:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.04453","created_at":"2026-05-18T01:11:01Z"},{"alias_kind":"pith_short_12","alias_value":"3OWGAYWAO62D","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3OWGAYWAO62D5EGK","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3OWGAYWA","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:3OWGAYWAO62D5EGKUY754QG4RM","target":"record","payload":{"canonical_record":{"source":{"id":"1607.04453","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-15T10:52:22Z","cross_cats_sorted":[],"title_canon_sha256":"805ad2ba81d4289d5a608c050859f7e774a1269f4aac7253ceaed7850ff7ccc4","abstract_canon_sha256":"b718a70b2454cfd8e03b6689d47dd082ee420679c819513cc134c7302996fc94"},"schema_version":"1.0"},"canonical_sha256":"dbac6062c077b43e90caa63fde40dc8b27d907b39effd58c5275b8f0106a6701","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:01.198173Z","signature_b64":"777qlrkHYGZbf7QabKkzwPFgmHhMniuuyjxxCTtjWlXS4C0gfKO3xyJX3WayUhDPJ51XMa2YInNVzCVWGJU2DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbac6062c077b43e90caa63fde40dc8b27d907b39effd58c5275b8f0106a6701","last_reissued_at":"2026-05-18T01:11:01.197691Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:01.197691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.04453","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":"2026-05-18T01:11:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vyOVrwd0rrWAfSxZe5EfqSxo+Vu7nQD3mGHxX9qARa8xHAKCu8LrJ4dpQRIc6Hx7SmGr9EcqrVIFS/KZrUwXAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T04:53:20.581246Z"},"content_sha256":"50a89a510cbccb59fdbbde75c18c18f28c07014f78bd754da8c1477a1ed8657b","schema_version":"1.0","event_id":"sha256:50a89a510cbccb59fdbbde75c18c18f28c07014f78bd754da8c1477a1ed8657b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:3OWGAYWAO62D5EGKUY754QG4RM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mean Field Games models of segregation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Marco Cirant, Martino Bardi, Yves Achdou","submitted_at":"2016-07-15T10:52:22Z","abstract_excerpt":"This paper introduces and analyses some models in the framework of Mean Field Games describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of Mean Field Game theory, in the stationary and in the evolutive case. Numerical methods are proposed, with several simulations. In the examples and in the numerical results, particular emphasis i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04453","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":"2026-05-18T01:11:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ikwisb3HBJO5WcTu+AVDSXPZuVER1mJLnUX6YpqMeVMDH/qnO3xAKNlazpXQY9sa0mX6LNJJ0kS3MpH27DyBBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T04:53:20.581910Z"},"content_sha256":"4fc1647de71ed1484e8449e5552b2933484a072756a574bdeff269c20c2dc1b6","schema_version":"1.0","event_id":"sha256:4fc1647de71ed1484e8449e5552b2933484a072756a574bdeff269c20c2dc1b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3OWGAYWAO62D5EGKUY754QG4RM/bundle.json","state_url":"https://pith.science/pith/3OWGAYWAO62D5EGKUY754QG4RM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3OWGAYWAO62D5EGKUY754QG4RM/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":"2026-06-04T04:53:20Z","links":{"resolver":"https://pith.science/pith/3OWGAYWAO62D5EGKUY754QG4RM","bundle":"https://pith.science/pith/3OWGAYWAO62D5EGKUY754QG4RM/bundle.json","state":"https://pith.science/pith/3OWGAYWAO62D5EGKUY754QG4RM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3OWGAYWAO62D5EGKUY754QG4RM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3OWGAYWAO62D5EGKUY754QG4RM","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":"b718a70b2454cfd8e03b6689d47dd082ee420679c819513cc134c7302996fc94","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-15T10:52:22Z","title_canon_sha256":"805ad2ba81d4289d5a608c050859f7e774a1269f4aac7253ceaed7850ff7ccc4"},"schema_version":"1.0","source":{"id":"1607.04453","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.04453","created_at":"2026-05-18T01:11:01Z"},{"alias_kind":"arxiv_version","alias_value":"1607.04453v1","created_at":"2026-05-18T01:11:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.04453","created_at":"2026-05-18T01:11:01Z"},{"alias_kind":"pith_short_12","alias_value":"3OWGAYWAO62D","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3OWGAYWAO62D5EGK","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3OWGAYWA","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:4fc1647de71ed1484e8449e5552b2933484a072756a574bdeff269c20c2dc1b6","target":"graph","created_at":"2026-05-18T01:11:01Z","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":"This paper introduces and analyses some models in the framework of Mean Field Games describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of Mean Field Game theory, in the stationary and in the evolutive case. Numerical methods are proposed, with several simulations. In the examples and in the numerical results, particular emphasis i","authors_text":"Marco Cirant, Martino Bardi, Yves Achdou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-15T10:52:22Z","title":"Mean Field Games models of segregation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04453","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":"sha256:50a89a510cbccb59fdbbde75c18c18f28c07014f78bd754da8c1477a1ed8657b","target":"record","created_at":"2026-05-18T01:11:01Z","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":"b718a70b2454cfd8e03b6689d47dd082ee420679c819513cc134c7302996fc94","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-07-15T10:52:22Z","title_canon_sha256":"805ad2ba81d4289d5a608c050859f7e774a1269f4aac7253ceaed7850ff7ccc4"},"schema_version":"1.0","source":{"id":"1607.04453","kind":"arxiv","version":1}},"canonical_sha256":"dbac6062c077b43e90caa63fde40dc8b27d907b39effd58c5275b8f0106a6701","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dbac6062c077b43e90caa63fde40dc8b27d907b39effd58c5275b8f0106a6701","first_computed_at":"2026-05-18T01:11:01.197691Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:01.197691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"777qlrkHYGZbf7QabKkzwPFgmHhMniuuyjxxCTtjWlXS4C0gfKO3xyJX3WayUhDPJ51XMa2YInNVzCVWGJU2DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:01.198173Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.04453","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50a89a510cbccb59fdbbde75c18c18f28c07014f78bd754da8c1477a1ed8657b","sha256:4fc1647de71ed1484e8449e5552b2933484a072756a574bdeff269c20c2dc1b6"],"state_sha256":"be222bed61a9e16fbf4a790fe7dc558ec313e88e5d95fd93dc7f7d4737d2d202"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BpWvjXBV+ZIX7mBuOdfonXssdQePH4iEkV2eB4uVXtAsIjXydNTihZYTbxT5T1DQ4ulTksQmydRymES19quxBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T04:53:20.589785Z","bundle_sha256":"3302d1f703ffe8aeff090cb2d9e40c38b3b7dbfc41677e50ce8dfce4d69e81fb"}}