{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:IN4TMLSLP3G7NB6GOCBNUTIAUJ","short_pith_number":"pith:IN4TMLSL","canonical_record":{"source":{"id":"1107.2814","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-07-14T13:29:22Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"b86da56ffa5cb85def71143dd8dbd3f3b26344abc1248f4ee8b26cb02cf74969","abstract_canon_sha256":"03fd5cdf75384884144e16fa8e796a75c538664c569c84c6c3cd1c7553537c47"},"schema_version":"1.0"},"canonical_sha256":"4379362e4b7ecdf687c67082da4d00a270a9a58eefd674f70c9a8deb73a7f549","source":{"kind":"arxiv","id":"1107.2814","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2814","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2814v2","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2814","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"pith_short_12","alias_value":"IN4TMLSLP3G7","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"IN4TMLSLP3G7NB6G","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"IN4TMLSL","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:IN4TMLSLP3G7NB6GOCBNUTIAUJ","target":"record","payload":{"canonical_record":{"source":{"id":"1107.2814","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-07-14T13:29:22Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"b86da56ffa5cb85def71143dd8dbd3f3b26344abc1248f4ee8b26cb02cf74969","abstract_canon_sha256":"03fd5cdf75384884144e16fa8e796a75c538664c569c84c6c3cd1c7553537c47"},"schema_version":"1.0"},"canonical_sha256":"4379362e4b7ecdf687c67082da4d00a270a9a58eefd674f70c9a8deb73a7f549","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:56.260217Z","signature_b64":"rCasl8FNPL2nMXjPAHpG6tBvufloci5s5G7smKD/0J3Wts/1ElC0+XNCHc+MYKcPWRGlfY+L0EI1iXzC4fphDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4379362e4b7ecdf687c67082da4d00a270a9a58eefd674f70c9a8deb73a7f549","last_reissued_at":"2026-05-18T04:12:56.259670Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:56.259670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.2814","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":"2026-05-18T04:12:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sc+9xN51lWF6bzKRKBLkkMXNL9NgEbJDrSFX9NHmyh4ouegvXGO6RNhAPNSkniPVHMpAtR1beJk5Yzn/2J2ACQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T17:46:11.184639Z"},"content_sha256":"ee9f891cd11a6489a7b8913b737f6780b77812fe83f83c9e6d646a96db20cc26","schema_version":"1.0","event_id":"sha256:ee9f891cd11a6489a7b8913b737f6780b77812fe83f83c9e6d646a96db20cc26"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:IN4TMLSLP3G7NB6GOCBNUTIAUJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Differential equation approximations of stochastic network processes: an operator semigroup approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.DS","authors_text":"Andr\\'as B\\'atkai, Eszter Sikolya, Istvan Z. Kiss, P\\'eter L. Simon","submitted_at":"2011-07-14T13:29:22Z","abstract_excerpt":"The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2814","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":"2026-05-18T04:12:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A1qRX6gNaJ4n/b6LzqQJhBuuh86nkaG+44pZ9P3Vp8DTTr1h+QBCBwA/pVYMjPo/ov7hOZjVFVFryrprutcZDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T17:46:11.185003Z"},"content_sha256":"0b02faefc04443c4aa81e0d889002ab1251b92caf856a84a85ffe2b7dd0fee47","schema_version":"1.0","event_id":"sha256:0b02faefc04443c4aa81e0d889002ab1251b92caf856a84a85ffe2b7dd0fee47"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IN4TMLSLP3G7NB6GOCBNUTIAUJ/bundle.json","state_url":"https://pith.science/pith/IN4TMLSLP3G7NB6GOCBNUTIAUJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IN4TMLSLP3G7NB6GOCBNUTIAUJ/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":"2026-06-05T17:46:11Z","links":{"resolver":"https://pith.science/pith/IN4TMLSLP3G7NB6GOCBNUTIAUJ","bundle":"https://pith.science/pith/IN4TMLSLP3G7NB6GOCBNUTIAUJ/bundle.json","state":"https://pith.science/pith/IN4TMLSLP3G7NB6GOCBNUTIAUJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IN4TMLSLP3G7NB6GOCBNUTIAUJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:IN4TMLSLP3G7NB6GOCBNUTIAUJ","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":"03fd5cdf75384884144e16fa8e796a75c538664c569c84c6c3cd1c7553537c47","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-07-14T13:29:22Z","title_canon_sha256":"b86da56ffa5cb85def71143dd8dbd3f3b26344abc1248f4ee8b26cb02cf74969"},"schema_version":"1.0","source":{"id":"1107.2814","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2814","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2814v2","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2814","created_at":"2026-05-18T04:12:56Z"},{"alias_kind":"pith_short_12","alias_value":"IN4TMLSLP3G7","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"IN4TMLSLP3G7NB6G","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"IN4TMLSL","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:0b02faefc04443c4aa81e0d889002ab1251b92caf856a84a85ffe2b7dd0fee47","target":"graph","created_at":"2026-05-18T04:12:56Z","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":"The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that","authors_text":"Andr\\'as B\\'atkai, Eszter Sikolya, Istvan Z. Kiss, P\\'eter L. Simon","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-07-14T13:29:22Z","title":"Differential equation approximations of stochastic network processes: an operator semigroup approach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2814","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:ee9f891cd11a6489a7b8913b737f6780b77812fe83f83c9e6d646a96db20cc26","target":"record","created_at":"2026-05-18T04:12:56Z","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":"03fd5cdf75384884144e16fa8e796a75c538664c569c84c6c3cd1c7553537c47","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-07-14T13:29:22Z","title_canon_sha256":"b86da56ffa5cb85def71143dd8dbd3f3b26344abc1248f4ee8b26cb02cf74969"},"schema_version":"1.0","source":{"id":"1107.2814","kind":"arxiv","version":2}},"canonical_sha256":"4379362e4b7ecdf687c67082da4d00a270a9a58eefd674f70c9a8deb73a7f549","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4379362e4b7ecdf687c67082da4d00a270a9a58eefd674f70c9a8deb73a7f549","first_computed_at":"2026-05-18T04:12:56.259670Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:56.259670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rCasl8FNPL2nMXjPAHpG6tBvufloci5s5G7smKD/0J3Wts/1ElC0+XNCHc+MYKcPWRGlfY+L0EI1iXzC4fphDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:56.260217Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2814","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee9f891cd11a6489a7b8913b737f6780b77812fe83f83c9e6d646a96db20cc26","sha256:0b02faefc04443c4aa81e0d889002ab1251b92caf856a84a85ffe2b7dd0fee47"],"state_sha256":"b7681945e449d0b88714834a178a05ec718aad5bfcb6b37cdbd4bbb06de39d37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gB8BNIngcD1Ks3Pzmq+CZy20J7aeuuEeRIhVTXCxUYawM3umshkvgEzdtJ/FmK6+CnTRg2tG1tbl9IoX5NM+Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T17:46:11.187308Z","bundle_sha256":"184b23545f62c0d76ad69040533c747fa7e5b68976425b76882481349a31d766"}}