{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:GYRSO3ND4S7TX7RXXPXR2AMRLN","short_pith_number":"pith:GYRSO3ND","canonical_record":{"source":{"id":"1303.3907","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T21:10:25Z","cross_cats_sorted":["physics.soc-ph","q-bio.MN","q-bio.NC"],"title_canon_sha256":"9055dd79735db6eccf07f3c4b9cc85a42bf95f7a6b740a4b90dd432d02db670e","abstract_canon_sha256":"3582560baaafdb622581f0e7c7bd28571c3a96105aaad38012710a2ba2d5eeaa"},"schema_version":"1.0"},"canonical_sha256":"3623276da3e4bf3bfe37bbef1d01915b76b124561db3008afd5c0672106c841d","source":{"kind":"arxiv","id":"1303.3907","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3907","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3907v1","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3907","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"pith_short_12","alias_value":"GYRSO3ND4S7T","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"GYRSO3ND4S7TX7RX","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"GYRSO3ND","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:GYRSO3ND4S7TX7RXXPXR2AMRLN","target":"record","payload":{"canonical_record":{"source":{"id":"1303.3907","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T21:10:25Z","cross_cats_sorted":["physics.soc-ph","q-bio.MN","q-bio.NC"],"title_canon_sha256":"9055dd79735db6eccf07f3c4b9cc85a42bf95f7a6b740a4b90dd432d02db670e","abstract_canon_sha256":"3582560baaafdb622581f0e7c7bd28571c3a96105aaad38012710a2ba2d5eeaa"},"schema_version":"1.0"},"canonical_sha256":"3623276da3e4bf3bfe37bbef1d01915b76b124561db3008afd5c0672106c841d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:56.885134Z","signature_b64":"Bl6cksmYZvPwvto4W4T+0Fa0I/oPmaSJq4SLtP/mSiZ4zBHQD6nNyVLPe8RTqtj0B9m/osZhPCgIWXI7Sv/2DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3623276da3e4bf3bfe37bbef1d01915b76b124561db3008afd5c0672106c841d","last_reissued_at":"2026-05-18T00:06:56.884471Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:56.884471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.3907","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":"2026-05-18T00:06:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nsFocgea86ak2eI9d4RAwsYtaJ7Wq+Os2bvMlIGFJZKsF1/abUbABaa6RLAbIn/vCQLnb7XAKTI510LaAhL/Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:12:28.627703Z"},"content_sha256":"d1396fe76bb5374c4ef5ca1f3a0268e97686dea1c3f277ba4fd6048833d3b523","schema_version":"1.0","event_id":"sha256:d1396fe76bb5374c4ef5ca1f3a0268e97686dea1c3f277ba4fd6048833d3b523"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:GYRSO3ND4S7TX7RXXPXR2AMRLN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular dynamical systems on networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["physics.soc-ph","q-bio.MN","q-bio.NC"],"primary_cat":"math.DS","authors_text":"Eugene Lerman, Lee DeVille","submitted_at":"2013-03-15T21:10:25Z","abstract_excerpt":"We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3907","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":"2026-05-18T00:06:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gcMGW4YMbmbBs47nDnGYzt0VOfZCcRrsl86aeF2jVshoUh/sG4Jsvr1/H93YUbyW2dfv7CkSL7E/NB/NYqzmBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:12:28.628300Z"},"content_sha256":"e038ed1bbd92d1042b2fa9142a5022615cd9c932a0a4291695538683a213d59f","schema_version":"1.0","event_id":"sha256:e038ed1bbd92d1042b2fa9142a5022615cd9c932a0a4291695538683a213d59f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GYRSO3ND4S7TX7RXXPXR2AMRLN/bundle.json","state_url":"https://pith.science/pith/GYRSO3ND4S7TX7RXXPXR2AMRLN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GYRSO3ND4S7TX7RXXPXR2AMRLN/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":"2026-05-28T20:12:28Z","links":{"resolver":"https://pith.science/pith/GYRSO3ND4S7TX7RXXPXR2AMRLN","bundle":"https://pith.science/pith/GYRSO3ND4S7TX7RXXPXR2AMRLN/bundle.json","state":"https://pith.science/pith/GYRSO3ND4S7TX7RXXPXR2AMRLN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GYRSO3ND4S7TX7RXXPXR2AMRLN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:GYRSO3ND4S7TX7RXXPXR2AMRLN","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":"3582560baaafdb622581f0e7c7bd28571c3a96105aaad38012710a2ba2d5eeaa","cross_cats_sorted":["physics.soc-ph","q-bio.MN","q-bio.NC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T21:10:25Z","title_canon_sha256":"9055dd79735db6eccf07f3c4b9cc85a42bf95f7a6b740a4b90dd432d02db670e"},"schema_version":"1.0","source":{"id":"1303.3907","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3907","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3907v1","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3907","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"pith_short_12","alias_value":"GYRSO3ND4S7T","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"GYRSO3ND4S7TX7RX","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"GYRSO3ND","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:e038ed1bbd92d1042b2fa9142a5022615cd9c932a0a4291695538683a213d59f","target":"graph","created_at":"2026-05-18T00:06:56Z","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":"We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller o","authors_text":"Eugene Lerman, Lee DeVille","cross_cats":["physics.soc-ph","q-bio.MN","q-bio.NC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T21:10:25Z","title":"Modular dynamical systems on networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3907","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:d1396fe76bb5374c4ef5ca1f3a0268e97686dea1c3f277ba4fd6048833d3b523","target":"record","created_at":"2026-05-18T00:06:56Z","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":"3582560baaafdb622581f0e7c7bd28571c3a96105aaad38012710a2ba2d5eeaa","cross_cats_sorted":["physics.soc-ph","q-bio.MN","q-bio.NC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-15T21:10:25Z","title_canon_sha256":"9055dd79735db6eccf07f3c4b9cc85a42bf95f7a6b740a4b90dd432d02db670e"},"schema_version":"1.0","source":{"id":"1303.3907","kind":"arxiv","version":1}},"canonical_sha256":"3623276da3e4bf3bfe37bbef1d01915b76b124561db3008afd5c0672106c841d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3623276da3e4bf3bfe37bbef1d01915b76b124561db3008afd5c0672106c841d","first_computed_at":"2026-05-18T00:06:56.884471Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:56.884471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bl6cksmYZvPwvto4W4T+0Fa0I/oPmaSJq4SLtP/mSiZ4zBHQD6nNyVLPe8RTqtj0B9m/osZhPCgIWXI7Sv/2DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:56.885134Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.3907","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d1396fe76bb5374c4ef5ca1f3a0268e97686dea1c3f277ba4fd6048833d3b523","sha256:e038ed1bbd92d1042b2fa9142a5022615cd9c932a0a4291695538683a213d59f"],"state_sha256":"47d212ddcdc0bbbe31b7c4551120222d22e9a2d0eabdd3e85a2be35f44d88691"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UmSv/8k7UjBAPsyM/AahZuCxXqbEOOeEUosJbKdgFfYX6VA6igrRwBCuqxcVZKg5sAHvoxhaUigsBh9wkIWjDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T20:12:28.631039Z","bundle_sha256":"a1e73693d414b21fc05072195580beae08389a60c71cc683c16cb9432fc05e16"}}