{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6HBHD24C2AXBTOV5KYKHTBWXN6","short_pith_number":"pith:6HBHD24C","canonical_record":{"source":{"id":"1510.07021","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-23T19:36:00Z","cross_cats_sorted":[],"title_canon_sha256":"97c485384bc71e7bc548d2e1de16dd19dc5096e34e568e822d852e72d5230510","abstract_canon_sha256":"c7fe4816c1a8eae05ed0328e61834380569c9d588cd480840641c7e85945e20b"},"schema_version":"1.0"},"canonical_sha256":"f1c271eb82d02e19babd56147986d76fbfe95bc5cd12238a49cefd29e077cbf7","source":{"kind":"arxiv","id":"1510.07021","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07021","created_at":"2026-05-18T00:45:50Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07021v2","created_at":"2026-05-18T00:45:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07021","created_at":"2026-05-18T00:45:50Z"},{"alias_kind":"pith_short_12","alias_value":"6HBHD24C2AXB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6HBHD24C2AXBTOV5","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6HBHD24C","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6HBHD24C2AXBTOV5KYKHTBWXN6","target":"record","payload":{"canonical_record":{"source":{"id":"1510.07021","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-23T19:36:00Z","cross_cats_sorted":[],"title_canon_sha256":"97c485384bc71e7bc548d2e1de16dd19dc5096e34e568e822d852e72d5230510","abstract_canon_sha256":"c7fe4816c1a8eae05ed0328e61834380569c9d588cd480840641c7e85945e20b"},"schema_version":"1.0"},"canonical_sha256":"f1c271eb82d02e19babd56147986d76fbfe95bc5cd12238a49cefd29e077cbf7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:50.268372Z","signature_b64":"+9YdQXk4QLAYLVkap1lUrxyCqlKcql30frWjiYdubonRiK5h2fh+J8D/C9K871hZ0ZdPMu5tGNT7QPO/eoc1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1c271eb82d02e19babd56147986d76fbfe95bc5cd12238a49cefd29e077cbf7","last_reissued_at":"2026-05-18T00:45:50.267792Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:50.267792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.07021","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":"2026-05-18T00:45:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GtOUoEIyavPIFYD0NQtZ069fethLgdTZl3ziAO7GBlt8pCybNbCeM1TcMa8BRhvVphhRmvu95gV0N8E7n2SIBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:52:05.435077Z"},"content_sha256":"4c2269f26613f3a1920fc705ad15252b59b0e3b465aa42ae2f12deae1b2d128c","schema_version":"1.0","event_id":"sha256:4c2269f26613f3a1920fc705ad15252b59b0e3b465aa42ae2f12deae1b2d128c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6HBHD24C2AXBTOV5KYKHTBWXN6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical Connectivity and Fastest Convergence Rates of Distributed Consensus with Switching Topologies and Additive Noises","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Chen Chen, Ge Chen, George Yin, Le Yi Wang","submitted_at":"2015-10-23T19:36:00Z","abstract_excerpt":"Consensus conditions and convergence speeds are crucial for distributed consensus algorithms of networked systems. Based on a basic first-order average-consensus protocol with time-varying topologies and additive noises, this paper first investigates its critical consensus condition on network topology by stochastic approximation frameworks. A new joint-connectivity condition called extensible joint-connectivity that contains a parameter $\\delta$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of $\\delta$ for consensus is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07021","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":"2026-05-18T00:45:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sjFJWufh2rJcggsnEmKJfrAq7/jU/cpgS7tKvB7msnEeJ4ai52QvHt11AR+wLOLBinws3fCYfAR6XkR+6ORwAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:52:05.435495Z"},"content_sha256":"9cd584eb86beccc7a36e59d3d79883046ee9deb678a63a6e2de6eb2785c26700","schema_version":"1.0","event_id":"sha256:9cd584eb86beccc7a36e59d3d79883046ee9deb678a63a6e2de6eb2785c26700"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6HBHD24C2AXBTOV5KYKHTBWXN6/bundle.json","state_url":"https://pith.science/pith/6HBHD24C2AXBTOV5KYKHTBWXN6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6HBHD24C2AXBTOV5KYKHTBWXN6/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":"2026-05-27T15:52:05Z","links":{"resolver":"https://pith.science/pith/6HBHD24C2AXBTOV5KYKHTBWXN6","bundle":"https://pith.science/pith/6HBHD24C2AXBTOV5KYKHTBWXN6/bundle.json","state":"https://pith.science/pith/6HBHD24C2AXBTOV5KYKHTBWXN6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6HBHD24C2AXBTOV5KYKHTBWXN6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6HBHD24C2AXBTOV5KYKHTBWXN6","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":"c7fe4816c1a8eae05ed0328e61834380569c9d588cd480840641c7e85945e20b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-23T19:36:00Z","title_canon_sha256":"97c485384bc71e7bc548d2e1de16dd19dc5096e34e568e822d852e72d5230510"},"schema_version":"1.0","source":{"id":"1510.07021","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07021","created_at":"2026-05-18T00:45:50Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07021v2","created_at":"2026-05-18T00:45:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07021","created_at":"2026-05-18T00:45:50Z"},{"alias_kind":"pith_short_12","alias_value":"6HBHD24C2AXB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6HBHD24C2AXBTOV5","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6HBHD24C","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:9cd584eb86beccc7a36e59d3d79883046ee9deb678a63a6e2de6eb2785c26700","target":"graph","created_at":"2026-05-18T00:45:50Z","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":"Consensus conditions and convergence speeds are crucial for distributed consensus algorithms of networked systems. Based on a basic first-order average-consensus protocol with time-varying topologies and additive noises, this paper first investigates its critical consensus condition on network topology by stochastic approximation frameworks. A new joint-connectivity condition called extensible joint-connectivity that contains a parameter $\\delta$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of $\\delta$ for consensus is","authors_text":"Chen Chen, Ge Chen, George Yin, Le Yi Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-23T19:36:00Z","title":"Critical Connectivity and Fastest Convergence Rates of Distributed Consensus with Switching Topologies and Additive Noises"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07021","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":"sha256:4c2269f26613f3a1920fc705ad15252b59b0e3b465aa42ae2f12deae1b2d128c","target":"record","created_at":"2026-05-18T00:45:50Z","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":"c7fe4816c1a8eae05ed0328e61834380569c9d588cd480840641c7e85945e20b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-23T19:36:00Z","title_canon_sha256":"97c485384bc71e7bc548d2e1de16dd19dc5096e34e568e822d852e72d5230510"},"schema_version":"1.0","source":{"id":"1510.07021","kind":"arxiv","version":2}},"canonical_sha256":"f1c271eb82d02e19babd56147986d76fbfe95bc5cd12238a49cefd29e077cbf7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1c271eb82d02e19babd56147986d76fbfe95bc5cd12238a49cefd29e077cbf7","first_computed_at":"2026-05-18T00:45:50.267792Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:50.267792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+9YdQXk4QLAYLVkap1lUrxyCqlKcql30frWjiYdubonRiK5h2fh+J8D/C9K871hZ0ZdPMu5tGNT7QPO/eoc1BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:50.268372Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07021","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c2269f26613f3a1920fc705ad15252b59b0e3b465aa42ae2f12deae1b2d128c","sha256:9cd584eb86beccc7a36e59d3d79883046ee9deb678a63a6e2de6eb2785c26700"],"state_sha256":"1384b0c963c03f7295b80e1a8da1bf9f46a96dc3529a2c74dee4bc95ccb58745"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c/I11zdIhm7rZ5L/s7C3PBk4YDFA1qfVHAUW/qDxD5q6QpNVeFGolW5LG8XFxVnVRpSQ1GjHZlYoXnMDdJHaBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T15:52:05.438212Z","bundle_sha256":"866e59e5a4b26341697e1808246d8c785f332aa4d22ecf281fa0e6bda464fb5f"}}