{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:25TTWQ4MLAJCFCFQQ6XGNBYUMS","short_pith_number":"pith:25TTWQ4M","canonical_record":{"source":{"id":"1202.6389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-28T21:45:09Z","cross_cats_sorted":["cs.IT","cs.SI","math.IT"],"title_canon_sha256":"1af21476b7843b84f12a3f765b2e9d026f26c35ef72a563c607074a091a19a8a","abstract_canon_sha256":"17ad720f1bdb089aa111e4a7d939cf5e83bf5486a88683d1172efb4ec51833a5"},"schema_version":"1.0"},"canonical_sha256":"d7673b438c58122288b087ae668714648d27efb65cfcc58fba3de29b7007dbfc","source":{"kind":"arxiv","id":"1202.6389","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.6389","created_at":"2026-05-18T01:58:18Z"},{"alias_kind":"arxiv_version","alias_value":"1202.6389v1","created_at":"2026-05-18T01:58:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.6389","created_at":"2026-05-18T01:58:18Z"},{"alias_kind":"pith_short_12","alias_value":"25TTWQ4MLAJC","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"25TTWQ4MLAJCFCFQ","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"25TTWQ4M","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:25TTWQ4MLAJCFCFQQ6XGNBYUMS","target":"record","payload":{"canonical_record":{"source":{"id":"1202.6389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-28T21:45:09Z","cross_cats_sorted":["cs.IT","cs.SI","math.IT"],"title_canon_sha256":"1af21476b7843b84f12a3f765b2e9d026f26c35ef72a563c607074a091a19a8a","abstract_canon_sha256":"17ad720f1bdb089aa111e4a7d939cf5e83bf5486a88683d1172efb4ec51833a5"},"schema_version":"1.0"},"canonical_sha256":"d7673b438c58122288b087ae668714648d27efb65cfcc58fba3de29b7007dbfc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:18.484567Z","signature_b64":"wYxQuCsn9dAqTmdAhCCoV0rPlKMaebVmJaEV2oHOMzybcVWgB1m6aDYYBKnacyiX5I+v86QLy+kQg5iZCrikCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7673b438c58122288b087ae668714648d27efb65cfcc58fba3de29b7007dbfc","last_reissued_at":"2026-05-18T01:58:18.484064Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:18.484064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.6389","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":"2026-05-18T01:58:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cnNE1IzCR54m0JFIgTzPn4tdfKNiG6LIuBH3jVNf6rNE/yTUtXxr+Rrrv2Kq9E7ktxsdI6oeMx5FfW2JTudDDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:07:14.689783Z"},"content_sha256":"9bc3d86ea89a73729f6bdcd3fbf47540749475d77b8f8b3a1af82f2650b0e1be","schema_version":"1.0","event_id":"sha256:9bc3d86ea89a73729f6bdcd3fbf47540749475d77b8f8b3a1af82f2650b0e1be"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:25TTWQ4MLAJCFCFQQ6XGNBYUMS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.SI","math.IT"],"primary_cat":"math.PR","authors_text":"Bruno Sinopoli, Dragana Bajovic, Joao Xavier, Jose M. F. Moura","submitted_at":"2012-02-28T21:45:09Z","abstract_excerpt":"Distributed consensus and other linear systems with system stochastic matrices $W_k$ emerge in various settings, like opinion formation in social networks, rendezvous of robots, and distributed inference in sensor networks. The matrices $W_k$ are often random, due to, e.g., random packet dropouts in wireless sensor networks. Key in analyzing the performance of such systems is studying convergence of matrix products $W_kW_{k-1}... W_1$. In this paper, we find the exact exponential rate $I$ for the convergence in probability of the product of such matrices when time $k$ grows large, under the as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.6389","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":"2026-05-18T01:58:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+4ERvLdibLO/+HvudHgXpGf3cwRGN/H3/Eqt7Sw3bS7ZuBlDEfUGS3Qlu1E+BjyH/Ku7jycjh89TFwG7QQXoCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:07:14.690374Z"},"content_sha256":"872e46f212c5b168b845cb758c5db59cbb3820370b7790158d53969666101f04","schema_version":"1.0","event_id":"sha256:872e46f212c5b168b845cb758c5db59cbb3820370b7790158d53969666101f04"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/25TTWQ4MLAJCFCFQQ6XGNBYUMS/bundle.json","state_url":"https://pith.science/pith/25TTWQ4MLAJCFCFQQ6XGNBYUMS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/25TTWQ4MLAJCFCFQQ6XGNBYUMS/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":"2026-05-26T19:07:14Z","links":{"resolver":"https://pith.science/pith/25TTWQ4MLAJCFCFQQ6XGNBYUMS","bundle":"https://pith.science/pith/25TTWQ4MLAJCFCFQQ6XGNBYUMS/bundle.json","state":"https://pith.science/pith/25TTWQ4MLAJCFCFQQ6XGNBYUMS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/25TTWQ4MLAJCFCFQQ6XGNBYUMS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:25TTWQ4MLAJCFCFQQ6XGNBYUMS","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":"17ad720f1bdb089aa111e4a7d939cf5e83bf5486a88683d1172efb4ec51833a5","cross_cats_sorted":["cs.IT","cs.SI","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-28T21:45:09Z","title_canon_sha256":"1af21476b7843b84f12a3f765b2e9d026f26c35ef72a563c607074a091a19a8a"},"schema_version":"1.0","source":{"id":"1202.6389","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.6389","created_at":"2026-05-18T01:58:18Z"},{"alias_kind":"arxiv_version","alias_value":"1202.6389v1","created_at":"2026-05-18T01:58:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.6389","created_at":"2026-05-18T01:58:18Z"},{"alias_kind":"pith_short_12","alias_value":"25TTWQ4MLAJC","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"25TTWQ4MLAJCFCFQ","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"25TTWQ4M","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:872e46f212c5b168b845cb758c5db59cbb3820370b7790158d53969666101f04","target":"graph","created_at":"2026-05-18T01:58:18Z","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":"Distributed consensus and other linear systems with system stochastic matrices $W_k$ emerge in various settings, like opinion formation in social networks, rendezvous of robots, and distributed inference in sensor networks. The matrices $W_k$ are often random, due to, e.g., random packet dropouts in wireless sensor networks. Key in analyzing the performance of such systems is studying convergence of matrix products $W_kW_{k-1}... W_1$. In this paper, we find the exact exponential rate $I$ for the convergence in probability of the product of such matrices when time $k$ grows large, under the as","authors_text":"Bruno Sinopoli, Dragana Bajovic, Joao Xavier, Jose M. F. Moura","cross_cats":["cs.IT","cs.SI","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-28T21:45:09Z","title":"Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.6389","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":"sha256:9bc3d86ea89a73729f6bdcd3fbf47540749475d77b8f8b3a1af82f2650b0e1be","target":"record","created_at":"2026-05-18T01:58:18Z","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":"17ad720f1bdb089aa111e4a7d939cf5e83bf5486a88683d1172efb4ec51833a5","cross_cats_sorted":["cs.IT","cs.SI","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-28T21:45:09Z","title_canon_sha256":"1af21476b7843b84f12a3f765b2e9d026f26c35ef72a563c607074a091a19a8a"},"schema_version":"1.0","source":{"id":"1202.6389","kind":"arxiv","version":1}},"canonical_sha256":"d7673b438c58122288b087ae668714648d27efb65cfcc58fba3de29b7007dbfc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d7673b438c58122288b087ae668714648d27efb65cfcc58fba3de29b7007dbfc","first_computed_at":"2026-05-18T01:58:18.484064Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:58:18.484064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wYxQuCsn9dAqTmdAhCCoV0rPlKMaebVmJaEV2oHOMzybcVWgB1m6aDYYBKnacyiX5I+v86QLy+kQg5iZCrikCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:58:18.484567Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.6389","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9bc3d86ea89a73729f6bdcd3fbf47540749475d77b8f8b3a1af82f2650b0e1be","sha256:872e46f212c5b168b845cb758c5db59cbb3820370b7790158d53969666101f04"],"state_sha256":"9c8044dc30a7c936f90f2d1b823673fea15d2bed692848ce50ddd0e3f3311508"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b3Q4qIwHC+3JQp62Vgkv1cnzkvs+aIUU83Ul5DxdRBlBA6UUQWotIbcbFoMbRQ8tZKJq/locpCI5rCbXlC/CAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T19:07:14.693394Z","bundle_sha256":"084f5d6d1e20f36bc6f2ad13d33441ff0f610d775ec50ab052fab313559dde34"}}