{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7Y62C3EKU2WYY5MBBWP34UWX5J","short_pith_number":"pith:7Y62C3EK","canonical_record":{"source":{"id":"1011.5691","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-26T02:02:04Z","cross_cats_sorted":[],"title_canon_sha256":"648d1bd9bdc64f21e13ecbcdc46a114dee85386a02d234841acb75d91a59a68e","abstract_canon_sha256":"553ff9c9216de5aaf7741a760a1040eff750dfdcfa3900f3c67e1e23c2fc3b88"},"schema_version":"1.0"},"canonical_sha256":"fe3da16c8aa6ad8c75810d9fbe52d7ea61a846e995d0681af34ca054e48aeb14","source":{"kind":"arxiv","id":"1011.5691","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.5691","created_at":"2026-05-18T04:34:38Z"},{"alias_kind":"arxiv_version","alias_value":"1011.5691v1","created_at":"2026-05-18T04:34:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.5691","created_at":"2026-05-18T04:34:38Z"},{"alias_kind":"pith_short_12","alias_value":"7Y62C3EKU2WY","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7Y62C3EKU2WYY5MB","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7Y62C3EK","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7Y62C3EKU2WYY5MBBWP34UWX5J","target":"record","payload":{"canonical_record":{"source":{"id":"1011.5691","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-26T02:02:04Z","cross_cats_sorted":[],"title_canon_sha256":"648d1bd9bdc64f21e13ecbcdc46a114dee85386a02d234841acb75d91a59a68e","abstract_canon_sha256":"553ff9c9216de5aaf7741a760a1040eff750dfdcfa3900f3c67e1e23c2fc3b88"},"schema_version":"1.0"},"canonical_sha256":"fe3da16c8aa6ad8c75810d9fbe52d7ea61a846e995d0681af34ca054e48aeb14","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:38.653441Z","signature_b64":"OCtSGiwgOn38a92Ez6kF/Tw/LfqmXE7avnsRQaVzW+2saNZtQLK/4KUwGiiHFJaV3iQ7kPUoBgdORqQKaw6fAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe3da16c8aa6ad8c75810d9fbe52d7ea61a846e995d0681af34ca054e48aeb14","last_reissued_at":"2026-05-18T04:34:38.653021Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:38.653021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.5691","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":"2026-05-18T04:34:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/JrCJdwddl7d9DiHA2YNMF8HW9k2QF+tT1r3Q3Ie5ntGzMCdxBJj2WwFP4U1eVvJFPh4r7Yd+5yFDfTx7pXuDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T17:32:14.008441Z"},"content_sha256":"6829b04b4ae8e9f65e6fb8791acee4a603096b55677f55f2ea72445114f0e255","schema_version":"1.0","event_id":"sha256:6829b04b4ae8e9f65e6fb8791acee4a603096b55677f55f2ea72445114f0e255"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7Y62C3EKU2WYY5MBBWP34UWX5J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The cone percolation on $\\bbT_d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"F\\'abio P. Machado, Mauricio Zuluaga, Valdivino V. Junior","submitted_at":"2010-11-26T02:02:04Z","abstract_excerpt":"We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour spreads out trough an infinite number of individuals. We present sharp lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.5691","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":"2026-05-18T04:34:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jAeOebY9gNT3oE91MI3M3VIPg+o8KsFVUbfdEA/h3252PbePdTR9+uCOg/FuTvpBDxhd7FZ/0X2MWnyVBRKeDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T17:32:14.008781Z"},"content_sha256":"03b88d5952491e0b15af0589a0469a810f7f8a343fe567a0f86462b770376b3c","schema_version":"1.0","event_id":"sha256:03b88d5952491e0b15af0589a0469a810f7f8a343fe567a0f86462b770376b3c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7Y62C3EKU2WYY5MBBWP34UWX5J/bundle.json","state_url":"https://pith.science/pith/7Y62C3EKU2WYY5MBBWP34UWX5J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7Y62C3EKU2WYY5MBBWP34UWX5J/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":"2026-06-25T17:32:14Z","links":{"resolver":"https://pith.science/pith/7Y62C3EKU2WYY5MBBWP34UWX5J","bundle":"https://pith.science/pith/7Y62C3EKU2WYY5MBBWP34UWX5J/bundle.json","state":"https://pith.science/pith/7Y62C3EKU2WYY5MBBWP34UWX5J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7Y62C3EKU2WYY5MBBWP34UWX5J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7Y62C3EKU2WYY5MBBWP34UWX5J","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":"553ff9c9216de5aaf7741a760a1040eff750dfdcfa3900f3c67e1e23c2fc3b88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-26T02:02:04Z","title_canon_sha256":"648d1bd9bdc64f21e13ecbcdc46a114dee85386a02d234841acb75d91a59a68e"},"schema_version":"1.0","source":{"id":"1011.5691","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.5691","created_at":"2026-05-18T04:34:38Z"},{"alias_kind":"arxiv_version","alias_value":"1011.5691v1","created_at":"2026-05-18T04:34:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.5691","created_at":"2026-05-18T04:34:38Z"},{"alias_kind":"pith_short_12","alias_value":"7Y62C3EKU2WY","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7Y62C3EKU2WYY5MB","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7Y62C3EK","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:03b88d5952491e0b15af0589a0469a810f7f8a343fe567a0f86462b770376b3c","target":"graph","created_at":"2026-05-18T04:34:38Z","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 study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour spreads out trough an infinite number of individuals. We present sharp lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual.","authors_text":"F\\'abio P. Machado, Mauricio Zuluaga, Valdivino V. Junior","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-26T02:02:04Z","title":"The cone percolation on $\\bbT_d$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.5691","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":"sha256:6829b04b4ae8e9f65e6fb8791acee4a603096b55677f55f2ea72445114f0e255","target":"record","created_at":"2026-05-18T04:34:38Z","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":"553ff9c9216de5aaf7741a760a1040eff750dfdcfa3900f3c67e1e23c2fc3b88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-26T02:02:04Z","title_canon_sha256":"648d1bd9bdc64f21e13ecbcdc46a114dee85386a02d234841acb75d91a59a68e"},"schema_version":"1.0","source":{"id":"1011.5691","kind":"arxiv","version":1}},"canonical_sha256":"fe3da16c8aa6ad8c75810d9fbe52d7ea61a846e995d0681af34ca054e48aeb14","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe3da16c8aa6ad8c75810d9fbe52d7ea61a846e995d0681af34ca054e48aeb14","first_computed_at":"2026-05-18T04:34:38.653021Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:34:38.653021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OCtSGiwgOn38a92Ez6kF/Tw/LfqmXE7avnsRQaVzW+2saNZtQLK/4KUwGiiHFJaV3iQ7kPUoBgdORqQKaw6fAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:34:38.653441Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.5691","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6829b04b4ae8e9f65e6fb8791acee4a603096b55677f55f2ea72445114f0e255","sha256:03b88d5952491e0b15af0589a0469a810f7f8a343fe567a0f86462b770376b3c"],"state_sha256":"6b38f74f5c24cdb8e2244e524d510df03dcbab420b25cdeb1fa12d0296ef3689"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FxUfiq3KKdRn0D0HfOoejjP3UOfr3z5R9sba595QDqfUNFt/W+FEO3pEdCJj5vC+P25AOVHMTaCJIAz2PR14AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T17:32:14.010636Z","bundle_sha256":"c3109871c32110e7cd9cacc8d376ce2bb90293dab458ab8ff6776b9844f36e7a"}}