{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:3HRX3AXXMI5BIOTWAYJXAYOEBD","short_pith_number":"pith:3HRX3AXX","canonical_record":{"source":{"id":"1410.5347","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-20T16:39:56Z","cross_cats_sorted":[],"title_canon_sha256":"67d95e904e4bb31b6466d51953c6c109727fc957278b1a4cf5f954362783d2ec","abstract_canon_sha256":"9dcf2428cd3db81797510553cdec60d0d2eec7945a847d344ca1143fbb27e76c"},"schema_version":"1.0"},"canonical_sha256":"d9e37d82f7623a143a7606137061c408d9248b72ebdad0e43de21eae2b52c318","source":{"kind":"arxiv","id":"1410.5347","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5347","created_at":"2026-05-18T00:04:47Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5347v2","created_at":"2026-05-18T00:04:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5347","created_at":"2026-05-18T00:04:47Z"},{"alias_kind":"pith_short_12","alias_value":"3HRX3AXXMI5B","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3HRX3AXXMI5BIOTW","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3HRX3AXX","created_at":"2026-05-18T12:28:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:3HRX3AXXMI5BIOTWAYJXAYOEBD","target":"record","payload":{"canonical_record":{"source":{"id":"1410.5347","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-20T16:39:56Z","cross_cats_sorted":[],"title_canon_sha256":"67d95e904e4bb31b6466d51953c6c109727fc957278b1a4cf5f954362783d2ec","abstract_canon_sha256":"9dcf2428cd3db81797510553cdec60d0d2eec7945a847d344ca1143fbb27e76c"},"schema_version":"1.0"},"canonical_sha256":"d9e37d82f7623a143a7606137061c408d9248b72ebdad0e43de21eae2b52c318","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:47.815181Z","signature_b64":"aoY43G2CyBrLL82BTLOYjzlW8gzzHfV2Itf6WqOsk3dLr+Cl8m+OykMAbDiOTk7aTQPDZCk4uArKIRvuMKgkDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9e37d82f7623a143a7606137061c408d9248b72ebdad0e43de21eae2b52c318","last_reissued_at":"2026-05-18T00:04:47.814497Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:47.814497Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.5347","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":"2026-05-18T00:04:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qr0D8E9HfwZ3hj/qjiR6ksDD7pPdGsp2xkw/sYaDB4dsfGnPpKYPCdP5rpSJs0X4ApRWJrwz2vYUjL+5VbGmBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T19:09:34.567617Z"},"content_sha256":"41212ee55bb6126d8709af3e9585b64c573a5cb64e7f26fbfebc1f1e069fbbee","schema_version":"1.0","event_id":"sha256:41212ee55bb6126d8709af3e9585b64c573a5cb64e7f26fbfebc1f1e069fbbee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:3HRX3AXXMI5BIOTWAYJXAYOEBD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boolean Percolation on Doubling Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Cristian F. Coletti, Daniel Miranda, Sebastian P. Grynberg","submitted_at":"2014-10-20T16:39:56Z","abstract_excerpt":"We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5347","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":"2026-05-18T00:04:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5BkTwXIhvxlB1gV3JraoCo3kNA8+Qgm7YrgLpgKO2LfAwDBn4Y8a/bHgA63Cut+a1US0itzqQSr3lAB0kgjNCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T19:09:34.567960Z"},"content_sha256":"509a1ddda8b3888b55c4316cd826bf70a25da674007630cc86ffeab869ff402e","schema_version":"1.0","event_id":"sha256:509a1ddda8b3888b55c4316cd826bf70a25da674007630cc86ffeab869ff402e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3HRX3AXXMI5BIOTWAYJXAYOEBD/bundle.json","state_url":"https://pith.science/pith/3HRX3AXXMI5BIOTWAYJXAYOEBD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3HRX3AXXMI5BIOTWAYJXAYOEBD/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":"2026-06-25T19:09:34Z","links":{"resolver":"https://pith.science/pith/3HRX3AXXMI5BIOTWAYJXAYOEBD","bundle":"https://pith.science/pith/3HRX3AXXMI5BIOTWAYJXAYOEBD/bundle.json","state":"https://pith.science/pith/3HRX3AXXMI5BIOTWAYJXAYOEBD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3HRX3AXXMI5BIOTWAYJXAYOEBD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:3HRX3AXXMI5BIOTWAYJXAYOEBD","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":"9dcf2428cd3db81797510553cdec60d0d2eec7945a847d344ca1143fbb27e76c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-20T16:39:56Z","title_canon_sha256":"67d95e904e4bb31b6466d51953c6c109727fc957278b1a4cf5f954362783d2ec"},"schema_version":"1.0","source":{"id":"1410.5347","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5347","created_at":"2026-05-18T00:04:47Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5347v2","created_at":"2026-05-18T00:04:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5347","created_at":"2026-05-18T00:04:47Z"},{"alias_kind":"pith_short_12","alias_value":"3HRX3AXXMI5B","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3HRX3AXXMI5BIOTW","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3HRX3AXX","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:509a1ddda8b3888b55c4316cd826bf70a25da674007630cc86ffeab869ff402e","target":"graph","created_at":"2026-05-18T00:04:47Z","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 consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.","authors_text":"Cristian F. Coletti, Daniel Miranda, Sebastian P. Grynberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-20T16:39:56Z","title":"Boolean Percolation on Doubling Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5347","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":"sha256:41212ee55bb6126d8709af3e9585b64c573a5cb64e7f26fbfebc1f1e069fbbee","target":"record","created_at":"2026-05-18T00:04:47Z","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":"9dcf2428cd3db81797510553cdec60d0d2eec7945a847d344ca1143fbb27e76c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-10-20T16:39:56Z","title_canon_sha256":"67d95e904e4bb31b6466d51953c6c109727fc957278b1a4cf5f954362783d2ec"},"schema_version":"1.0","source":{"id":"1410.5347","kind":"arxiv","version":2}},"canonical_sha256":"d9e37d82f7623a143a7606137061c408d9248b72ebdad0e43de21eae2b52c318","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d9e37d82f7623a143a7606137061c408d9248b72ebdad0e43de21eae2b52c318","first_computed_at":"2026-05-18T00:04:47.814497Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:47.814497Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aoY43G2CyBrLL82BTLOYjzlW8gzzHfV2Itf6WqOsk3dLr+Cl8m+OykMAbDiOTk7aTQPDZCk4uArKIRvuMKgkDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:47.815181Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.5347","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41212ee55bb6126d8709af3e9585b64c573a5cb64e7f26fbfebc1f1e069fbbee","sha256:509a1ddda8b3888b55c4316cd826bf70a25da674007630cc86ffeab869ff402e"],"state_sha256":"17c610dbeecb53005824c0bc6926de7cdb953c8ec82937ce32af82130e1a93ed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pN+tv4NC14JVS0W9/9fJ0uUPOgKwHD9fgKn9x7wcSjSlsguCT+Y9rW092aV1D71vEH3QwYBa5JU+luW8A18tBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T19:09:34.570063Z","bundle_sha256":"a983fea93b4f2b03a34874fb1cc7edbe765a25cf03c579211385f7d9b05c9e65"}}