{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:V3CIO4NVUVYGGW54JWBW4FRJ6U","short_pith_number":"pith:V3CIO4NV","canonical_record":{"source":{"id":"1503.06346","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-21T20:07:03Z","cross_cats_sorted":[],"title_canon_sha256":"172cd0d68073c0bef1aab9eda56a5a4842e8031eceb75ce1cabcdc8177d65463","abstract_canon_sha256":"f683e4e81fe545c2dbd619cb76f57ed97ad83d035136637b9a6af8832fb648c3"},"schema_version":"1.0"},"canonical_sha256":"aec48771b5a570635bbc4d836e1629f5348936889c98c6ad43341a1c5a6085c0","source":{"kind":"arxiv","id":"1503.06346","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.06346","created_at":"2026-05-18T02:20:02Z"},{"alias_kind":"arxiv_version","alias_value":"1503.06346v2","created_at":"2026-05-18T02:20:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06346","created_at":"2026-05-18T02:20:02Z"},{"alias_kind":"pith_short_12","alias_value":"V3CIO4NVUVYG","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"V3CIO4NVUVYGGW54","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"V3CIO4NV","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:V3CIO4NVUVYGGW54JWBW4FRJ6U","target":"record","payload":{"canonical_record":{"source":{"id":"1503.06346","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-21T20:07:03Z","cross_cats_sorted":[],"title_canon_sha256":"172cd0d68073c0bef1aab9eda56a5a4842e8031eceb75ce1cabcdc8177d65463","abstract_canon_sha256":"f683e4e81fe545c2dbd619cb76f57ed97ad83d035136637b9a6af8832fb648c3"},"schema_version":"1.0"},"canonical_sha256":"aec48771b5a570635bbc4d836e1629f5348936889c98c6ad43341a1c5a6085c0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:02.529666Z","signature_b64":"YrXVzfetOEvYW5Rznd70iitENZG8JvEpwYZovdR2SmR2yuytIOI4CclBBVCdI54SKFeBZ6pjsGKUOviVbC8GAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aec48771b5a570635bbc4d836e1629f5348936889c98c6ad43341a1c5a6085c0","last_reissued_at":"2026-05-18T02:20:02.528923Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:02.528923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.06346","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":"2026-05-18T02:20:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5YZ1AZYdTihuVJXKpA5LPZ3FxQdHsC6RqmNZj/tsZUdSVgbOlOIusDuuRmm1+YU/aYUjpgHDDQE4Nvktz4eqCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:05:05.238796Z"},"content_sha256":"df03fc49986772b6b4f4fa7e8c890ccc0c448f7d2a6d368d9a4571e7ae2b61f2","schema_version":"1.0","event_id":"sha256:df03fc49986772b6b4f4fa7e8c890ccc0c448f7d2a6d368d9a4571e7ae2b61f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:V3CIO4NVUVYGGW54JWBW4FRJ6U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Jigsaw Percolation on Erdos-Renyi Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Erik Slivken","submitted_at":"2015-03-21T20:07:03Z","abstract_excerpt":"We extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erd\\H{o}s-R\\'enyi random graphs. Let $p_{\\text{ppl}}$ and $p_{\\text{puz}}$ denote the probability that an edge exists in the respective people and puzzle graphs and define $p_{\\text{eff}}= p_{\\text{ppl}}p_{\\text{puz}}$, the effective probability. We show for constants $c_1>1$ and $c_2> \\pi^2/6$ and $c_3<e^{-5}$ if $min(p_{\\text{ppl}},p_{\\text{puz}}) > c_1 \\log n /n$ the critical effective probability $p^c_{\\text{eff}}$, satisfies $c_3 < p^c_{\\text{eff}}n\\log n < c_2.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06346","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":"2026-05-18T02:20:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gSkDKvnOzGTWbBs376BO3t/mUb9RbaIJngrTXE5Ug0/w/U2Iqpwa2ZvK7j0hXQ1vFv6MkYk4gwPigiRiVAtlDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:05:05.239492Z"},"content_sha256":"262c7950e88ad5a2539b0fb87a922b6ee8bbac954a9650b399ed0ec7076ffa70","schema_version":"1.0","event_id":"sha256:262c7950e88ad5a2539b0fb87a922b6ee8bbac954a9650b399ed0ec7076ffa70"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V3CIO4NVUVYGGW54JWBW4FRJ6U/bundle.json","state_url":"https://pith.science/pith/V3CIO4NVUVYGGW54JWBW4FRJ6U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V3CIO4NVUVYGGW54JWBW4FRJ6U/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":"2026-05-29T18:05:05Z","links":{"resolver":"https://pith.science/pith/V3CIO4NVUVYGGW54JWBW4FRJ6U","bundle":"https://pith.science/pith/V3CIO4NVUVYGGW54JWBW4FRJ6U/bundle.json","state":"https://pith.science/pith/V3CIO4NVUVYGGW54JWBW4FRJ6U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V3CIO4NVUVYGGW54JWBW4FRJ6U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:V3CIO4NVUVYGGW54JWBW4FRJ6U","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":"f683e4e81fe545c2dbd619cb76f57ed97ad83d035136637b9a6af8832fb648c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-21T20:07:03Z","title_canon_sha256":"172cd0d68073c0bef1aab9eda56a5a4842e8031eceb75ce1cabcdc8177d65463"},"schema_version":"1.0","source":{"id":"1503.06346","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.06346","created_at":"2026-05-18T02:20:02Z"},{"alias_kind":"arxiv_version","alias_value":"1503.06346v2","created_at":"2026-05-18T02:20:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06346","created_at":"2026-05-18T02:20:02Z"},{"alias_kind":"pith_short_12","alias_value":"V3CIO4NVUVYG","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"V3CIO4NVUVYGGW54","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"V3CIO4NV","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:262c7950e88ad5a2539b0fb87a922b6ee8bbac954a9650b399ed0ec7076ffa70","target":"graph","created_at":"2026-05-18T02:20:02Z","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 extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erd\\H{o}s-R\\'enyi random graphs. Let $p_{\\text{ppl}}$ and $p_{\\text{puz}}$ denote the probability that an edge exists in the respective people and puzzle graphs and define $p_{\\text{eff}}= p_{\\text{ppl}}p_{\\text{puz}}$, the effective probability. We show for constants $c_1>1$ and $c_2> \\pi^2/6$ and $c_3<e^{-5}$ if $min(p_{\\text{ppl}},p_{\\text{puz}}) > c_1 \\log n /n$ the critical effective probability $p^c_{\\text{eff}}$, satisfies $c_3 < p^c_{\\text{eff}}n\\log n < c_2.$","authors_text":"Erik Slivken","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-21T20:07:03Z","title":"Jigsaw Percolation on Erdos-Renyi Random Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06346","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:df03fc49986772b6b4f4fa7e8c890ccc0c448f7d2a6d368d9a4571e7ae2b61f2","target":"record","created_at":"2026-05-18T02:20:02Z","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":"f683e4e81fe545c2dbd619cb76f57ed97ad83d035136637b9a6af8832fb648c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-21T20:07:03Z","title_canon_sha256":"172cd0d68073c0bef1aab9eda56a5a4842e8031eceb75ce1cabcdc8177d65463"},"schema_version":"1.0","source":{"id":"1503.06346","kind":"arxiv","version":2}},"canonical_sha256":"aec48771b5a570635bbc4d836e1629f5348936889c98c6ad43341a1c5a6085c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aec48771b5a570635bbc4d836e1629f5348936889c98c6ad43341a1c5a6085c0","first_computed_at":"2026-05-18T02:20:02.528923Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:20:02.528923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YrXVzfetOEvYW5Rznd70iitENZG8JvEpwYZovdR2SmR2yuytIOI4CclBBVCdI54SKFeBZ6pjsGKUOviVbC8GAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:20:02.529666Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.06346","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:df03fc49986772b6b4f4fa7e8c890ccc0c448f7d2a6d368d9a4571e7ae2b61f2","sha256:262c7950e88ad5a2539b0fb87a922b6ee8bbac954a9650b399ed0ec7076ffa70"],"state_sha256":"393d6d3b770132b269f1c631e3fd2a677335350a96f4a7403686ea0b5015eb45"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bGRXE8SicYAZ38jWuQO/kcFGgCE1gOuTnj0blzpIHjWh5mvihpnhxZXwl7tX2jUPdP29ZumsvyXLS62dubgUAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T18:05:05.243960Z","bundle_sha256":"6d23dbfc021e8ebff1ea143c3464feb90fd25dcbc1d46c73f7200033532e8ef1"}}