{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:E7T7WQBVFYILXGURMGPFR4E6XD","short_pith_number":"pith:E7T7WQBV","canonical_record":{"source":{"id":"1403.6558","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-26T02:12:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"20e04774e562a8cab5bba58b7a6801a9737013151da1aa62ff917ca1e73c2845","abstract_canon_sha256":"6eb1fe95ea5fb4a497e6ec2b25d98ccf498c3f21c6f17be04a0b3f143fb7e576"},"schema_version":"1.0"},"canonical_sha256":"27e7fb40352e10bb9a91619e58f09eb8c1cfdf443722e62e86036466c83df4f1","source":{"kind":"arxiv","id":"1403.6558","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.6558","created_at":"2026-05-18T00:41:42Z"},{"alias_kind":"arxiv_version","alias_value":"1403.6558v2","created_at":"2026-05-18T00:41:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.6558","created_at":"2026-05-18T00:41:42Z"},{"alias_kind":"pith_short_12","alias_value":"E7T7WQBVFYIL","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"E7T7WQBVFYILXGUR","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"E7T7WQBV","created_at":"2026-05-18T12:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:E7T7WQBVFYILXGURMGPFR4E6XD","target":"record","payload":{"canonical_record":{"source":{"id":"1403.6558","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-26T02:12:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"20e04774e562a8cab5bba58b7a6801a9737013151da1aa62ff917ca1e73c2845","abstract_canon_sha256":"6eb1fe95ea5fb4a497e6ec2b25d98ccf498c3f21c6f17be04a0b3f143fb7e576"},"schema_version":"1.0"},"canonical_sha256":"27e7fb40352e10bb9a91619e58f09eb8c1cfdf443722e62e86036466c83df4f1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:42.338640Z","signature_b64":"pWQzG+j1zg4VjRzTwj1nLNondZEAKu+cLHQ6wYV2Cp3tGqoKz6wvJmCu4Z3yVruN9kxijznPWoWwPf/oEJWMAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27e7fb40352e10bb9a91619e58f09eb8c1cfdf443722e62e86036466c83df4f1","last_reissued_at":"2026-05-18T00:41:42.337834Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:42.337834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1403.6558","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":"2026-05-18T00:41:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aLLrzcJocazs5AKDrBA7MdY7UlHF8FJnqV+bYyHHK0S47eR+C3xi6ngiiPdNA5dnCW7SH6NCP5K7GH4omVbNBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T10:26:16.628986Z"},"content_sha256":"9c7daeee5ebfc68ec27a6475e7444046099d316208b69ff981c7538f3fed6fd8","schema_version":"1.0","event_id":"sha256:9c7daeee5ebfc68ec27a6475e7444046099d316208b69ff981c7538f3fed6fd8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:E7T7WQBVFYILXGURMGPFR4E6XD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exploring hypergraphs with martingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"B\\'ela Bollob\\'as, Oliver Riordan","submitted_at":"2014-03-26T02:12:20Z","abstract_excerpt":"Recently, we adapted exploration and martingale arguments of Nachmias and Peres, in turn based on ideas of Martin-L\\\"of, Karp and Aldous, to prove asymptotic normality of the number $L_1$ of vertices in the largest component $C$ of the random $r$-uniform hypergraph throughout the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of $L_1$, and joint asymptotic normality of $L_1$ and the number $M_1$ of edges of $C$. These results are used in a separate paper \"Counting connected hypergraphs via the probabilistic m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.6558","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":"2026-05-18T00:41:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8AvwyS6OJkpBUwjxSvX05I1u26fAGL8P2B17J3M9Ri9XzYM/Sm2Ij6L2dDxHvsmDhinP59Cmh2tHwTZaxQ5gDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T10:26:16.629339Z"},"content_sha256":"f7addba6778901be70388aa56025add15f55ba80602d35e312194968e17296ad","schema_version":"1.0","event_id":"sha256:f7addba6778901be70388aa56025add15f55ba80602d35e312194968e17296ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/bundle.json","state_url":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E7T7WQBVFYILXGURMGPFR4E6XD/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":"2026-06-02T10:26:16Z","links":{"resolver":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD","bundle":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/bundle.json","state":"https://pith.science/pith/E7T7WQBVFYILXGURMGPFR4E6XD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E7T7WQBVFYILXGURMGPFR4E6XD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:E7T7WQBVFYILXGURMGPFR4E6XD","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":"6eb1fe95ea5fb4a497e6ec2b25d98ccf498c3f21c6f17be04a0b3f143fb7e576","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-26T02:12:20Z","title_canon_sha256":"20e04774e562a8cab5bba58b7a6801a9737013151da1aa62ff917ca1e73c2845"},"schema_version":"1.0","source":{"id":"1403.6558","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.6558","created_at":"2026-05-18T00:41:42Z"},{"alias_kind":"arxiv_version","alias_value":"1403.6558v2","created_at":"2026-05-18T00:41:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.6558","created_at":"2026-05-18T00:41:42Z"},{"alias_kind":"pith_short_12","alias_value":"E7T7WQBVFYIL","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"E7T7WQBVFYILXGUR","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"E7T7WQBV","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:f7addba6778901be70388aa56025add15f55ba80602d35e312194968e17296ad","target":"graph","created_at":"2026-05-18T00:41:42Z","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":"Recently, we adapted exploration and martingale arguments of Nachmias and Peres, in turn based on ideas of Martin-L\\\"of, Karp and Aldous, to prove asymptotic normality of the number $L_1$ of vertices in the largest component $C$ of the random $r$-uniform hypergraph throughout the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of $L_1$, and joint asymptotic normality of $L_1$ and the number $M_1$ of edges of $C$. These results are used in a separate paper \"Counting connected hypergraphs via the probabilistic m","authors_text":"B\\'ela Bollob\\'as, Oliver Riordan","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-26T02:12:20Z","title":"Exploring hypergraphs with martingales"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.6558","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":"sha256:9c7daeee5ebfc68ec27a6475e7444046099d316208b69ff981c7538f3fed6fd8","target":"record","created_at":"2026-05-18T00:41:42Z","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":"6eb1fe95ea5fb4a497e6ec2b25d98ccf498c3f21c6f17be04a0b3f143fb7e576","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-26T02:12:20Z","title_canon_sha256":"20e04774e562a8cab5bba58b7a6801a9737013151da1aa62ff917ca1e73c2845"},"schema_version":"1.0","source":{"id":"1403.6558","kind":"arxiv","version":2}},"canonical_sha256":"27e7fb40352e10bb9a91619e58f09eb8c1cfdf443722e62e86036466c83df4f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"27e7fb40352e10bb9a91619e58f09eb8c1cfdf443722e62e86036466c83df4f1","first_computed_at":"2026-05-18T00:41:42.337834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:42.337834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pWQzG+j1zg4VjRzTwj1nLNondZEAKu+cLHQ6wYV2Cp3tGqoKz6wvJmCu4Z3yVruN9kxijznPWoWwPf/oEJWMAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:42.338640Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.6558","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9c7daeee5ebfc68ec27a6475e7444046099d316208b69ff981c7538f3fed6fd8","sha256:f7addba6778901be70388aa56025add15f55ba80602d35e312194968e17296ad"],"state_sha256":"c47c35fd180f2e511d7c32a4c4cab075b85a2441f2973badf070624bfe8d0e71"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j6tJEOv1SiD2sB0/AfoWnylg5Rn8AWqA9NwU0zEt+hHtlGuQlEnWaT+XOzTwChueyJC1D+x5cOGnEOYUy/ZfAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T10:26:16.631174Z","bundle_sha256":"a23f9ad988aa6b045c7a826ee752448894a94b0dec13caac4409ce35d264d5ee"}}