{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:UQPH7MXHUTOKZG67VPLYAV36TY","short_pith_number":"pith:UQPH7MXH","canonical_record":{"source":{"id":"1202.6112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-28T04:09:03Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"e4aad695b28f395ef36f16dbbae63f8747ba69a43104700633611cb038e0a057","abstract_canon_sha256":"5ec6f6d730c90c8a8186975d66778ba34b1c9a597b54ac3884acf207d0da02fe"},"schema_version":"1.0"},"canonical_sha256":"a41e7fb2e7a4dcac9bdfabd780577e9e0e977bef62a37b63b25111f641c4afa1","source":{"kind":"arxiv","id":"1202.6112","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.6112","created_at":"2026-05-18T04:00:00Z"},{"alias_kind":"arxiv_version","alias_value":"1202.6112v1","created_at":"2026-05-18T04:00:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.6112","created_at":"2026-05-18T04:00:00Z"},{"alias_kind":"pith_short_12","alias_value":"UQPH7MXHUTOK","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"UQPH7MXHUTOKZG67","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"UQPH7MXH","created_at":"2026-05-18T12:27:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:UQPH7MXHUTOKZG67VPLYAV36TY","target":"record","payload":{"canonical_record":{"source":{"id":"1202.6112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-28T04:09:03Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"e4aad695b28f395ef36f16dbbae63f8747ba69a43104700633611cb038e0a057","abstract_canon_sha256":"5ec6f6d730c90c8a8186975d66778ba34b1c9a597b54ac3884acf207d0da02fe"},"schema_version":"1.0"},"canonical_sha256":"a41e7fb2e7a4dcac9bdfabd780577e9e0e977bef62a37b63b25111f641c4afa1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:00.738517Z","signature_b64":"WZy8bGsFvf++33Qh8BiGWQwbgcfZjwAo6BLbT/eQ/x1RbBfEvzkGBW6vW84lwLO///7tT7t8NB0CdueC4hfoDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a41e7fb2e7a4dcac9bdfabd780577e9e0e977bef62a37b63b25111f641c4afa1","last_reissued_at":"2026-05-18T04:00:00.737256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:00.737256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.6112","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":"2026-05-18T04:00:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LeCoOB1s5F/tLm6q3tG2wQaMzv1cJ8pI74P4yNwWDt8wKaC1k228F84hgt3KAoZj97LRQmb71xWNreG8v/3mCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T22:24:56.005477Z"},"content_sha256":"0b37d3f1b43e811dc6a7391ba79d5efe50a18a6df408f1843ea2d08c21b43fcd","schema_version":"1.0","event_id":"sha256:0b37d3f1b43e811dc6a7391ba79d5efe50a18a6df408f1843ea2d08c21b43fcd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:UQPH7MXHUTOKZG67VPLYAV36TY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Anatomy of the giant component: The strictly supercritical regime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Eyal Lubetzky, Jian Ding, Yuval Peres","submitted_at":"2012-02-28T04:09:03Z","abstract_excerpt":"In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erd\\H{o}s-R\\'enyi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+\\epsilon)/n$ where $\\epsilon^3 n \\to\\infty$ and $\\epsilon=o(1)$, in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e., the largest component of $G(n,p)$ for $p = \\lambda/n$ where $\\lambda>1$ is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.6112","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":"2026-05-18T04:00:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v7UgbnW5GjDmEK7TBf2llzfRHL4KW2mVNuQgiYJlj3hAIQGkEJ05C0EihgR/iV0ieEvSzscOM0Elsg1gA5VlDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T22:24:56.005837Z"},"content_sha256":"9b06395fdb8d8912afa38e6ff384f3102a63a32f401d976e1110ce3d4bce0ff3","schema_version":"1.0","event_id":"sha256:9b06395fdb8d8912afa38e6ff384f3102a63a32f401d976e1110ce3d4bce0ff3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UQPH7MXHUTOKZG67VPLYAV36TY/bundle.json","state_url":"https://pith.science/pith/UQPH7MXHUTOKZG67VPLYAV36TY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UQPH7MXHUTOKZG67VPLYAV36TY/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":"2026-05-29T22:24:56Z","links":{"resolver":"https://pith.science/pith/UQPH7MXHUTOKZG67VPLYAV36TY","bundle":"https://pith.science/pith/UQPH7MXHUTOKZG67VPLYAV36TY/bundle.json","state":"https://pith.science/pith/UQPH7MXHUTOKZG67VPLYAV36TY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UQPH7MXHUTOKZG67VPLYAV36TY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:UQPH7MXHUTOKZG67VPLYAV36TY","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":"5ec6f6d730c90c8a8186975d66778ba34b1c9a597b54ac3884acf207d0da02fe","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-28T04:09:03Z","title_canon_sha256":"e4aad695b28f395ef36f16dbbae63f8747ba69a43104700633611cb038e0a057"},"schema_version":"1.0","source":{"id":"1202.6112","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.6112","created_at":"2026-05-18T04:00:00Z"},{"alias_kind":"arxiv_version","alias_value":"1202.6112v1","created_at":"2026-05-18T04:00:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.6112","created_at":"2026-05-18T04:00:00Z"},{"alias_kind":"pith_short_12","alias_value":"UQPH7MXHUTOK","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"UQPH7MXHUTOKZG67","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"UQPH7MXH","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:9b06395fdb8d8912afa38e6ff384f3102a63a32f401d976e1110ce3d4bce0ff3","target":"graph","created_at":"2026-05-18T04:00:00Z","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":"In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erd\\H{o}s-R\\'enyi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+\\epsilon)/n$ where $\\epsilon^3 n \\to\\infty$ and $\\epsilon=o(1)$, in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e., the largest component of $G(n,p)$ for $p = \\lambda/n$ where $\\lambda>1$ is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degr","authors_text":"Eyal Lubetzky, Jian Ding, Yuval Peres","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-28T04:09:03Z","title":"Anatomy of the giant component: The strictly supercritical regime"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.6112","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":"sha256:0b37d3f1b43e811dc6a7391ba79d5efe50a18a6df408f1843ea2d08c21b43fcd","target":"record","created_at":"2026-05-18T04:00:00Z","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":"5ec6f6d730c90c8a8186975d66778ba34b1c9a597b54ac3884acf207d0da02fe","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-28T04:09:03Z","title_canon_sha256":"e4aad695b28f395ef36f16dbbae63f8747ba69a43104700633611cb038e0a057"},"schema_version":"1.0","source":{"id":"1202.6112","kind":"arxiv","version":1}},"canonical_sha256":"a41e7fb2e7a4dcac9bdfabd780577e9e0e977bef62a37b63b25111f641c4afa1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a41e7fb2e7a4dcac9bdfabd780577e9e0e977bef62a37b63b25111f641c4afa1","first_computed_at":"2026-05-18T04:00:00.737256Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:00.737256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WZy8bGsFvf++33Qh8BiGWQwbgcfZjwAo6BLbT/eQ/x1RbBfEvzkGBW6vW84lwLO///7tT7t8NB0CdueC4hfoDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:00.738517Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.6112","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b37d3f1b43e811dc6a7391ba79d5efe50a18a6df408f1843ea2d08c21b43fcd","sha256:9b06395fdb8d8912afa38e6ff384f3102a63a32f401d976e1110ce3d4bce0ff3"],"state_sha256":"e0fcadee1259adeae5321a0e1e654809a60b89602410a5c882349258d5774390"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BOrPF+fk1Ak6HAU9DqQ6P3Z253v/09/11ILXqcdSmDyffsX1um7gZwJpjWbNk0iXttZ7xQVA9AEmNJDFZAGZDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T22:24:56.008207Z","bundle_sha256":"d398afff4e03fc3066188c7ad89bee7c12b8b195704931a83a0bb58abc6be7de"}}