{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:45VMFAQMEZOSDGCGP7B7KGKUC3","short_pith_number":"pith:45VMFAQM","canonical_record":{"source":{"id":"1203.2209","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-09T23:27:45Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"66df7fd21f5dc7f6d0925022b5b0644915e0bb996c5906c233b79e5d77ecc000","abstract_canon_sha256":"57aad640341260f32698cc8ac015a8fe974ad45fd66fcbc1f88dab32a110a3f2"},"schema_version":"1.0"},"canonical_sha256":"e76ac2820c265d2198467fc3f5195416d39ef4a3b742a686a81878acabd82658","source":{"kind":"arxiv","id":"1203.2209","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.2209","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"arxiv_version","alias_value":"1203.2209v2","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.2209","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"pith_short_12","alias_value":"45VMFAQMEZOS","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"45VMFAQMEZOSDGCG","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"45VMFAQM","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:45VMFAQMEZOSDGCGP7B7KGKUC3","target":"record","payload":{"canonical_record":{"source":{"id":"1203.2209","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-09T23:27:45Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"66df7fd21f5dc7f6d0925022b5b0644915e0bb996c5906c233b79e5d77ecc000","abstract_canon_sha256":"57aad640341260f32698cc8ac015a8fe974ad45fd66fcbc1f88dab32a110a3f2"},"schema_version":"1.0"},"canonical_sha256":"e76ac2820c265d2198467fc3f5195416d39ef4a3b742a686a81878acabd82658","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:53.037039Z","signature_b64":"Zdz12g8B8hgLEO3rkWnGjYKzn5bMuSoeJpgFoW5U80198XKc8MfSyxbzyjgwT3tAcVtRlewwI2apgxVY0Dj3Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e76ac2820c265d2198467fc3f5195416d39ef4a3b742a686a81878acabd82658","last_reissued_at":"2026-05-18T03:45:53.036445Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:53.036445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.2209","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":"2026-05-18T03:45:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mfzl1jMVo0nw3PgW2CbsTpEYn6UNlRtCJlY5KlnGP+yKEldWJQdPfi+U6CNFP7U3gk4X8sLHUNP+617cZ6aOAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:01:15.451576Z"},"content_sha256":"73f17a48119ed5d4458a883308e152bcd8207acb3e97b70826ff5a5c526b98fa","schema_version":"1.0","event_id":"sha256:73f17a48119ed5d4458a883308e152bcd8207acb3e97b70826ff5a5c526b98fa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:45VMFAQMEZOSDGCGP7B7KGKUC3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the robustness of random k-cores","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Cristiane M. Sato","submitted_at":"2012-03-09T23:27:45Z","abstract_excerpt":"The k-core of a graph is its maximal subgraph with minimum degree at least k. In this paper, we address robustness questions about k-cores. Given a k-core, remove one edge uniformly at random and find its new k-core. We are interested in how many vertices are deleted from the original k-core to find the new one. This can be seem as a measure of robustness of the original k-core. We prove that, if the initial k-core is chosen uniformly at random from the k-cores with n vertices and m edges, its robustness depends essentially on its average degree c. We prove that, if c converges to k, then the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.2209","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":"2026-05-18T03:45:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H7+hXCPHUlbdrt2p8KBulYJSAe0b4qyxSzI+1YwoeppNMY8PHnM62unqNNRpp9JNZeYCu51FVTvE4kXZN9EfCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:01:15.451929Z"},"content_sha256":"8d1e35ebb902241e3666777790dfbe2197dd95a3899fb34d330f3ac0356c67a0","schema_version":"1.0","event_id":"sha256:8d1e35ebb902241e3666777790dfbe2197dd95a3899fb34d330f3ac0356c67a0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/45VMFAQMEZOSDGCGP7B7KGKUC3/bundle.json","state_url":"https://pith.science/pith/45VMFAQMEZOSDGCGP7B7KGKUC3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/45VMFAQMEZOSDGCGP7B7KGKUC3/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":"2026-06-01T12:01:15Z","links":{"resolver":"https://pith.science/pith/45VMFAQMEZOSDGCGP7B7KGKUC3","bundle":"https://pith.science/pith/45VMFAQMEZOSDGCGP7B7KGKUC3/bundle.json","state":"https://pith.science/pith/45VMFAQMEZOSDGCGP7B7KGKUC3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/45VMFAQMEZOSDGCGP7B7KGKUC3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:45VMFAQMEZOSDGCGP7B7KGKUC3","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":"57aad640341260f32698cc8ac015a8fe974ad45fd66fcbc1f88dab32a110a3f2","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-09T23:27:45Z","title_canon_sha256":"66df7fd21f5dc7f6d0925022b5b0644915e0bb996c5906c233b79e5d77ecc000"},"schema_version":"1.0","source":{"id":"1203.2209","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.2209","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"arxiv_version","alias_value":"1203.2209v2","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.2209","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"pith_short_12","alias_value":"45VMFAQMEZOS","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"45VMFAQMEZOSDGCG","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"45VMFAQM","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:8d1e35ebb902241e3666777790dfbe2197dd95a3899fb34d330f3ac0356c67a0","target":"graph","created_at":"2026-05-18T03:45:53Z","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":"The k-core of a graph is its maximal subgraph with minimum degree at least k. In this paper, we address robustness questions about k-cores. Given a k-core, remove one edge uniformly at random and find its new k-core. We are interested in how many vertices are deleted from the original k-core to find the new one. This can be seem as a measure of robustness of the original k-core. We prove that, if the initial k-core is chosen uniformly at random from the k-cores with n vertices and m edges, its robustness depends essentially on its average degree c. We prove that, if c converges to k, then the ","authors_text":"Cristiane M. Sato","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-09T23:27:45Z","title":"On the robustness of random k-cores"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.2209","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":"sha256:73f17a48119ed5d4458a883308e152bcd8207acb3e97b70826ff5a5c526b98fa","target":"record","created_at":"2026-05-18T03:45:53Z","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":"57aad640341260f32698cc8ac015a8fe974ad45fd66fcbc1f88dab32a110a3f2","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-09T23:27:45Z","title_canon_sha256":"66df7fd21f5dc7f6d0925022b5b0644915e0bb996c5906c233b79e5d77ecc000"},"schema_version":"1.0","source":{"id":"1203.2209","kind":"arxiv","version":2}},"canonical_sha256":"e76ac2820c265d2198467fc3f5195416d39ef4a3b742a686a81878acabd82658","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e76ac2820c265d2198467fc3f5195416d39ef4a3b742a686a81878acabd82658","first_computed_at":"2026-05-18T03:45:53.036445Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:53.036445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Zdz12g8B8hgLEO3rkWnGjYKzn5bMuSoeJpgFoW5U80198XKc8MfSyxbzyjgwT3tAcVtRlewwI2apgxVY0Dj3Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:53.037039Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.2209","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73f17a48119ed5d4458a883308e152bcd8207acb3e97b70826ff5a5c526b98fa","sha256:8d1e35ebb902241e3666777790dfbe2197dd95a3899fb34d330f3ac0356c67a0"],"state_sha256":"490de4aa95b1e3bf5e634613425407b04ed694e23ca952470b87155de95208a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WeDkYrmkxfU+RywoxVe4YrdPPBIoroHJK3eFxeZNTVXtAgrjGD3kH2bW+VLOppajAdt6gQRmYwAg4BvQWt6QBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T12:01:15.453905Z","bundle_sha256":"6ea17dcf1c8b6ee060342440ff74289c1f83d714f0717f895dd005bb2292fea4"}}