{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:N6NZNQSDK4STIMATLXYCW5GZCQ","short_pith_number":"pith:N6NZNQSD","canonical_record":{"source":{"id":"1611.03075","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-09T20:48:20Z","cross_cats_sorted":["cs.DM","math.PR"],"title_canon_sha256":"7d394680ed17e2d25bf66838ab7e55d424c3db2a6f90a34c39f44ed1fe57dd26","abstract_canon_sha256":"037b4a70c858c5007646a9e8924fa6bdf1102e65451b28ffa1330bb5e5886ed8"},"schema_version":"1.0"},"canonical_sha256":"6f9b96c24357253430135df02b74d9143bb5046aa59f057ff712685dcb729219","source":{"kind":"arxiv","id":"1611.03075","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03075","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03075v2","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03075","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"pith_short_12","alias_value":"N6NZNQSDK4ST","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"N6NZNQSDK4STIMAT","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"N6NZNQSD","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:N6NZNQSDK4STIMATLXYCW5GZCQ","target":"record","payload":{"canonical_record":{"source":{"id":"1611.03075","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-09T20:48:20Z","cross_cats_sorted":["cs.DM","math.PR"],"title_canon_sha256":"7d394680ed17e2d25bf66838ab7e55d424c3db2a6f90a34c39f44ed1fe57dd26","abstract_canon_sha256":"037b4a70c858c5007646a9e8924fa6bdf1102e65451b28ffa1330bb5e5886ed8"},"schema_version":"1.0"},"canonical_sha256":"6f9b96c24357253430135df02b74d9143bb5046aa59f057ff712685dcb729219","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:51.132189Z","signature_b64":"i0cnAO5XH3OBhA609NdkWb2tgUJ/pEjjwUAb1qBWH7Vuju3iDd62H3e/sGjMRdwrS4ghboJFWVKj7nXtP9eCDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f9b96c24357253430135df02b74d9143bb5046aa59f057ff712685dcb729219","last_reissued_at":"2026-05-18T00:32:51.131561Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:51.131561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.03075","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":"2026-05-18T00:32:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mDUMMhp52xHXrTs0f19aQUaJn+r5iNE3bphCfufUVewxrB1WgNXlFIhTWmi0GvW0VBrwLCkKOR3eLsbO02zZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:11:43.104836Z"},"content_sha256":"9bb733277bab458478b9e5e59383dd7d706e905c704a74b86ba899c8e612b187","schema_version":"1.0","event_id":"sha256:9bb733277bab458478b9e5e59383dd7d706e905c704a74b86ba899c8e612b187"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:N6NZNQSDK4STIMATLXYCW5GZCQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Phase transitions of extremal cuts for the configuration model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.PR"],"primary_cat":"math.CO","authors_text":"Debankur Mukherjee, Souvik Dhara, Subhabrata Sen","submitted_at":"2016-11-09T20:48:20Z","abstract_excerpt":"The $k$-section width and the Max-Cut for the configuration model are shown to exhibit phase transitions according to the values of certain parameters of the asymptotic degree distribution. These transitions mirror those observed on Erd\\H{o}s-R\\'enyi random graphs, established by Luczak and McDiarmid (2001), and Coppersmith et al. (2004), respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03075","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":"2026-05-18T00:32:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J6AR6oMf5beuGp0GnR9yhAHkDbCqJtDTVg/YI7S91ErcULT7BSyNH9GvJBewxPFoCywcP96+JcxNs6S+WhBdDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:11:43.105181Z"},"content_sha256":"c0706eec834c28a7e1874dab9cc3beefc6652095d1224c8d510c9a88779b53bf","schema_version":"1.0","event_id":"sha256:c0706eec834c28a7e1874dab9cc3beefc6652095d1224c8d510c9a88779b53bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N6NZNQSDK4STIMATLXYCW5GZCQ/bundle.json","state_url":"https://pith.science/pith/N6NZNQSDK4STIMATLXYCW5GZCQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N6NZNQSDK4STIMATLXYCW5GZCQ/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":"2026-05-28T19:11:43Z","links":{"resolver":"https://pith.science/pith/N6NZNQSDK4STIMATLXYCW5GZCQ","bundle":"https://pith.science/pith/N6NZNQSDK4STIMATLXYCW5GZCQ/bundle.json","state":"https://pith.science/pith/N6NZNQSDK4STIMATLXYCW5GZCQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N6NZNQSDK4STIMATLXYCW5GZCQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:N6NZNQSDK4STIMATLXYCW5GZCQ","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":"037b4a70c858c5007646a9e8924fa6bdf1102e65451b28ffa1330bb5e5886ed8","cross_cats_sorted":["cs.DM","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-09T20:48:20Z","title_canon_sha256":"7d394680ed17e2d25bf66838ab7e55d424c3db2a6f90a34c39f44ed1fe57dd26"},"schema_version":"1.0","source":{"id":"1611.03075","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03075","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03075v2","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03075","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"pith_short_12","alias_value":"N6NZNQSDK4ST","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"N6NZNQSDK4STIMAT","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"N6NZNQSD","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:c0706eec834c28a7e1874dab9cc3beefc6652095d1224c8d510c9a88779b53bf","target":"graph","created_at":"2026-05-18T00:32:51Z","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 $k$-section width and the Max-Cut for the configuration model are shown to exhibit phase transitions according to the values of certain parameters of the asymptotic degree distribution. These transitions mirror those observed on Erd\\H{o}s-R\\'enyi random graphs, established by Luczak and McDiarmid (2001), and Coppersmith et al. (2004), respectively.","authors_text":"Debankur Mukherjee, Souvik Dhara, Subhabrata Sen","cross_cats":["cs.DM","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-09T20:48:20Z","title":"Phase transitions of extremal cuts for the configuration model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03075","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":"sha256:9bb733277bab458478b9e5e59383dd7d706e905c704a74b86ba899c8e612b187","target":"record","created_at":"2026-05-18T00:32:51Z","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":"037b4a70c858c5007646a9e8924fa6bdf1102e65451b28ffa1330bb5e5886ed8","cross_cats_sorted":["cs.DM","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-09T20:48:20Z","title_canon_sha256":"7d394680ed17e2d25bf66838ab7e55d424c3db2a6f90a34c39f44ed1fe57dd26"},"schema_version":"1.0","source":{"id":"1611.03075","kind":"arxiv","version":2}},"canonical_sha256":"6f9b96c24357253430135df02b74d9143bb5046aa59f057ff712685dcb729219","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f9b96c24357253430135df02b74d9143bb5046aa59f057ff712685dcb729219","first_computed_at":"2026-05-18T00:32:51.131561Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:51.131561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i0cnAO5XH3OBhA609NdkWb2tgUJ/pEjjwUAb1qBWH7Vuju3iDd62H3e/sGjMRdwrS4ghboJFWVKj7nXtP9eCDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:51.132189Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.03075","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9bb733277bab458478b9e5e59383dd7d706e905c704a74b86ba899c8e612b187","sha256:c0706eec834c28a7e1874dab9cc3beefc6652095d1224c8d510c9a88779b53bf"],"state_sha256":"89d74ccf0704f242beef9c72304f81568cb24d4b783efac8c225a8019c5e3620"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F1tnQSiPX30e5AVaNY8KVVQPg7FMLyl478xV+k6hb6ZoHxhBCNnDNiGLuAwr8EbciWEBO/rMJkqoWGxucp/0BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T19:11:43.107149Z","bundle_sha256":"b7d3d505321519890911fb4b6aca5d316ce8634eecfc74756aadcd466a7a9ba2"}}