{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:DMXDYI5VCRPKLRS5C4FR2VZ4FM","short_pith_number":"pith:DMXDYI5V","canonical_record":{"source":{"id":"1310.6921","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-25T13:56:36Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"d38bbc5e26ba378eee7bf7b198007364f5117100b5d4cdf46820004b22147b2b","abstract_canon_sha256":"6847face5b1121d8241319be6ecbd88f85c880582a9a55f9c41d26e6e9cbee36"},"schema_version":"1.0"},"canonical_sha256":"1b2e3c23b5145ea5c65d170b1d573c2b0b6695211f01875aed987a997e5f8d60","source":{"kind":"arxiv","id":"1310.6921","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.6921","created_at":"2026-05-18T02:55:38Z"},{"alias_kind":"arxiv_version","alias_value":"1310.6921v3","created_at":"2026-05-18T02:55:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.6921","created_at":"2026-05-18T02:55:38Z"},{"alias_kind":"pith_short_12","alias_value":"DMXDYI5VCRPK","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DMXDYI5VCRPKLRS5","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DMXDYI5V","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:DMXDYI5VCRPKLRS5C4FR2VZ4FM","target":"record","payload":{"canonical_record":{"source":{"id":"1310.6921","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-25T13:56:36Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"d38bbc5e26ba378eee7bf7b198007364f5117100b5d4cdf46820004b22147b2b","abstract_canon_sha256":"6847face5b1121d8241319be6ecbd88f85c880582a9a55f9c41d26e6e9cbee36"},"schema_version":"1.0"},"canonical_sha256":"1b2e3c23b5145ea5c65d170b1d573c2b0b6695211f01875aed987a997e5f8d60","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:38.646905Z","signature_b64":"zr/qa1OfPHZDAKQVl4hTDHZUvQUP/0pfdy4RSfzczq2swzykv1FqJdojWgUTI/FcnCNDslVBq/c5/IsCO1CsAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b2e3c23b5145ea5c65d170b1d573c2b0b6695211f01875aed987a997e5f8d60","last_reissued_at":"2026-05-18T02:55:38.646345Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:38.646345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.6921","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/Te9PqwFAOdMhhUAiULn89NXKCPY5fne/bD/ImMkUQi+qhF4AUqPtqdsiiVmdxo/UfYbI5M0F72MU22s0VytAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T19:48:29.274512Z"},"content_sha256":"249db0ab21bed1d047be1227221672af78451552b3e13f48924514d97fa47984","schema_version":"1.0","event_id":"sha256:249db0ab21bed1d047be1227221672af78451552b3e13f48924514d97fa47984"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:DMXDYI5VCRPKLRS5C4FR2VZ4FM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Comparing rankings by means of competitivity graphs: structural properties and computation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Esther Garcia, Francisco Pedroche, Miguel Romance, Regino Criado","submitted_at":"2013-10-25T13:56:36Z","abstract_excerpt":"In this paper we introduce a new technique to analyze families of rankings focused on the study of structural properties of a new type of graphs. Given a finite number of elements and a family of rankings of those elements, we say that two elements compete when they exchange their relative positions in at least two rankings. This allows us to define an undirected graph by connecting elements that compete. We call this graph a competitivity graph. We study the relationship of competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6921","kind":"arxiv","version":3},"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":"2026-05-18T02:55:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8++Q7WCqAmkx/jezbeQJ8VWKUF8Ae7rXL/+7a3MvkHAwXSVSjpnNddMseMxn3nv8OYDZ+P5X+eMIDlAxX39+CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T19:48:29.275225Z"},"content_sha256":"234d4cc1a1118274dbbd848add1471b53b468f21d9d75a41e38cd78fcfcf9a0e","schema_version":"1.0","event_id":"sha256:234d4cc1a1118274dbbd848add1471b53b468f21d9d75a41e38cd78fcfcf9a0e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DMXDYI5VCRPKLRS5C4FR2VZ4FM/bundle.json","state_url":"https://pith.science/pith/DMXDYI5VCRPKLRS5C4FR2VZ4FM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DMXDYI5VCRPKLRS5C4FR2VZ4FM/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":"2026-05-30T19:48:29Z","links":{"resolver":"https://pith.science/pith/DMXDYI5VCRPKLRS5C4FR2VZ4FM","bundle":"https://pith.science/pith/DMXDYI5VCRPKLRS5C4FR2VZ4FM/bundle.json","state":"https://pith.science/pith/DMXDYI5VCRPKLRS5C4FR2VZ4FM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DMXDYI5VCRPKLRS5C4FR2VZ4FM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:DMXDYI5VCRPKLRS5C4FR2VZ4FM","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":"6847face5b1121d8241319be6ecbd88f85c880582a9a55f9c41d26e6e9cbee36","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-25T13:56:36Z","title_canon_sha256":"d38bbc5e26ba378eee7bf7b198007364f5117100b5d4cdf46820004b22147b2b"},"schema_version":"1.0","source":{"id":"1310.6921","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.6921","created_at":"2026-05-18T02:55:38Z"},{"alias_kind":"arxiv_version","alias_value":"1310.6921v3","created_at":"2026-05-18T02:55:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.6921","created_at":"2026-05-18T02:55:38Z"},{"alias_kind":"pith_short_12","alias_value":"DMXDYI5VCRPK","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DMXDYI5VCRPKLRS5","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DMXDYI5V","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:234d4cc1a1118274dbbd848add1471b53b468f21d9d75a41e38cd78fcfcf9a0e","target":"graph","created_at":"2026-05-18T02:55:38Z","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 this paper we introduce a new technique to analyze families of rankings focused on the study of structural properties of a new type of graphs. Given a finite number of elements and a family of rankings of those elements, we say that two elements compete when they exchange their relative positions in at least two rankings. This allows us to define an undirected graph by connecting elements that compete. We call this graph a competitivity graph. We study the relationship of competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chor","authors_text":"Esther Garcia, Francisco Pedroche, Miguel Romance, Regino Criado","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-25T13:56:36Z","title":"Comparing rankings by means of competitivity graphs: structural properties and computation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6921","kind":"arxiv","version":3},"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":"sha256:249db0ab21bed1d047be1227221672af78451552b3e13f48924514d97fa47984","target":"record","created_at":"2026-05-18T02:55:38Z","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":"6847face5b1121d8241319be6ecbd88f85c880582a9a55f9c41d26e6e9cbee36","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-25T13:56:36Z","title_canon_sha256":"d38bbc5e26ba378eee7bf7b198007364f5117100b5d4cdf46820004b22147b2b"},"schema_version":"1.0","source":{"id":"1310.6921","kind":"arxiv","version":3}},"canonical_sha256":"1b2e3c23b5145ea5c65d170b1d573c2b0b6695211f01875aed987a997e5f8d60","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b2e3c23b5145ea5c65d170b1d573c2b0b6695211f01875aed987a997e5f8d60","first_computed_at":"2026-05-18T02:55:38.646345Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:38.646345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zr/qa1OfPHZDAKQVl4hTDHZUvQUP/0pfdy4RSfzczq2swzykv1FqJdojWgUTI/FcnCNDslVBq/c5/IsCO1CsAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:38.646905Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.6921","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:249db0ab21bed1d047be1227221672af78451552b3e13f48924514d97fa47984","sha256:234d4cc1a1118274dbbd848add1471b53b468f21d9d75a41e38cd78fcfcf9a0e"],"state_sha256":"58b2e75990b699d5c925d7c2675a4037c75f37d0606700cfde37f2de98919575"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UHmEttfZLYcPvp7UsHrmvqE+09KzEUHvrndL2IzyuRBNoY3Xs6OXMSbQ7cIM9ZZasnLxBp8o7xAVFenT3KIhCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T19:48:29.278946Z","bundle_sha256":"0eae5a15e85a53509a0036f618d86de08ad6651e0b137538df6a439ab7f1f890"}}