{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:GTRAMWENZ4SBRIJN5DTVBNDTZQ","short_pith_number":"pith:GTRAMWEN","schema_version":"1.0","canonical_sha256":"34e206588dcf2418a12de8e750b473cc002fdf8ff46f2b9952728060effd9590","source":{"kind":"arxiv","id":"2608.01929","version":1},"attestation_state":"computed","paper":{"title":"Markov and lattice bases for Forman-Ricci curvature of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","physics.soc-ph"],"primary_cat":"math.CO","authors_text":"Giulio Zucal, Jane Ivy Coons","submitted_at":"2026-08-03T09:02:15Z","abstract_excerpt":"Discrete Forman-Ricci curvature is a quantity associated to each edge of a graph that describes its local geometry. It has proven to be a useful tool in network analysis in a variety of applications. Recent work by Roost et al.\\ (2024) proposed the use of Markov bases to sample from the space of graphs with prescribed vertex degrees and curvatures. In the present work, we further develop the algebraic and combinatorial theory of these Markov bases. We show that the degree of an indispensable Markov move grows at least quadratically in the maximum degree of the graph. In light of this result, a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2608.01929","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-03T09:02:15Z","cross_cats_sorted":["cs.DM","physics.soc-ph"],"title_canon_sha256":"e464248d40b918ad9ab5fe8c32054bb34e6d869f692c450c5d589fe6034e5c24","abstract_canon_sha256":"e1578eba71b813ed71a9eb47f1878cf021d6a54dd322ca2e6d9079aab143945e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:07:45.709286Z","signature_b64":"37X3LvcbfyTACNDzbeyaS3a3xfq2KIyJs9tJzSJ1WKfC5ieiHeOApzsKPDC8/KAx4O5WaS2gkRFm0v+2uvLJCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34e206588dcf2418a12de8e750b473cc002fdf8ff46f2b9952728060effd9590","last_reissued_at":"2026-08-04T02:07:45.707825Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:07:45.707825Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Markov and lattice bases for Forman-Ricci curvature of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","physics.soc-ph"],"primary_cat":"math.CO","authors_text":"Giulio Zucal, Jane Ivy Coons","submitted_at":"2026-08-03T09:02:15Z","abstract_excerpt":"Discrete Forman-Ricci curvature is a quantity associated to each edge of a graph that describes its local geometry. It has proven to be a useful tool in network analysis in a variety of applications. Recent work by Roost et al.\\ (2024) proposed the use of Markov bases to sample from the space of graphs with prescribed vertex degrees and curvatures. In the present work, we further develop the algebraic and combinatorial theory of these Markov bases. We show that the degree of an indispensable Markov move grows at least quadratically in the maximum degree of the graph. In light of this result, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01929","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.01929/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.01929","created_at":"2026-08-04T02:07:45.709268+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.01929v1","created_at":"2026-08-04T02:07:45.709268+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01929","created_at":"2026-08-04T02:07:45.709268+00:00"},{"alias_kind":"pith_short_12","alias_value":"GTRAMWENZ4SB","created_at":"2026-08-04T02:07:45.709268+00:00"},{"alias_kind":"pith_short_16","alias_value":"GTRAMWENZ4SBRIJN","created_at":"2026-08-04T02:07:45.709268+00:00"},{"alias_kind":"pith_short_8","alias_value":"GTRAMWEN","created_at":"2026-08-04T02:07:45.709268+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ","json":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ.json","graph_json":"https://pith.science/api/pith-number/GTRAMWENZ4SBRIJN5DTVBNDTZQ/graph.json","events_json":"https://pith.science/api/pith-number/GTRAMWENZ4SBRIJN5DTVBNDTZQ/events.json","paper":"https://pith.science/paper/GTRAMWEN"},"agent_actions":{"view_html":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ","download_json":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ.json","view_paper":"https://pith.science/paper/GTRAMWEN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.01929&json=true","fetch_graph":"https://pith.science/api/pith-number/GTRAMWENZ4SBRIJN5DTVBNDTZQ/graph.json","fetch_events":"https://pith.science/api/pith-number/GTRAMWENZ4SBRIJN5DTVBNDTZQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ/action/storage_attestation","attest_author":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ/action/author_attestation","sign_citation":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ/action/citation_signature","submit_replication":"https://pith.science/pith/GTRAMWENZ4SBRIJN5DTVBNDTZQ/action/replication_record"}},"created_at":"2026-08-04T02:07:45.709268+00:00","updated_at":"2026-08-04T02:07:45.709268+00:00"}