{"paper":{"title":"Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CO"],"primary_cat":"math.DG","authors_text":"Shiping Liu, Yunyan Yang","submitted_at":"2026-08-06T12:06:40Z","abstract_excerpt":"In this note, we prove that, on weighted graphs, the Lin--Lu--Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\\geq 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier--Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin--Lu--Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05939","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/2608.05939/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"}