{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T05:36:50Z","timestamp":1776317810501,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,11]],"date-time":"2023-06-11T00:00:00Z","timestamp":1686441600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key R and D Program of China","award":["2020YFA0713100"],"award-info":[{"award-number":["2020YFA0713100"]}]},{"name":"National Key R and D Program of China","award":["12031017"],"award-info":[{"award-number":["12031017"]}]},{"name":"National Key R and D Program of China","award":["URB-2021-02"],"award-info":[{"award-number":["URB-2021-02"]}]},{"name":"National Key R and D Program of China","award":["RPG-2021-080"],"award-info":[{"award-number":["RPG-2021-080"]}]},{"name":"National Key R and D Program of China","award":["2022660219"],"award-info":[{"award-number":["2022660219"]}]},{"name":"National Natural Science Foundation of China","award":["2020YFA0713100"],"award-info":[{"award-number":["2020YFA0713100"]}]},{"name":"National Natural Science Foundation of China","award":["12031017"],"award-info":[{"award-number":["12031017"]}]},{"name":"National Natural Science Foundation of China","award":["URB-2021-02"],"award-info":[{"award-number":["URB-2021-02"]}]},{"name":"National Natural Science Foundation of China","award":["RPG-2021-080"],"award-info":[{"award-number":["RPG-2021-080"]}]},{"name":"National Natural Science Foundation of China","award":["2022660219"],"award-info":[{"award-number":["2022660219"]}]},{"name":"London Mathematical Society","award":["2020YFA0713100"],"award-info":[{"award-number":["2020YFA0713100"]}]},{"name":"London Mathematical Society","award":["12031017"],"award-info":[{"award-number":["12031017"]}]},{"name":"London Mathematical Society","award":["URB-2021-02"],"award-info":[{"award-number":["URB-2021-02"]}]},{"name":"London Mathematical Society","award":["RPG-2021-080"],"award-info":[{"award-number":["RPG-2021-080"]}]},{"name":"London Mathematical Society","award":["2022660219"],"award-info":[{"award-number":["2022660219"]}]},{"name":"Leverhulme Trust Research Project","award":["2020YFA0713100"],"award-info":[{"award-number":["2020YFA0713100"]}]},{"name":"Leverhulme Trust Research Project","award":["12031017"],"award-info":[{"award-number":["12031017"]}]},{"name":"Leverhulme Trust Research Project","award":["URB-2021-02"],"award-info":[{"award-number":["URB-2021-02"]}]},{"name":"Leverhulme Trust Research Project","award":["RPG-2021-080"],"award-info":[{"award-number":["RPG-2021-080"]}]},{"name":"Leverhulme Trust Research Project","award":["2022660219"],"award-info":[{"award-number":["2022660219"]}]},{"name":"Shuimu Scholar Program of Tsinghua University","award":["2020YFA0713100"],"award-info":[{"award-number":["2020YFA0713100"]}]},{"name":"Shuimu Scholar Program of Tsinghua University","award":["12031017"],"award-info":[{"award-number":["12031017"]}]},{"name":"Shuimu Scholar Program of Tsinghua University","award":["URB-2021-02"],"award-info":[{"award-number":["URB-2021-02"]}]},{"name":"Shuimu Scholar Program of Tsinghua University","award":["RPG-2021-080"],"award-info":[{"award-number":["RPG-2021-080"]}]},{"name":"Shuimu Scholar Program of Tsinghua University","award":["2022660219"],"award-info":[{"award-number":["2022660219"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry\u2013\u00c9mery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the corresponding paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, and hypercubes) and exhibit various properties of this flow. One particular aspect of our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the Python functions and routines freely available in an ancillary file on arXiv or via github. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.<\/jats:p>","DOI":"10.3390\/axioms12060577","type":"journal-article","created":{"date-parts":[[2023,6,12]],"date-time":"2023-06-12T02:28:42Z","timestamp":1686536922000},"page":"577","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Bakry\u2013\u00c9mery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5488-2705","authenticated-orcid":false,"given":"David","family":"Cushing","sequence":"first","affiliation":[{"name":"Department of Mathematics, The University of Manchester, Manchester M13 9PL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Supanat","family":"Kamtue","sequence":"additional","affiliation":[{"name":"Yau Mathematical Sciences Center, Tsinghua University, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9987-1935","authenticated-orcid":false,"given":"Shiping","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6123-9507","authenticated-orcid":false,"given":"Florentin","family":"M\u00fcnch","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9630-7901","authenticated-orcid":false,"given":"Norbert","family":"Peyerimhoff","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ben","family":"Snodgrass","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cushing, D., Kamtue, S., Liu, S., M\u00fcnch, F., Peyerimhoff, N., and Snodgrass, H.B. 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