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Bakry-\'Emery curvature sharpness and curvature flow in finite weighted graphs. I. Theory

David CushingSupanat KamtueShiping LiuFlorentin M\"unchNorbert PeyerimhoffHugo Benedict Snodgrass
Apr 2022
摘要
In this sequence of two papers, we introduce a curvature flow on (mixed)weighted graphs which is based on the Bakry-\'Emery calculus. The flow isdescribed via a time-continuous evolution through the weighting schemes. Byadapting this flow to preserve the Markovian property, its limits turn out tobe curvature sharp. Our aim is to present the flow in the most general case ofnot necessarily reversible random walks allowing laziness, including vanishingtransition probabilities along some edges ("degenerate" edges). This approachrequires to extend all concepts (in particular, the Bakry-\'Emery curvaturerelated notions) to this general case and it leads to a distinction between theunderlying topology (a mixed combinatorial graph) and the weighting scheme(given by transition rates). We present various results about curvature sharpvertices and weighted graphs as well as some fundamental properties of this newcurvature flow. This paper is accompanied by a second paper discussing thecurvature flow implementation in Python for practical use. In this second paperwe present examples and exhibit further properties of the flow.
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