Exact and Asymptotic Results on Coarse Ricci Curvature of Graphs

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dc.contributor.authorBhattacharya, Bhaswar B
dc.contributor.authorMukherjee, Sumit
dc.date2023-05-17T20:25:05.000
dc.date.accessioned2023-05-23T03:37:51Z
dc.date.available2023-05-23T03:37:51Z
dc.date.issued2015-01-06
dc.date.submitted2018-07-19T11:22:49-07:00
dc.description.abstractRicci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we obtain the exact formulas for Ollivier’s Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature that hold for a wide class of graphs, and extend earlier results where the Ricci-curvature for graphs with girth 6 was obtained. We also prove a general lower bound on the Ricci-curvature in terms of the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize the Ricci-flat graphs of girth 5. Moreover, using our general lower bound and the Birkhoff–von Neumann theorem, we give the first necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci-curvature of random bipartite graphs G (n, n, p) and random graphs G (n, p), in various regimes of p.
dc.identifier.urihttps://repository.upenn.edu/handle/20.500.14332/48001
dc.legacy.articleid1671
dc.legacy.fields10.1016/j.disc.2014.08.012
dc.legacy.fulltexturlhttps://repository.upenn.edu/cgi/viewcontent.cgi?article=1671&context=statistics_papers&unstamped=1
dc.rights<p>© 2015. This manuscript version is made available under the CC-BY-NC-ND 4.0 license <a href="http://creativecommons.org/licenses/by-nc-nd/4.0/">http://creativecommons.org/licenses/by-nc-nd/4.0/</a></p>
dc.source.beginpage23
dc.source.endpage42
dc.source.issue598
dc.source.issue1
dc.source.journalStatistics Papers
dc.source.journaltitleDiscrete Mathematics
dc.source.peerreviewedtrue
dc.source.statuspublished
dc.source.volume338
dc.subject.othergraph curvature
dc.subject.otheroptimal transportation
dc.subject.otherrandom graphs
dc.subject.otherWasserstein's distance
dc.subject.otherApplied Mathematics
dc.subject.otherBusiness
dc.subject.otherDiscrete Mathematics and Combinatorics
dc.subject.otherStatistics and Probability
dc.titleExact and Asymptotic Results on Coarse Ricci Curvature of Graphs
dc.typeReport
digcom.contributor.authorBhattacharya, Bhaswar B
digcom.contributor.authorMukherjee, Sumit
digcom.identifierstatistics_papers/598
digcom.identifier.contextkey12512102
digcom.identifier.submissionpathstatistics_papers/598
digcom.typereport
dspace.entity.typePublication
upenn.schoolDepartmentCenterStatistics Papers
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