A Finite-Size Crossover in OllivierRicci Curvature of Random Regular Graphs

A Finite-Size Crossover in OllivierRicci Curvature of Random Regular Graphs | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 16 March 2026 V1 Latest version Share on A Finite-Size Crossover in OllivierRicci Curvature of Random Regular Graphs Authors : Demetrios Chiuratto Agourakis 0009-0001-8671-8878 [email protected] and Marli Gerenutti Authors Info & Affiliations https://doi.org/10.22541/au.177368669.93268105/v1 86 views 76 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We study the sign of mean OllivierRicci curvature (ORC) on random k-regular graphs as a function of the density parameter η = k 2 /N. Exact linear programming computation of the Wasserstein-1 distance across N ∈ {50, 100, 200, 500, 1000} reveals a nite-size crossover at η c (N) = 3.75 − 14.62/ √ N (bootstrap 95% CI: η ∞ c ∈ [3.61, 3.84]; R 2 = 0.995). The transition slope scales as N −0.20, indicating a smooth crossover rather than a sharp phase transition. Erd®sRényi and BarabásiAlbert graphs exhibit the same phenomenon at shifted η c, with the ordering η c (BA) < η c (ER) < η c (reg) explained by degree heterogeneity. A semi-analytical framework based on the Hehl matching formula reproduces the crossover with MAE = 0.009 and predicts a clustering-dependent phase boundary η c (N, C). Sphere-embedded ORC on the CayleyDickson tower (S 3 through S 127) saturates at a positive asymptote κ∞ > 0 with power-law exponent β = 0.283 ± 0.034, below the Johnson Lindenstrauss bound. A Lean 4 formalization provides machine-checked proofs of regime exclusivity (η < η c ⇒ κ η c ⇒ κ > 0). We validate the crossover on 6 real-world semantic networks spanning 4 languages, achieving 6/6 correct phase classication. Supplementary Material File (paper1_theory.pdf) Download 691.33 KB Information & Authors Information Version history V1 Version 1 16 March 2026 Copyright This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Keywords formal verication nite-size scaling ollivierricci curvature phase transition random regular graphs wasserstein distance Authors Affiliations Demetrios Chiuratto Agourakis 0009-0001-8671-8878 [email protected] Faculdade São Leopoldo Mandic Biomaterials and Regenerative Medicine Post-Graduate Program, Pontifícia Universidade Católica de São Paulo (PUC-SP), Sorocaba, SP, Brazil View all articles by this author Marli Gerenutti Biomaterials and Regenerative Medicine Post-Graduate Program, Pontifícia Universidade Católica de São Paulo (PUC-SP), Sorocaba, SP, Brazil View all articles by this author Metrics & Citations Metrics Article Usage 86 views 76 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Demetrios Chiuratto Agourakis, Marli Gerenutti. A Finite-Size Crossover in OllivierRicci Curvature of Random Regular Graphs. Authorea . 16 March 2026. DOI: https://doi.org/10.22541/au.177368669.93268105/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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