{"data":{"abstract":{"value":"<p>Turing patterns, arising from the interplay between competing species of diffusive particles, have long been an important concept for describing nonequilibrium self-organization in nature and have been extensively investigated in many chemical and biological systems. Historically, these patterns have been studied in extended systems and lattices. Recently, the Turing instability was found to produce topological patterns in networks with scale-free degree distributions and the small-world property, although with an apparent absence of geometric organization. While hints of explicitly geometric patterns in simple network models (e.g., Watts-Strogatz) have been found, the question of the exact nature and morphology of geometric Turing patterns in heterogeneous complex networks remained unresolved. In this work, we study the Turing instability in the framework of geometric random graph models, where the network topology is explained using an underlying geometric space. We demonstrate that not only can geometric patterns be observed, their wavelength can also be estimated by studying the eigenvectors of the annealed graph Laplacian. Finally, we show that Turing patterns can be found in geometric embeddings of real networks. These results indicate that there is a profound connection between the function of a network and its hidden geometry, even when the associated dynamical processes are exclusively determined by the network topology.</p>","format":"html"},"articleType":"article","authors":[{"type":"Person","name":"Jasper van der Kolk","firstname":"Jasper","surname":"van der Kolk","affiliationIds":["a1","a2"]},{"type":"Person","name":"Guillermo García-Pérez","firstname":"Guillermo","surname":"García-Pérez","affiliationIds":["a3"]},{"type":"Person","name":"Nikos E. Kouvaris","firstname":"Nikos E.","surname":"Kouvaris","affiliationIds":["a4"]},{"type":"Person","name":"M. Ángeles Serrano","firstname":"M. Ángeles","surname":"Serrano","affiliationIds":["a1","a2","a5"]},{"type":"Person","name":"Marián Boguñá","firstname":"Marián","surname":"Boguñá","affiliationIds":["a1","a2"]}],"affiliations":[{"name":"Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain","id":"a1"},{"name":"Universitat de Barcelona Institute of Complex Systems (UBICS), Martí i Franquès 1, E-08028 Barcelona, Spain","id":"a2"},{"name":"Algorithmiq Ltd, Kanavakatu 3C, FI-00160 Helsinki, Finland","id":"a3"},{"name":"Dribia Data Research Sociedad Limitada, Carrer Llacuna 162, 08018 Barcelona, Spain","id":"a4"},{"name":"Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís Companys 23, E-08010 Barcelona, Spain","id":"a5"}],"notes":[{"format":"html","value":"<p>jasper.vanderkolk@ub.edu</p>","label":"*","id":"n1","type":"contrib"},{"format":"html","value":"<p>marian.serrano@ub.edu</p>","label":"†","id":"n2","type":"contrib"},{"format":"html","value":"<p>marian.boguna@ub.edu</p>","label":"‡","id":"n3","type":"contrib"}],"date":"2023-06-22","fundings":[{"funderId":"http://dx.doi.org/10.13039/501100011033","funderName":"Agencia Estatal de Investigación","awards":["PID2019–106290 GB-C22"]},{"funderId":"http://dx.doi.org/10.13039/501100002809","funderName":"Generalitat de Catalunya","awards":["2021SGR00856"]},{"funderId":"http://dx.doi.org/10.13039/501100023561","funderName":"Ministerio de Universidades","awards":[]}],"type":"article","metadata_last_modified_at":"2023-06-22T14:20:38+0000","last_modified_at":"2023-06-22T14:20:38+0000","id":"10.1103/PhysRevX.13.021038","identifiers":{"doi":"10.1103/PhysRevX.13.021038","arxiv":"arXiv:2211.11311"},"issue":{"number":"2"},"pageStart":"021038","hasArticleId":true,"numPages":18,"classificationSchemes":{"physh":{"concepts":[{"id":"c485f44d-2cc4-4e6d-a20b-0899fb9e53a1","facet":{"id":"f45b3c40-959c-4e90-ba0e-38232980802a"},"primary":false},{"id":"1c4f20f1-446b-4875-abc0-744966470931","facet":{"id":"f45b3c40-959c-4e90-ba0e-38232980802a"},"primary":false},{"id":"b4d51608-61b8-4e73-8186-ba02e5a41a85","facet":{"id":"f45b3c40-959c-4e90-ba0e-38232980802a"},"primary":false},{"id":"0dc06882-40b4-4da9-b489-9706a97e3e25","facet":{"id":"f45b3c40-959c-4e90-ba0e-38232980802a"},"primary":false},{"id":"82d0ef36-63e0-4a7c-8dc3-a5292d2d338e","facet":{"id":"f45b3c40-959c-4e90-ba0e-38232980802a"},"primary":false},{"id":"21802c03-692c-4093-a35c-11955f447cd4","facet":{"id":"f45b3c40-959c-4e90-ba0e-38232980802a"},"primary":false},{"id":"8e35265c-459a-43fc-9eb8-73e1dd82b47c","facet":{"id":"b96dac97-ab85-4320-892d-9b245caf097f"},"primary":true}],"disciplines":["c9174048-b368-4b5b-b065-20094708fa4d","e4211828-5d99-4bc4-b986-eb2bbbc8b248","6f025065-4b96-4156-b146-2c285d0994c4"]}},"publisher":{"name":"APS"},"rights":{"rightsStatement":"Published by the American Physical Society","copyrightYear":2023,"copyrightHolders":[],"creativeCommons":true,"licenses":[{"url":"https://creativecommons.org/licenses/by/4.0/","licenseStatement":"Published by the American Physical Society under the terms of the <a href=\"https://creativecommons.org/licenses/by/4.0/\">Creative Commons Attribution 4.0 International</a> license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI."}]},"journal":{"id":"PRX","abbreviatedName":"Phys. Rev. X","name":"Physical Review X"},"title":{"value":"Emergence of Geometric Turing Patterns in Complex Networks","format":"html"},"volume":{"number":"13"}}}