{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,3]],"date-time":"2026-09-03T05:21:52Z","timestamp":1788412912285,"version":"build-2803163510"},"reference-count":44,"publisher":"National Academy of Sciences","issue":"29","license":[{"start":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:00:00Z","timestamp":1783900800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000181","name":"DOD | AF | AMC | AFRL | Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-24-1-003"],"award-info":[{"award-number":["FA9550-24-1-003"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,7,21]]},"abstract":"\n Cities must allocate limited resources to maintain mobility, with uncertainties about the resulting state of the system. Analyzing roughly 3,000 bus routes with more than 4 billion yearly riders across 19 metropolitan areas worldwide, we uncover a robust scaling law of the form\n \n \n \n f<\/mml:mi>\n \u223c<\/mml:mo>\n \n \n (<\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u03b1<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula>\n with exponent\n \n \n \n \u03b1<\/mml:mi>\n \u2208<\/mml:mo>\n [<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n 2<\/mml:mn>\n \/<\/mml:mo>\n 3<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula>\n , linking the service frequency\n \n \n f<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n to passenger demand\n \n \n d<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n and route duration\n \n \n t<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n . We show that this scaling emerges from a simple optimization principle: Cities implicitly minimize total passenger waiting time under a fixed operational budget when both schedule frequency and crowding are taken into account. This mechanism produces two universal regimes: a frequency-dominated regime with\n \n \n \n \u03b1<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula>\n when crowding is negligible and a capacity-dominated regime with\n \n \n \n \u03b1<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n \/<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula>\n when most routes are overloaded. Intermediate exponents arise when only part of the network operates near capacity. Furthermore, we find that the benefits of additional investment are highly uneven across systems. For instance, our model suggests that a\n \n \n \n 20<\/mml:mn>\n %<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula>\n budget increase yields nearly a 5-min reduction in daily waiting time per passenger in Boston, compared to only about 1 min in Paris. These findings place urban transit within a broader class of constrained capacity-allocation problems, while highlighting a distinct regime in which prescribed route demands shape the allocation of limited service resources. The resulting scaling laws show how simple optimization principles can generate systematic exponents in complex transport systems, beyond the dissipation-based frameworks usually considered in physical and biological flow networks.\n <\/jats:p>","DOI":"10.1073\/pnas.2535998123","type":"journal-article","created":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T19:01:17Z","timestamp":1783969277000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":1,"title":["Crowding controls the scaling of bus frequency with demand"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7355-355X","authenticated-orcid":false,"given":"Siddharth","family":"Patwardhan","sequence":"first","affiliation":[{"name":"Center for Science of Science and Innovation, Kellogg School of Management, Northwestern University","place":["Evanston"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6419-2756","authenticated-orcid":false,"given":"\u015eirag","family":"Erkol","sequence":"additional","affiliation":[{"name":"Center for Science of Science and Innovation, Kellogg School of Management, Northwestern University","place":["Evanston"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8352-1287","authenticated-orcid":false,"given":"Filippo","family":"Radicchi","sequence":"additional","affiliation":[{"name":"Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University","place":["Bloomington"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0890-9713","authenticated-orcid":false,"given":"Marc","family":"Barthelemy","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00jjx8s55","id-type":"ROR","asserted-by":"publisher"}],"name":"Institut de Physique Th\u00e9orique, Universit\u00e9 Paris-Saclay, Commissariat \u00e0 l\u2019\u00e9nergie atomique et aux \u00e9nergies alternatives, CNRS","place":["Gif-sur-Yvette, France"]},{"id":[{"id":"https:\/\/ror.org\/02d9dg697","id-type":"ROR","asserted-by":"publisher"}],"name":"Centre d\u2019Analyse et de Math\u00e9matique Sociales (CNRS\/\u00c9cole des Hautes \u00c9tudes en Sciences Sociales)","place":["Paris, France"]},{"id":[{"id":"https:\/\/ror.org\/023dz9m50","id-type":"ROR","asserted-by":"publisher"}],"name":"Complexity Science Hub","place":["Vienna, Austria"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,7,13]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejtl.2025.100160"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-021-02418-5"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1177\/0361198118822292"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0161738"},{"key":"e_1_3_4_5_2","first-page":"591","article-title":"Optimization and scale economies in urban bus transportation","volume":"62","author":"Mohring H.","year":"1972","unstructured":"H. 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