{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T14:30:48Z","timestamp":1786113048030,"version":"build-2736575974"},"reference-count":57,"publisher":"National Academy of Sciences","issue":"32","license":[{"start":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T00:00:00Z","timestamp":1785888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,8,11]]},"abstract":"\n Graphs provide a natural framework to describe processes shaped by pairwise interactions among agents. But many dynamical systems involve interactions within groups of three or more agents. Here, we develop the \u201c\n \n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n -hyperedge approximation,\u201d an analytical framework for stochastic processes on regular hypergraphs, in which each individual belongs to\n \n \n k<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n groups of size\n \n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n . The framework accommodates both higher-order interactions that determine payoffs and higher-order processes for updating states in response to payoffs. For evolutionary games on hypergraphs, our analysis generalizes the classical\n \n \n \n b<\/mml:mi>\n \/<\/mml:mo>\n c<\/mml:mi>\n ><\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula>\n rule for cooperation to the\n \n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n -player donation game; and it yields critical benefit-to-cost ratios for the nonlinear\n \n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula>\n -player public goods game, which remains bounded as the degree grows. Applied to neutral complex contagions, where inheritance of states occurs within hyperedges rather than along parent\u2013offspring edges, the framework gives a closed-form fixation probability, showing how a single complexity parameter governs the spread of rare types. Coupling the two processes produces a unified stochastic model of payoff-biased complex contagions in structured populations. Together, these results extend pair approximation from graphs to hypergraphs, accommodating multiway interactions and group-level inheritance with no pairwise analog.\n <\/jats:p>","DOI":"10.1073\/pnas.2619078123","type":"journal-article","created":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T17:36:00Z","timestamp":1785951360000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Hyperedge approximation for stochastic processes on higher-order networks"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6044-707X","authenticated-orcid":false,"given":"Anzhi","family":"Sheng","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/026vcq606","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Decision and Control Systems, KTH Royal Institute of Technology","place":["Stockholm, Sweden"]},{"id":[{"id":"https:\/\/ror.org\/026vcq606","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Mathematics, KTH Royal Institute of Technology","place":["Stockholm, Sweden"]},{"id":[{"id":"https:\/\/ror.org\/00b30xv10","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Biology, University of Pennsylvania","place":["Philadelphia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9110-4635","authenticated-orcid":false,"given":"Alex","family":"McAvoy","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0130frc33","id-type":"ROR","asserted-by":"publisher"}],"name":"School of Data Science and Society, University of North Carolina at Chapel Hill","place":["Chapel Hill"]},{"id":[{"id":"https:\/\/ror.org\/0130frc33","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Mathematics, University of North Carolina at Chapel Hill","place":["Chapel Hill"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4965-2459","authenticated-orcid":false,"given":"Ye","family":"Tian","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/026vcq606","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Decision and Control Systems, KTH Royal Institute of Technology","place":["Stockholm, Sweden"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Silun","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/026vcq606","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Mathematics, KTH Royal Institute of Technology","place":["Stockholm, Sweden"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6367-6302","authenticated-orcid":false,"given":"Angela","family":"Fontan","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/026vcq606","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Decision and Control Systems, KTH Royal Institute of Technology","place":["Stockholm, Sweden"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2349-6304","authenticated-orcid":false,"given":"Joshua B.","family":"Plotkin","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00b30xv10","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Biology, University of Pennsylvania","place":["Philadelphia"]},{"id":[{"id":"https:\/\/ror.org\/00b30xv10","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Mathematical Biology, University of Pennsylvania","place":["Philadelphia"]},{"id":[{"id":"https:\/\/ror.org\/02rqvrp93","id-type":"ROR","asserted-by":"publisher"}],"name":"Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University","place":["Nakano, Tokyo, Japan"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,8,5]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.74.47"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1137\/S003614450342480"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0400087101"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1038\/nphys2761"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.66.016104"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature04605"},{"key":"e_1_3_4_7_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2113468118"},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1006\/tpbi.1994.1032"},{"key":"e_1_3_4_9_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-019-0459-y"},{"key":"e_1_3_4_10_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-021-01371-4"},{"key":"e_1_3_4_11_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature23273"},{"key":"e_1_3_4_12_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature22898"},{"key":"e_1_3_4_13_2","doi-asserted-by":"crossref","unstructured":"A. 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