{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T17:00:33Z","timestamp":1790874033805,"version":"4.1.0"},"reference-count":27,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T00:00:00Z","timestamp":1770854400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007767","name":"Max-Planck-Institut f\u00fcr Physik Komplexer Systeme","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007767","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. E"],"accepted":{"date-parts":[[2026,1,28]]},"abstract":"<jats:p>\n                    Triadic percolation turns bond percolation into a dynamical problem governed by an effective one-dimensional unimodal map. We show that the geometry of superstable cycles provides a direct, map-agnostic probe of local nonlinearity: specifically, the distance from the map's maximum to a distinguished next-to-maximum point on the attracting\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <a:msup>\n                        <a:mn>2<\/a:mn>\n                        <a:mi>n<\/a:mi>\n                      <\/a:msup>\n                    <\/a:math>\n                    cycle (which coincides with a preimage of the maximum at\n                    <b:math xmlns:b=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <b:msup>\n                        <b:mn>2<\/b:mn>\n                        <b:mi>n<\/b:mi>\n                      <\/b:msup>\n                    <\/b:math>\n                    superstability) scales as\n                    <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <c:msup>\n                        <c:mrow>\n                          <c:mo>|<\/c:mo>\n                          <c:mi mathvariant=\"normal\">\u0394<\/c:mi>\n                          <c:mi>p<\/c:mi>\n                          <c:mo>|<\/c:mo>\n                        <\/c:mrow>\n                        <c:mi>\u03b3<\/c:mi>\n                      <\/c:msup>\n                    <\/c:math>\n                    , with\n                    <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <e:mrow>\n                        <e:mi>\u03b3<\/e:mi>\n                        <e:mo>=<\/e:mo>\n                        <e:mn>1<\/e:mn>\n                        <e:mo>\/<\/e:mo>\n                        <e:mi>z<\/e:mi>\n                      <\/e:mrow>\n                    <\/e:math>\n                    , where\n                    <f:math xmlns:f=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <f:mi>z<\/f:mi>\n                    <\/f:math>\n                    is the nonflat order of the maximum. This prediction is verified across canonical unimodal families and heterogeneous triadic ensembles, with Lyapunov spectra corroborating the one-dimensional reduction. A derivative condition on the activation kernel fixes the local nonlinearity order\n                    <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <g:mi>z<\/g:mi>\n                    <\/g:math>\n                    (and thus, under standard unimodal-map hypotheses, the associated\n                    <h:math xmlns:h=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <h:mi>z<\/h:mi>\n                    <\/h:math>\n                    -logistic universality class) and gives conditions under which\n                    <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <i:mrow>\n                        <i:mi>z<\/i:mi>\n                        <i:mo>&gt;<\/i:mo>\n                        <i:mn>2<\/i:mn>\n                      <\/i:mrow>\n                    <\/i:math>\n                    can be realized. The diagnostic operates directly on orbit data under standard regularity assumptions, providing a practical tool to classify universality in higher-order networks.\n                  <\/jats:p>","DOI":"10.1103\/b29t-62kv","type":"journal-article","created":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T21:26:45Z","timestamp":1769635605000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Superstable geometry in triadic percolation"],"prefix":"10.1103","volume":"113","author":[{"given":"Fatemeh","family":"Aghaei","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01bf9rw71","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Physics of Complex Systems"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8864-4748","authenticated-orcid":true,"given":"Abbas Ali","family":"Saberi","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02yrs2n53","id-type":"ROR","asserted-by":"publisher"}],"name":"Constructor University"},{"id":[{"id":"https:\/\/ror.org\/01bf9rw71","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Physics of Complex Systems"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6921-6094","authenticated-orcid":true,"given":"Holger","family":"Kantz","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01bf9rw71","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Physics of Complex Systems"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5926-4276","authenticated-orcid":true,"given":"J\u00fcrgen","family":"Kurths","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03e8s1d88","id-type":"ROR","asserted-by":"publisher"}],"name":"Potsdam Institute for Climate Impact Research"},{"id":[{"id":"https:\/\/ror.org\/01hcx6992","id-type":"ROR","asserted-by":"publisher"}],"name":"Humboldt University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,2,12]]},"reference":[{"key":"b29t-62kvCc1R1","volume-title":"Introduction to Percolation Theory","author":"D. Stauffer","year":"1994","type":"book"},{"key":"b29t-62kvCc2R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2015.03.003","type":"journal-article"},{"key":"b29t-62kvCc3R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.64.026118","type":"journal-article"},{"key":"b29t-62kvCc4R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-37019-5","type":"journal-article"},{"key":"b29t-62kvCc5R1","doi-asserted-by":"publisher","DOI":"10.1093\/pnasnexus\/pgae270","type":"journal-article"},{"key":"b29t-62kvCc6R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.110.064315","type":"journal-article"},{"key":"b29t-62kvCc7R1","doi-asserted-by":"publisher","DOI":"10.1103\/yvtg-wnn4","type":"journal-article"},{"key":"b29t-62kvCc8R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2020.05.004","type":"journal-article"},{"key":"b29t-62kvCc9R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-021-01371-4","type":"journal-article"},{"key":"b29t-62kvCc10R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2023.04.002","type":"journal-article"},{"key":"b29t-62kvCc11R1","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms12285","type":"journal-article"},{"key":"b29t-62kvCc12R1","doi-asserted-by":"publisher","DOI":"10.1038\/nature23273","type":"journal-article"},{"key":"b29t-62kvCc13R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-025-66577-z","type":"journal-article"},{"key":"b29t-62kvCc14R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01020332","type":"journal-article"},{"key":"b29t-62kvCc15R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01107909","type":"journal-article"},{"key":"b29t-62kvCc16R1","volume-title":"Dynamical Systems (College Park, MD, 1986\u201387)","author":"J. Milnor","year":"1988","type":"book"},{"key":"b29t-62kvCc17R1","volume-title":"Iterated Maps on the Interval as Dynamical Systems","author":"P. Collet","year":"1980","type":"book"},{"key":"b29t-62kvCc18R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-78043-1","type":"book","volume-title":"One-Dimensional Dynamics","author":"W. de Melo","year":"1993"},{"key":"b29t-62kvCc19R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1922831118","type":"journal-article"},{"key":"b29t-62kvCc20R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.ado8049","type":"journal-article"},{"key":"b29t-62kvCc21R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.134.037401","type":"journal-article"},{"key":"b29t-62kvCc22R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.adt2404","type":"journal-article"},{"key":"b29t-62kvCc23R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.66.016128","type":"journal-article"},{"key":"b29t-62kvCc24R1","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(94)90991-1","type":"journal-article"},{"key":"b29t-62kvCc25R1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(93)90009-P","type":"journal-article"},{"key":"b29t-62kvCc26R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02128236","type":"journal-article"},{"key":"b29t-62kvCc27R1","unstructured":"https:\/\/www.pks.mpg.de\/asg2024.","type":"other"}],"container-title":["Physical Review E"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/b29t-62kv","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/b29t-62kv\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T16:10:56Z","timestamp":1790871056000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/b29t-62kv"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,12]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2]]}},"URL":"https:\/\/doi.org\/10.1103\/b29t-62kv","relation":{},"ISSN":["2470-0045","2470-0053"],"issn-type":[{"value":"2470-0045","type":"print"},{"value":"2470-0053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,12]]},"assertion":[{"value":"2025-11-18","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"Published by the American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2026","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92026 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"024306"}}