{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T05:30:49Z","timestamp":1778563849835,"version":"3.51.4"},"reference-count":64,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Ministry of Science, Innovation and Universities","award":["RTI2018-093732-B-C22"],"award-info":[{"award-number":["RTI2018-093732-B-C22"]}]},{"name":"Spanish Ministry of Science, Innovation and Universities","award":["MDM-2017-0711"],"award-info":[{"award-number":["MDM-2017-0711"]}]},{"name":"Spanish Ministry of Science, Innovation and Universities","award":["FA8655-20-1-7020"],"award-info":[{"award-number":["FA8655-20-1-7020"]}]},{"name":"Spanish Ministry of Science, Innovation and Universities","award":["952026"],"award-info":[{"award-number":["952026"]}]},{"name":"Agency AEI and FEDER (EU)","award":["RTI2018-093732-B-C22"],"award-info":[{"award-number":["RTI2018-093732-B-C22"]}]},{"name":"Agency AEI and FEDER (EU)","award":["MDM-2017-0711"],"award-info":[{"award-number":["MDM-2017-0711"]}]},{"name":"Agency AEI and FEDER (EU)","award":["FA8655-20-1-7020"],"award-info":[{"award-number":["FA8655-20-1-7020"]}]},{"name":"Agency AEI and FEDER (EU)","award":["952026"],"award-info":[{"award-number":["952026"]}]},{"name":"Maria de Maeztu program for Units of Excellence in R&amp;D","award":["RTI2018-093732-B-C22"],"award-info":[{"award-number":["RTI2018-093732-B-C22"]}]},{"name":"Maria de Maeztu program for Units of Excellence in R&amp;D","award":["MDM-2017-0711"],"award-info":[{"award-number":["MDM-2017-0711"]}]},{"name":"Maria de Maeztu program for Units of Excellence in R&amp;D","award":["FA8655-20-1-7020"],"award-info":[{"award-number":["FA8655-20-1-7020"]}]},{"name":"Maria de Maeztu program for Units of Excellence in R&amp;D","award":["952026"],"award-info":[{"award-number":["952026"]}]},{"DOI":"10.13039\/100000181","name":"AFOSR","doi-asserted-by":"publisher","award":["RTI2018-093732-B-C22"],"award-info":[{"award-number":["RTI2018-093732-B-C22"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"AFOSR","doi-asserted-by":"publisher","award":["MDM-2017-0711"],"award-info":[{"award-number":["MDM-2017-0711"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"AFOSR","doi-asserted-by":"publisher","award":["FA8655-20-1-7020"],"award-info":[{"award-number":["FA8655-20-1-7020"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"AFOSR","doi-asserted-by":"publisher","award":["952026"],"award-info":[{"award-number":["952026"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"name":"EU H2020 Humane AI-Net","award":["RTI2018-093732-B-C22"],"award-info":[{"award-number":["RTI2018-093732-B-C22"]}]},{"name":"EU H2020 Humane AI-Net","award":["MDM-2017-0711"],"award-info":[{"award-number":["MDM-2017-0711"]}]},{"name":"EU H2020 Humane AI-Net","award":["FA8655-20-1-7020"],"award-info":[{"award-number":["FA8655-20-1-7020"]}]},{"name":"EU H2020 Humane AI-Net","award":["952026"],"award-info":[{"award-number":["952026"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The conventional voter model is modified so that an agent\u2019s switching rate depends on the \u2018age\u2019 of the agent\u2014that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age-dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of a spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.<\/jats:p>","DOI":"10.3390\/e24101331","type":"journal-article","created":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T21:22:23Z","timestamp":1663795343000},"page":"1331","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Analytical and Numerical Treatment of Continuous Ageing in the Voter Model"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2815-3263","authenticated-orcid":false,"given":"Joseph W.","family":"Baron","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5118-5725","authenticated-orcid":false,"given":"Antonio F.","family":"Peralta","sequence":"additional","affiliation":[{"name":"Department of Network and Data Science, Central European University, A-1100 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3402-2163","authenticated-orcid":false,"given":"Tobias","family":"Galla","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2046-7620","authenticated-orcid":false,"given":"Ra\u00fal","family":"Toral","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1103\/RevModPhys.81.591","article-title":"Statistical physics of social dynamics","volume":"81","author":"Castellano","year":"2009","journal-title":"Rev. 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