{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T02:01:12Z","timestamp":1774576872745,"version":"3.50.1"},"reference-count":25,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T00:00:00Z","timestamp":1773100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"name":"National Natural Science Foundation of People\u2019s Republic of China","award":["62366049"],"award-info":[{"award-number":["62366049"]}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["2025D01C32"],"award-info":[{"award-number":["2025D01C32"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,3,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Addressing the overlooked stochastic, language-specific interactions on heterogeneous networks, we put forward a multilingual stochastic IE2SR rumor model with variable contact rates. We first develop a deterministic IE2SR model and prove the existence and uniqueness of its equilibrium. Subsequently, we extend it to a stochastic version, deriving explicit conditions for rumor existence and extinction using the strong law of large numbers. Numerical simulations confirm our theoretical findings and reveal that both contact rates and stochastic noise significantly impact rumor thresholds, offering practical insights for rumor control in multilingual environments.<\/jats:p>","DOI":"10.1093\/comnet\/cnag003","type":"journal-article","created":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T01:40:37Z","timestamp":1774575637000},"source":"Crossref","is-referenced-by-count":0,"title":["Stochastic IE2SR for multilingual rumor dynamics: varying contact rates on heterogeneous networks"],"prefix":"10.1093","volume":"14","author":[{"given":"Liuqin","family":"Huang","sequence":"first","affiliation":[{"name":"School of Science, Jimei University, No. 185, Yinjiang Road, Jimei District , Xiamen 361021,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3224-5558","authenticated-orcid":false,"given":"Jinling","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Jimei University, No. 185, Yinjiang Road, Jimei District , Xiamen 361021,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiarong","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, No. 666, Shengli Road, Tianshan District , Urumqi 830046,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haijun","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, No. 666, Shengli Road, Tianshan District , Urumqi 830046,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2026,3,26]]},"reference":[{"key":"2026032621403267100_cnag003-B1","doi-asserted-by":"crossref","first-page":"e001691","DOI":"10.47405\/mjssh.v7i8.1691","article-title":"Systematic review of the relationship between trust in government and rumor-related behavior during the COVID-19","volume":"7","author":"Chen","year":"2022","journal-title":"MJSSH"},{"key":"2026032621403267100_cnag003-B2","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1108\/DTA-06-2021-0160","article-title":"A deep neural networks-based fusion model for covid-19 rumor detection from online social media","volume":"56","author":"Lu","year":"2022","journal-title":"DTA"},{"key":"2026032621403267100_cnag003-B3","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1093\/imamat\/1.1.42","article-title":"Stochastic rumours","volume":"1","author":"Daley","year":"1965","journal-title":"IMA J Appl Math"},{"key":"2026032621403267100_cnag003-B4","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.ins.2021.02.047","article-title":"Dynamical behaviors and control measures of rumor-spreading model in consideration of the infected media and time delay","volume":"564","author":"Cheng","year":"2021","journal-title":"Inf Sci"},{"key":"2026032621403267100_cnag003-B5","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1016\/j.physa.2012.09.030","article-title":"Sir rumor spreading model in the new media age","volume":"392","author":"Zhao","year":"2013","journal-title":"Phys A"},{"key":"2026032621403267100_cnag003-B6","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1016\/j.physa.2016.11.067","article-title":"The analysis of an seir rumor propagation model on heterogeneous network","volume":"469","author":"Liu","year":"2017","journal-title":"Phys A"},{"key":"2026032621403267100_cnag003-B7","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1186\/s13662-017-1315-y","article-title":"Global stability of a seir rumor spreading model with demographics on scale-free networks","volume":"2017","author":"Wan","year":"2017","journal-title":"Adv Differ Equ"},{"key":"2026032621403267100_cnag003-B8","doi-asserted-by":"crossref","first-page":"2444","DOI":"10.1016\/j.physa.2011.12.008","article-title":"Sihr rumor spreading model in social networks","volume":"391","author":"Zhao","year":"2012","journal-title":"Phys A"},{"key":"2026032621403267100_cnag003-B9","doi-asserted-by":"crossref","first-page":"128689","DOI":"10.1016\/j.jmaa.2024.128689","article-title":"Dynamic analysis of a rumor propagation model based on a familiarity mechanism and refuters","volume":"541","author":"Jie","year":"2025","journal-title":"J Math Anal Appl"},{"key":"2026032621403267100_cnag003-B10","doi-asserted-by":"crossref","first-page":"3925","DOI":"10.1007\/s11071-022-07947-w","article-title":"Dynamic analysis of a SIDRW rumor propagation model considering the effect of media reports and rumor refuters","volume":"111","author":"Pan","year":"2023","journal-title":"Nonlinear Dyn"},{"key":"2026032621403267100_cnag003-B11","doi-asserted-by":"crossref","first-page":"120981","DOI":"10.1016\/j.physa.2019.04.217","article-title":"Dynamic behavior of an i2s2r rumor propagation model on weighted contract networks","volume":"536","author":"Zhang","year":"2019","journal-title":"Phys A"},{"key":"2026032621403267100_cnag003-B12","doi-asserted-by":"crossref","first-page":"5068","DOI":"10.3934\/mbe.2024224","article-title":"The dynamics and control of a multi-lingual rumor propagation model with non-smooth inhibition mechanism","volume":"21","author":"Liao","year":"2024","journal-title":"Math Biosci Eng"},{"key":"2026032621403267100_cnag003-B13","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.chaos.2019.05.027","article-title":"Global dynamics of the multi-lingual sir rumor spreading model with cross-transmitted mechanism","volume":"126","author":"Wang","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"2026032621403267100_cnag003-B14","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1186\/s13662-020-03081-2","article-title":"Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment","volume":"2020","author":"Yang","year":"2020","journal-title":"Adv Differ Equ"},{"key":"2026032621403267100_cnag003-B15","first-page":"1","article-title":"Dynamic analysis and optimal control of a novel fractional-order 2i2sr rumor spreading model","volume":"28","author":"Ye","year":"2023","journal-title":"NAMC"},{"key":"2026032621403267100_cnag003-B16","doi-asserted-by":"crossref","first-page":"2933","DOI":"10.1007\/s11071-020-05621-7","article-title":"The spread and control of rumors in a multilingual environment","volume":"100","author":"Yu","year":"2020","journal-title":"Nonlinear Dyn"},{"key":"2026032621403267100_cnag003-B17","doi-asserted-by":"crossref","first-page":"16875","DOI":"10.1007\/s11071-023-08768-1","article-title":"A comparative study of deterministic and stochastic dynamics of rumor propagation model with counter-rumor spreader","volume":"111","author":"Ghosh","year":"2023","journal-title":"Nonlinear Dyn"},{"key":"2026032621403267100_cnag003-B18","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.12677\/AAM.2020.912271","article-title":"Stochastic rumor propagation model under the influence of official media","volume":"9","author":"Hu","year":"2020","journal-title":"Adv Appl Math"},{"key":"2026032621403267100_cnag003-B19","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1016\/j.physa.2017.08.125","article-title":"Dynamic analysis of a stochastic rumor propagation model","volume":"490","author":"Jia","year":"2018","journal-title":"Physica A"},{"key":"2026032621403267100_cnag003-B20","doi-asserted-by":"crossref","first-page":"113637","DOI":"10.1016\/j.chaos.2023.113637","article-title":"Dynamic analysis of the IFCD rumor propagation model under stochastic disturbance on heterogeneous networks","volume":"173","author":"Tong","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"2026032621403267100_cnag003-B21","doi-asserted-by":"crossref","first-page":"031005","DOI":"10.1115\/1.4053269","article-title":"Dynamics of a rumor propagation model with stochastic perturbation on homogeneous social networks","volume":"17","author":"Zhang","year":"2022","journal-title":"J Comput Nonlinear Dyn"},{"key":"2026032621403267100_cnag003-B22","doi-asserted-by":"crossref","volume-title":"Stochastic Differential Equations and Applications","author":"Mao","DOI":"10.1533\/9780857099402"},{"key":"2026032621403267100_cnag003-B23","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/j.ins.2020.05.037","article-title":"Dynamical analysis of rumor spreading model in multi-lingual environment and heterogeneous complex networks","volume":"536","author":"Li","year":"2020","journal-title":"Inf Sci"},{"key":"2026032621403267100_cnag003-B24","doi-asserted-by":"crossref","first-page":"035108","DOI":"10.1103\/PhysRevE.65.035108","article-title":"Epidemic dynamics in finite size scale-free networks","volume":"65","author":"Pastor-Satorras","year":"2002","journal-title":"Phys Rev E Stat Nonlin Soft Matter Phys"},{"key":"2026032621403267100_cnag003-B25","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Van den Driessche","year":"2002","journal-title":"Math Biosci"}],"container-title":["Journal of Complex Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/comnet\/article-pdf\/14\/2\/cnag003\/67558107\/cnag003.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/comnet\/article-pdf\/14\/2\/cnag003\/67558107\/cnag003.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T01:40:40Z","timestamp":1774575640000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/comnet\/article\/doi\/10.1093\/comnet\/cnag003\/8542447"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,10]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3,10]]}},"URL":"https:\/\/doi.org\/10.1093\/comnet\/cnag003","relation":{},"ISSN":["2051-1329"],"issn-type":[{"value":"2051-1329","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,4]]},"published":{"date-parts":[[2026,3,10]]},"article-number":"cnag003"}}