{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T07:07:26Z","timestamp":1768892846080,"version":"3.49.0"},"reference-count":27,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2015,1,19]],"date-time":"2015-01-19T00:00:00Z","timestamp":1421625600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61374180"],"award-info":[{"award-number":["61374180"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61373136"],"award-info":[{"award-number":["61373136"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Research Foundation for Humanities and Social Sciences of Ministry of Education","award":["12YJAZH120"],"award-info":[{"award-number":["12YJAZH120"]}]},{"name":"Six Projects Sponsoring Talent Summits of Jiangsu Province","award":["RLD201212"],"award-info":[{"award-number":["RLD201212"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we propose a novel two-stage rumor spreading Susceptible-Infected-Authoritative-Removed (SIAR) model for complex homogeneous and heterogeneous networks. The interaction Markov chains (IMC) mean-field equations based on the SIAR model are derived to describe the dynamic interaction between the rumors and authoritative information. We use a Monte Carlo simulation method to characterize the dynamics of the Susceptible-Infected-Removed (SIR) and SIAR models, showing that the SIAR model with consideration of authoritative information gives a more realistic description of propagation features of rumors than the SIR model. The simulation results demonstrate that the critical threshold \u03bbc of the SIAR model has the tiniest increase than the threshold of SIR model. The sooner the authoritative information is introduced, the less negative impact the rumors will bring. We also get the result that heterogeneous networks are more prone to the spreading of rumors. Additionally, the inhibition of rumor spreading, as one of the characteristics of the new SIAR model itself, is instructive for later studies on the rumor spreading models and the controlling strategies.<\/jats:p>","DOI":"10.3390\/e17010471","type":"journal-article","created":{"date-parts":[[2015,1,19]],"date-time":"2015-01-19T10:53:54Z","timestamp":1421664834000},"page":"471-482","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":30,"title":["Modeling and Analyzing the Interaction between Network Rumors and Authoritative Information"],"prefix":"10.3390","volume":"17","author":[{"given":"Lingling","family":"Xia","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guoping","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yurong","family":"Song","sequence":"additional","affiliation":[{"name":"School of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bo","family":"Song","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,1,19]]},"reference":[{"key":"ref_1","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":"Physica A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3200","DOI":"10.1103\/PhysRevLett.86.3200","article-title":"Epidemic spreading in scale-free networks","volume":"86","author":"Vespignani","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Rizzo, A., Frasca, M., and Porfiri, M. (2014). Effect of individual behavior on epidemic spreading in activity-driven networks. Phys. Rev. E., 90.","DOI":"10.1103\/PhysRevE.90.042801"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Buscarino, A., Fortuna, L., Frasca, M., and Rizzo, A. (2014). Local and global epidemic outbreaks in populations moving in inhomogeneous environments. Phys. Rev. E., 90.","DOI":"10.1103\/PhysRevE.90.042813"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Fu, X., Small, M., and Chen, G. (2014). Propagation Dynamics on Complex Networks: Models, Methods and Stability Analysis, Wiley.","DOI":"10.1002\/9781118762783"},{"key":"ref_6","unstructured":"Guo, L., and Xu, X.M. (2006). Complex Networks, Science and Technology Education Press. In Chinese."},{"key":"ref_7","first-page":"91","article-title":"Modeling and analyzing of the interaction between worms and antiworms during network worm propagation","volume":"48","author":"Yang","year":"2005","journal-title":"Sci. China Ser. F."},{"key":"ref_8","first-page":"376","article-title":"Modeling and analysis of hybrid structured benign worms countering against worms","volume":"33","author":"Zhou","year":"2012","journal-title":"J. Chin. Comput. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ahn, Y.Y., Jeong, H., Masuda, N., and Noh, J.D. (2006). Epidemic dynamics of two species of interacting particles on scale-free networks. Phys. Rev. E., 74.","DOI":"10.1103\/PhysRevE.74.066113"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Trpevski, D., Tang, W.K.S., and Kocarev, L. (2010). Model for rumor spreading over networks. Phys. Rev. E., 81.","DOI":"10.1103\/PhysRevE.81.056102"},{"key":"ref_11","first-page":"424","article-title":"Rumor spreading model with skepticism mechanism in social networks","volume":"34","author":"Wang","year":"2012","journal-title":"J. Univ. Shanghai Sci. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3846","DOI":"10.1016\/j.physa.2013.04.018","article-title":"The spreading of opposite opinions on online social networks with authoritative nodes","volume":"392","author":"Yan","year":"2013","journal-title":"Physica A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Moreno, Y., Nekovee, M., and Pacheco, A.F. (2004). Dynamics of rumor spreading in complex networks. Phys. Rev. E., 69.","DOI":"10.1103\/PhysRevE.69.066130"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/j.physa.2006.07.017","article-title":"Theory of rumor spreading in complex social networks","volume":"374","author":"Nekovee","year":"2007","journal-title":"Physica A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1016\/j.physleta.2007.01.094","article-title":"Influence of network structure on rumor propagation","volume":"368","author":"Zhou","year":"2007","journal-title":"Phys. Lett. A."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2619","DOI":"10.1016\/j.physa.2011.03.010","article-title":"Rumor spreading model with consideration of forgetting mechanism: A case of online blogging LiveJournal","volume":"390","author":"Zhao","year":"2011","journal-title":"Physica A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6146","DOI":"10.1016\/j.physa.2013.07.080","article-title":"A rumor spreading model with variable forgetting rate","volume":"392","author":"Zhao","year":"2013","journal-title":"Physica A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1016\/j.physa.2012.10.031","article-title":"Rumor spreading model considering forgetting and remembering mechanisms in inhomogeneous networks","volume":"392","author":"Zhao","year":"2013","journal-title":"Physica A"},{"key":"ref_19","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":"Physica A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1088\/0253-6102\/59\/4\/21","article-title":"Rumor spreading model with trust mechanism in complex social networks","volume":"59","author":"Wang","year":"2013","journal-title":"Commun. Theor. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Daley, D.J., and Kendall, D.G. (1964). Epidemics and rumours. Nature, 204.","DOI":"10.1038\/2041118a0"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1080\/0022250X.1976.9989852","article-title":"Interactive Markov chains","volume":"4","author":"Conlisk","year":"1976","journal-title":"J. Math. Sociol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1109\/37.969135","article-title":"The influence model","volume":"21","author":"Asavathiratham","year":"2001","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_24","unstructured":"Said, M.R., Oppenheim, A.V., and Lauffenburger, D.A. (2003, January 6\u201310). Modeling cellular signal processing using interacting Markov chains, Hong Kong, China."},{"key":"ref_25","unstructured":"Peng, S.J., Li, Y.X., and Zheng, B.J. (2008, January 25\u201327). Steady states and critical behavior of epidemic spreading on complex networks, Chongqing, China."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/1\/471\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:41:47Z","timestamp":1760215307000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/1\/471"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,19]]},"references-count":27,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1]]}},"alternative-id":["e17010471"],"URL":"https:\/\/doi.org\/10.3390\/e17010471","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1,19]]}}}