{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T14:41:02Z","timestamp":1778769662095,"version":"3.51.4"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T00:00:00Z","timestamp":1740355200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key R&amp;D Program of China","award":["2022ZD0116800"],"award-info":[{"award-number":["2022ZD0116800"]}]},{"name":"National Key R&amp;D Program of China","award":["62141605"],"award-info":[{"award-number":["62141605"]}]},{"name":"National Key R&amp;D Program of China","award":["12201026"],"award-info":[{"award-number":["12201026"]}]},{"name":"National Key R&amp;D Program of China","award":["11922102"],"award-info":[{"award-number":["11922102"]}]},{"name":"National Key R&amp;D Program of China","award":["11871004"],"award-info":[{"award-number":["11871004"]}]},{"DOI":"10.13039\/501100001809","name":"Program of National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022ZD0116800"],"award-info":[{"award-number":["2022ZD0116800"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Program of National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62141605"],"award-info":[{"award-number":["62141605"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Program of National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201026"],"award-info":[{"award-number":["12201026"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Program of National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11922102"],"award-info":[{"award-number":["11922102"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Program of National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871004"],"award-info":[{"award-number":["11871004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Information and knowledge diffusion are important dynamical processes in complex social systems, in which the underlying topology of interactions among individuals is often modeled as networks. Recent studies have examined various information diffusion scenarios primarily focusing on the dynamics within one network; yet, relatively little scholarly attention has been paid to possible interactions among individuals beyond the focal network. Here, in this study, we account for this phenomenon by modeling the information diffusion dynamics with the involvement of independent spreaders in a susceptible\u2013exposed\u2013infectious\u2013recovered contagion process. Independent spreaders receive information using latent information transmission pathways without following the links in the focal network and can spread the information to remote areas of the network not well connected to the major components. We derive the mathematics of the critical epidemic thresholds on homogeneous and heterogeneous networks as a function of the infectious rate, exposure rate, recovery rate and the activeness of independent spreaders. We present simulation results on Small World and Scale-Free complex networks, and real-world social networks of Facebook artists and physicist collaborations. The result shows that the extent to which information or knowledge can spread might be more extensive than we can explain in terms of link contagion only. In addition, these results also help to explain how the activeness of independent spreaders can affect the diffusion process of information and knowledge in complex networks, which may have implications for studies exploring other dynamical processes.<\/jats:p>","DOI":"10.3390\/e27030234","type":"journal-article","created":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T06:47:17Z","timestamp":1740379637000},"page":"234","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Information and Knowledge Diffusion Dynamics in Complex Networks with Independent Spreaders"],"prefix":"10.3390","volume":"27","author":[{"given":"Yan","family":"Zhuang","sequence":"first","affiliation":[{"name":"School of Economics and Management, Beihang University, Beijing 100191, China"},{"name":"School of Software, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weihua","family":"Li","sequence":"additional","affiliation":[{"name":"LMIB, NLSDE, BDBC, and School of Artificial Intelligence, Beihang University, Beijing 100191, China"},{"name":"Hangzhou International Innovation Institute, Beihang University, Hangzhou 311115, China"},{"name":"Department of Strategic and Advanced Interdisciplinary Research, Pengcheng Laboratory, Shenzhen 518055, China"},{"name":"Zhongguancun Laboratory, Beijing 100080, China"},{"name":"Qianyuan Laboratory, Hangzhou 310024, China"},{"name":"Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4778-9566","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"Communist Youth League Committee, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1109\/MCOM.001.2300333","article-title":"Dynamics of Ideological Biases of Social Media Users","volume":"62","author":"Modi","year":"2024","journal-title":"IEEE Commun. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Flamino, J., Gong, B., Buchanan, F., and Szymanski, B.K. (2021). Characterizing Topics in Social Media Using Dynamics of Conversation. Entropy, 23.","DOI":"10.3390\/e23121642"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Xie, W., Wang, X., and Jia, T. (2022, January 4\u20136). Independent asymmetric embedding for information diffusion prediction on social networks. Proceedings of the 2022 IEEE 25th International Conference on Computer Supported Cooperative Work in Design (CSCWD), Hangzhou, China.","DOI":"10.1109\/CSCWD54268.2022.9776071"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1093\/comnet\/cnx032","article-title":"Loss of information in feedforward social networks","volume":"6","author":"Stolarczyk","year":"2017","journal-title":"J. Complex Netw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"e1601895","DOI":"10.1126\/sciadv.1601895","article-title":"A three-degree horizon of peace in the military alliance network","volume":"3","author":"Li","year":"2017","journal-title":"Sci. Adv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"18720","DOI":"10.1073\/pnas.1107583108","article-title":"Inferring the structure and dynamics of interactions in schooling fish","volume":"108","author":"Katz","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5408","DOI":"10.1038\/s41467-020-19086-0","article-title":"Vortex phase matching as a strategy for schooling in robots and in fish","volume":"11","author":"Li","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2862","DOI":"10.1016\/j.cub.2017.08.004","article-title":"Consistent individual differences drive collective behavior and group functioning of schooling fish","volume":"27","author":"Jolles","year":"2017","journal-title":"Curr. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"0078","DOI":"10.1038\/s41562-017-0078","article-title":"Quantifying patterns of research-interest evolution","volume":"1","author":"Jia","year":"2017","journal-title":"Nat. Hum. Behav."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5170","DOI":"10.1038\/s41467-019-13130-4","article-title":"Early coauthorship with top scientists predicts success in academic careers","volume":"10","author":"Li","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4907","DOI":"10.1038\/s41467-022-32604-6","article-title":"Untangling the network effects of productivity and prominence among scientists","volume":"13","author":"Li","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Barrat, A., Barthelemy, M., and Vespignani, A. (2008). Dynamical Processes on Complex Networks, Cambridge University Press.","DOI":"10.1017\/CBO9780511791383"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1103\/RevModPhys.87.925","article-title":"Epidemic processes in complex networks","volume":"87","author":"Castellano","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2485","DOI":"10.1038\/s41467-019-10431-6","article-title":"Simplicial models of social contagion","volume":"10","author":"Iacopini","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_15","first-page":"016128","article-title":"Spread of epidemic disease on networks","volume":"66","author":"Newman","year":"2002","journal-title":"Phys. Rev."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kiss, I.Z., Miller, J.C., and Simon, P.L. (2017). Mathematics of Epidemics on Networks, Springer.","DOI":"10.1007\/978-3-319-50806-1"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"389","DOI":"10.3934\/mbe.2008.5.389","article-title":"SEIR epidemiological model with varying infectivity and infinite delay","volume":"5","year":"2008","journal-title":"Math. Biosci. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"114386","DOI":"10.1016\/j.chaos.2023.114386","article-title":"A Parrondo paradoxical interplay of reciprocity and reputation in social dynamics","volume":"179","author":"Lai","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"113107","DOI":"10.1063\/5.0172121","article-title":"Opinion cascade under perception bias in social networks","volume":"33","author":"Yu","year":"2023","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"124813","DOI":"10.1016\/j.eswa.2024.124813","article-title":"The evolution dynamics of collective and individual opinions in social networks","volume":"255","author":"Dong","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_21","unstructured":"Wen, T., Zheng, R., Wu, T., Liu, Z., Zhou, M., Syed, T.A., Ghataoura, D., and Chen, Y.W. Formulating opinion dynamics from belief formation, diffusion and updating in social network group decision-making: Towards developing a holistic framework, Eur. J. Oper. Res., in press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"834","DOI":"10.1038\/s41562-021-01136-2","article-title":"A review and agenda for integrated disease models including social and behavioural factors","volume":"5","author":"Bedson","year":"2021","journal-title":"Nat. Hum. Behav."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2021.10.005","article-title":"Social physics","volume":"948","author":"Jusup","year":"2022","journal-title":"Phys. Rep."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"126558","DOI":"10.1016\/j.physa.2021.126558","article-title":"The coupled dynamics of information dissemination and SEIR-based epidemic spreading in multiplex networks","volume":"588","author":"Ma","year":"2022","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"161","DOI":"10.3934\/mbe.2006.3.161","article-title":"Epidemic threshold conditions for seasonally forced SEIR models","volume":"3","author":"Ma","year":"2005","journal-title":"Math. Biosci. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.physa.2013.11.021","article-title":"The rumor diffusion process with emerging independent spreaders in complex networks","volume":"397","author":"Li","year":"2014","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.physa.2017.09.052","article-title":"Information spreading in complex networks with participation of independent spreaders","volume":"492","author":"Ma","year":"2018","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"122921","DOI":"10.1016\/j.physa.2019.122921","article-title":"The SIS diffusion process in complex networks with independent spreaders","volume":"546","author":"Ding","year":"2020","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Winkelmann, S., Zonker, J., Sch\u00fctte, C., and Conrad, N.D. (2021). Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading. Math. Biosci., 336.","DOI":"10.1016\/j.mbs.2021.108619"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Liao, C., Squicciarini, A., Griffin, C., and Rajtmajer, S. (2015, January 25\u201328). A hybrid epidemic model for antinormative behavior in online social networks. Proceedings of the 2015 IEEE\/ACM International Conference on Advances in Social Networks Analysis and Mining 2015, Paris, France.","DOI":"10.1145\/2808797.2809334"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1375","DOI":"10.1038\/s41467-023-37118-3","article-title":"Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs","volume":"14","author":"Petri","year":"2023","journal-title":"Nat. Commun."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Bicker, J., Schmieding, R., Meyer-Hermann, M., and K\u00fchn, M.J. (2024). Hybrid metapopulation agent-based epidemiological models for efficient insight on the individual scale: A contribution to green computing. arXiv.","DOI":"10.1016\/j.idm.2024.12.015"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"112759","DOI":"10.1016\/j.chaos.2022.112759","article-title":"Interplay between exogenous triggers and endogenous behavioral changes in contagion processes on social networks","volume":"165","author":"Eminente","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"043358","DOI":"10.1103\/PhysRevResearch.2.043358","article-title":"Statistical physics of discovering exogenous and endogenous factors in a chain of events","volume":"2","author":"Koyama","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1038\/s41586-021-03914-4","article-title":"Burden and characteristics of COVID-19 in the United States during 2020","volume":"598","author":"Pei","year":"2021","journal-title":"Nature"},{"key":"ref_36","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_37","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_38","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"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Rozemberczki, B., Davies, R., Sarkar, R., and Sutton, C. (2019, January 27\u201330). GEMSEC: Graph Embedding with Self Clustering. Proceedings of the 2019 IEEE\/ACM International Conference on Advances in Social Networks Analysis and Mining 2019, Vancouver, BC, Canada.","DOI":"10.1145\/3341161.3342890"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1140\/epjds\/s13688-019-0199-3","article-title":"Reciprocity and impact in academic careers","volume":"8","author":"Li","year":"2019","journal-title":"Epj Data Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1108\/13673271311300831","article-title":"Knowledge sharing amongst academics in UK universities","volume":"17","author":"Fullwood","year":"2013","journal-title":"J. Knowl. Manag."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"De Domenico, M., Lima, A., Mougel, P., and Musolesi, M. (2013). The anatomy of a scientific rumor. Sci. Rep., 3.","DOI":"10.1038\/srep02980"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/3\/234\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:41:22Z","timestamp":1760028082000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/3\/234"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,24]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["e27030234"],"URL":"https:\/\/doi.org\/10.3390\/e27030234","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,24]]}}}