{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:41:56Z","timestamp":1760060516825,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T00:00:00Z","timestamp":1757116800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangdong Basic and Applied Basic Research Foundation","award":["2025A1515012130","ITS\/188\/20"],"award-info":[{"award-number":["2025A1515012130","ITS\/188\/20"]}]},{"name":"Innovation and Technology Commission (ITC)","award":["2025A1515012130","ITS\/188\/20"],"award-info":[{"award-number":["2025A1515012130","ITS\/188\/20"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the problem of identifying the source of a rumor in a network, given only a snapshot observation of infected nodes after the rumor has spread. Classical approaches, such as the maximum likelihood (ML) and joint maximum likelihood (JML) estimators based on the conventional Susceptible\u2013Infectious (SI) model, exhibit degeneracy, failing to uniquely identify the source even in simple network structures. To address these limitations, we propose a generalized estimator that incorporates independent random observation times. To capture the structure of information flow beyond graphs, our formulations consider rate constraints on the rumor and the multicast capacities for cyclic polylinking networks. Furthermore, we develop forward elimination and backward search algorithms for rate-constrained source detection and validate their effectiveness and scalability through comprehensive simulations. Our study establishes a rigorous and scalable foundation on the infodemic source detection.<\/jats:p>","DOI":"10.3390\/e27090936","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T12:04:55Z","timestamp":1757505895000},"page":"936","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Infodemic Source Detection with Information Flow: Foundations and Scalable Computation"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2757-7208","authenticated-orcid":false,"given":"Zimeng","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Computer Science, City University of Hong Kong, Hong Kong, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4165-2123","authenticated-orcid":false,"given":"Chao","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Computer Science, City University of Hong Kong, Hong Kong, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9779-7225","authenticated-orcid":false,"given":"Qiaoqiao","family":"Zhou","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science and Control Engineering, Shenzhen University of Advanced Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6624-9752","authenticated-orcid":false,"given":"Chee Wei","family":"Tan","sequence":"additional","affiliation":[{"name":"College of Computing and Data Science, Nanyang Technological University, Singapore 639978, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2006-0898","authenticated-orcid":false,"given":"Chung","family":"Chan","sequence":"additional","affiliation":[{"name":"Department of Computer Science, City University of Hong Kong, Hong Kong, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.5121\/ijcnc.2014.6508","article-title":"A computer virus propagation model using delay differential equations with probabilistic contagion and immunity","volume":"6","author":"Khan","year":"2014","journal-title":"Int. J. Comput. Netw. Commun."},{"key":"ref_2","first-page":"1","article-title":"Exploring Cyber Risk Contagion-A Boundless Threat","volume":"16","author":"Ai","year":"2023","journal-title":"Variance"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1093\/bib\/bbz005","article-title":"Computational medicine: Quantitative modeling of complex diseases","volume":"21","author":"Tiwary","year":"2019","journal-title":"Briefings Bioinform."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2747","DOI":"10.1109\/TSMC.2024.3518560","article-title":"Enhanced epidemic control: Community-based observer placement and source tracing","volume":"55","author":"Zhao","year":"2025","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2602","DOI":"10.1016\/j.tcs.2010.11.001","article-title":"Rumor spreading in social networks","volume":"412","author":"Chierichetti","year":"2011","journal-title":"Theor. Comput. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2130002","DOI":"10.1142\/S1793830921300022","article-title":"Schemes of propagation models and source estimators for rumor source detection in online social networks: A short survey of a decade of research","volume":"13","author":"Jin","year":"2021","journal-title":"Discret. Math. Algorithms Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3716498","article-title":"A survey on exploring real and virtual social network rumors: State-of-the-art and research challenges","volume":"57","author":"He","year":"2025","journal-title":"ACM Comput. Surv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.osnem.2018.12.001","article-title":"Source detection of rumor in social network\u2013a review","volume":"9","author":"Shelke","year":"2019","journal-title":"Online Soc. Netw. Media"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ahmad, T., Faisal, M.S., Rizwan, A., Alkanhel, R., Khan, P.W., and Muthanna, A. (2022). Efficient fake news detection mechanism using enhanced deep learning model. Appl. Sci., 12.","DOI":"10.3390\/app12031743"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4787","DOI":"10.1038\/s41467-018-06930-7","article-title":"The spread of low-credibility content by social bots","volume":"9","author":"Shao","year":"2018","journal-title":"Nat. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"544","DOI":"10.2471\/BLT.21.287654","article-title":"Infodemics and health misinformation: A systematic review of reviews","volume":"100","author":"Pizarro","year":"2022","journal-title":"Bull. World Health Organ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1057\/s41288-022-00266-6","article-title":"Cyber risk and cybersecurity: A systematic review of data availability","volume":"47","author":"Cremer","year":"2022","journal-title":"Geneva Pap. Risk Insurance. Issues Pract."},{"key":"ref_13","unstructured":"Bailey, N.T.J. (1975). The Mathematical Theory of Infectious Diseases and Its Applications, Charles Griffin & Company Ltd.. [2nd ed.]."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5163","DOI":"10.1109\/TIT.2011.2158885","article-title":"Rumors in a Network: Who\u2019s the Culprit?","volume":"57","author":"Shah","year":"2011","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","unstructured":"MacKay, D.J.C. (2003). Information Theory, Inference & Learning Algorithms, Cambridge University Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1145\/2318857.2254782","article-title":"Rumor centrality: A universal source detector","volume":"40","author":"Shah","year":"2012","journal-title":"SIGMETRICS Perform. Eval. Rev."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Athreya, K.B., and Ney, P.E. (1972). Multi-Type Branching Processes. Branching Processes, Springer.","DOI":"10.1007\/978-3-642-65371-1"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1109\/JSTSP.2022.3153168","article-title":"Epidemic Source Detection in Contact Tracing Networks: Epidemic Centrality in Graphs and Message-Passing Algorithms","volume":"16","author":"Yu","year":"2022","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1561\/1300000068","article-title":"Contagion Source Detection in Epidemic and Infodemic Outbreaks: Mathematical Analysis and Network Algorithms","volume":"13","author":"Tan","year":"2023","journal-title":"Found. Trends Netw."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Fan, T.H., and Wang, I.H. (2018, January 15\u201320). Rumor Source Detection: A Probabilistic Perspective. Proceedings of the 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Calgary, AB, Canada.","DOI":"10.1109\/ICASSP.2018.8461881"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1109\/18.850663","article-title":"Network information flow","volume":"46","author":"Ahlswede","year":"2000","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1109\/TIT.2002.807285","article-title":"Linear network coding","volume":"49","author":"Li","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1109\/TNET.2003.818197","article-title":"An algebraic approach to network coding","volume":"11","author":"Koetter","year":"2003","journal-title":"IEEE\/ACM Trans. Netw."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1973","DOI":"10.1109\/TIT.2005.847712","article-title":"Polynomial time algorithms for multicast network code construction","volume":"51","author":"Jaggi","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2345","DOI":"10.1109\/TIT.2006.874531","article-title":"On the capacity of information networks","volume":"52","author":"Harvey","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4413","DOI":"10.1109\/TIT.2006.881746","article-title":"A Random Linear Network Coding Approach to Multicast","volume":"52","author":"Ho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","unstructured":"Schrijver, A. (2003). Combinatorial Optimization: Polyhedra and Efficiency, Springer."},{"key":"ref_28","unstructured":"Yeung, R.W. (2008). Information Theory and Network Coding, Springer Science & Business Media."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1440","DOI":"10.1109\/18.681320","article-title":"On characterization of entropy function via information inequalities","volume":"44","author":"Zhang","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Zhao, C., Wang, Z., Zhou, Q., Tan, C.W., and Chan, C. (2024, January 7\u201312). Infodemic Source Detection: Enhanced Formulations with Information Flow. Proceedings of the 2024 IEEE International Symposium on Information Theory (ISIT), Athens, Greece.","DOI":"10.1109\/ISIT57864.2024.10619619"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10107-011-0446-2","article-title":"A flow model based on polylinking system","volume":"135","author":"Goemans","year":"2012","journal-title":"Math. Program."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/0095-8956(79)90011-X","article-title":"Matroids and linking systems","volume":"26","author":"Schrijver","year":"1979","journal-title":"J. Comb. Theory, Ser. B"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Chan, C., Shum, K.W., and Sun, Q.T. (2013, January 9\u201313). Combinatorial flow over cyclic linear networks. Proceedings of the 2013 IEEE Information Theory Workshop (ITW), Sevilla, Spain.","DOI":"10.1109\/ITW.2013.6691270"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chan, C. (2012, January 1\u20136). Matroidal undirected network. Proceedings of the 2012 IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, USA.","DOI":"10.1109\/ISIT.2012.6283513"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/BF01588971","article-title":"An analysis of approximations for maximizing submodular set functions\u2014I","volume":"14","author":"Nemhauser","year":"1978","journal-title":"Math. Program."},{"key":"ref_36","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_37","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_38","first-page":"290","article-title":"On random graph","volume":"6","author":"Renyi","year":"1959","journal-title":"Publ. Math."},{"key":"ref_39","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","author":"Erdos","year":"1960","journal-title":"Publ. Math. Inst. Hung. Acad. Sci"},{"key":"ref_40","first-page":"3","article-title":"Submodular function maximization","volume":"3","author":"Krause","year":"2014","journal-title":"Tractability"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/9\/936\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:41:25Z","timestamp":1760035285000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/9\/936"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,6]]},"references-count":40,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["e27090936"],"URL":"https:\/\/doi.org\/10.3390\/e27090936","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,9,6]]}}}