{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T05:20:10Z","timestamp":1763702410241,"version":"3.45.0"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T00:00:00Z","timestamp":1763510400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guizhou Provincial Basic Research Progra","award":["Qiankehejichu-MS[2025]:No.280"],"award-info":[{"award-number":["Qiankehejichu-MS[2025]:No.280"]}]},{"name":"the Youth Guidance Project of the Guizhou Province Basic Research Program","award":["Qiankehe Foundation [2024]: No.Youth 357"],"award-info":[{"award-number":["Qiankehe Foundation [2024]: No.Youth 357"]}]},{"name":"the Scientific Research Foundation for Young Talents of Higher Education Institutions of the Department of Education of Guizhou Province","award":["Qianjiao Technology [2022]: No.129"],"award-info":[{"award-number":["Qianjiao Technology [2022]: No.129"]}]},{"name":"National Natural Science Foundation of Henan Youth Program under Grant","award":["No.242300420670 and 252300420939"],"award-info":[{"award-number":["No.242300420670 and 252300420939"]}]},{"DOI":"10.13039\/100014472","name":"Scientific Research Foundation of Hunan Provincial Education Department","doi-asserted-by":"publisher","award":["No. 22B0869"],"award-info":[{"award-number":["No. 22B0869"]}],"id":[{"id":"10.13039\/100014472","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Affiliation networks, with their bipartite structure and non-binary features, pose unique challenges due to their complex relationships and diverse node attributes. These challenges differ from those in symmetric one-mode networks. To address them, we propose a generalized logistic affiliation network model. Despite the structural asymmetry, the model incorporates node attributes and includes parameters for actor activeness, event popularity, and symmetric patterns in actor\u2013event interactions. We study the theoretical properties of this model under an asymptotic framework, where the number of actors and events grows to infinity. Using maximum likelihood estimation, we show that the estimators for degree heterogeneity and node homophily converge to multivariate normal distributions under mild conditions. To validate the model and our theory, we conduct experiments on both simulated data and a movie-rating dataset.<\/jats:p>","DOI":"10.3390\/sym17112005","type":"journal-article","created":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T15:03:15Z","timestamp":1763564595000},"page":"2005","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Asymptotic Analysis of Generalized Logistic Affiliation Network Models with Node Attributes"],"prefix":"10.3390","volume":"17","author":[{"given":"Yifan","family":"Fan","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Zhongyuan University of Technology, Zhengzhou 450007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Si","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1137\/S003614450342480","article-title":"The structure and function of complex networks","volume":"45","author":"Newman","year":"2003","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"S20","DOI":"10.1016\/j.ijid.2021.12.047","article-title":"Improved contact tracing using network analysis and spatial-temporal proximity","volume":"116","author":"Myall","year":"2022","journal-title":"Int. J. Infect. Dis."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1089\/cmb.2022.0373","article-title":"Graph-based reconstruction and analysis of disease transmission networks using viral genomic data","volume":"30","author":"Ke","year":"2023","journal-title":"J. Comput. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1038\/nrg2085","article-title":"Exploring genetic interactions and networks with yeast","volume":"8","author":"Boone","year":"2007","journal-title":"Nat. Rev. Genet."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.gde.2019.03.002","article-title":"Genetic interaction networks in cancer cells","volume":"54","author":"Mair","year":"2019","journal-title":"Curr. Opin. Genet. Dev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1016\/j.physa.2006.04.047","article-title":"Bipartite graphs as models of complex networks","volume":"371","author":"Guillaume","year":"2006","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"giy014","DOI":"10.1093\/gigascience\/giy014","article-title":"Bipartite graphs in systems biology and medicine: A survey of methods and applications","volume":"7","author":"Pavlopoulos","year":"2018","journal-title":"GigaScience"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1679","DOI":"10.1016\/j.physa.2017.11.089","article-title":"A novel community detection method in bipartite networks","volume":"492","author":"Zhou","year":"2018","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1080\/01621459.1986.10478342","article-title":"Markov graphs","volume":"81","author":"Frank","year":"1986","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"066117","DOI":"10.1103\/PhysRevE.70.066117","article-title":"Statistical mechanics of networks","volume":"70","author":"Park","year":"2004","journal-title":"Phys. Rev. E"},{"doi-asserted-by":"crossref","unstructured":"Lusher, D., Koskinen, J., and Robins, G. (2013). Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications, Cambridge University Press.","key":"ref_11","DOI":"10.1017\/CBO9780511894701"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1038\/s42254-018-0002-6","article-title":"The statistical physics of real-world networks","volume":"1","author":"Cimini","year":"2019","journal-title":"Nat. Rev. Phys."},{"unstructured":"Hillar, C., and Wibisono, A. (2013). Maximum entropy distributions on graphs. arXiv.","key":"ref_13"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1111\/rssb.12444","article-title":"Analysis of networks via the sparse \u03b2-model","volume":"83","author":"Chen","year":"2021","journal-title":"J. R. Stat. Soc. Ser. Stat. Methodol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1111\/sjos.12650","article-title":"Time-varying \u03b2-model for dynamic directed networks","volume":"50","author":"Du","year":"2023","journal-title":"Scand. J. Stat."},{"unstructured":"Yan, T., Li, Y., Xu, J., Yang, Y., and Zhu, J. (2023). Likelihood ratio tests in random graph models with increasing dimensions. arXiv.","key":"ref_16"},{"unstructured":"Shao, M., Zhang, Y., Wang, Q., Zhang, Y., Luo, J., and Yan, T. (2021). L2 Regularized maximum likelihood for \u03b2-model in large and sparse networks. arXiv.","key":"ref_17"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1093\/biomet\/ass084","article-title":"A central limit theorem in the \u03b2-model for undirected random graphs with a diverging number of vertices","volume":"100","author":"Yan","year":"2013","journal-title":"Biometrika"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1214\/15-AOS1343","article-title":"Asymptotics in directed exponential random graph models with an increasing bi-degree sequence","volume":"44","author":"Yan","year":"2016","journal-title":"Ann. Stat."},{"key":"ref_20","first-page":"273","article-title":"Asymptotics in undirected random graph models parameterized by the strengths of vertices","volume":"26","author":"Yan","year":"2016","journal-title":"Stat. Sin."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"385","DOI":"10.4310\/SII.2020.v13.n3.a8","article-title":"Asymptotic Theory for Differentially Private Generalized \u03b2-models with Parameters Increasing","volume":"13","author":"Fan","year":"2020","journal-title":"Stat. Interface"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11163","DOI":"10.1080\/03610926.2016.1260741","article-title":"Affiliation networks with an increasing degree sequence","volume":"46","author":"Zhang","year":"2017","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1002\/cjs.11735","article-title":"Asymptotic theory in bipartite graph models with a growing number of parameters","volume":"51","author":"Fan","year":"2023","journal-title":"Can. J. Stat."},{"key":"ref_24","first-page":"1","article-title":"Two-mode networks: Inference with as many parameters as actors and differential privacy","volume":"23","author":"Wang","year":"2022","journal-title":"J. Mach. Learn. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s00362-021-01243-2","article-title":"Affiliation weighted networks with a differentially private degree sequence","volume":"63","author":"Luo","year":"2022","journal-title":"Stat. Pap."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6477","DOI":"10.1080\/03610926.2023.2246090","article-title":"Differentially private estimation in a class of bipartite graph models","volume":"53","author":"Pan","year":"2024","journal-title":"Commun. Stat. Theory Methods"},{"unstructured":"Sarkar, P., Siddiqi, S.M., and Gordon, G.J. (2007, January 21\u201324). A latent space approach to dynamic embedding of co-occurrence data. Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, San Juan, Puerto Rico.","key":"ref_27"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6629","DOI":"10.1073\/pnas.1606295113","article-title":"Interlocking directorates in Irish companies using a latent space model for bipartite networks","volume":"113","author":"Friel","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00180-025-01626-1","article-title":"Network embedding for bipartite networks with applications in interlocking directorates in Chinese companies","volume":"40","author":"Zhang","year":"2025","journal-title":"Comput. Stat."},{"key":"ref_30","first-page":"1","article-title":"Matched bipartite block model with covariates","volume":"20","author":"Razaee","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"105562","DOI":"10.1016\/j.jeconom.2023.105562","article-title":"Bipartite network influence analysis of a two-mode network","volume":"239","author":"Wu","year":"2024","journal-title":"J. Econom."},{"key":"ref_32","first-page":"7","article-title":"Bipartite mixed membership stochastic blockmodel","volume":"1050","author":"Qing","year":"2021","journal-title":"stat"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.3982\/ECTA12679","article-title":"An econometric model of network formation with degree heterogeneity","volume":"85","author":"Graham","year":"2017","journal-title":"Econometrica"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1080\/01621459.2018.1448829","article-title":"Statistical inference in a directed network model with covariates","volume":"114","author":"Yan","year":"2019","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s10463-022-00848-0","article-title":"Asymptotic theory in network models with covariates and a growing number of node parameters","volume":"75","author":"Wang","year":"2023","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2092","DOI":"10.1080\/01621459.2023.2244731","article-title":"PCABM: Pairwise covariates-adjusted block model for community detection","volume":"119","author":"Huang","year":"2024","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2827872","article-title":"The MovieLens Datasets: History and Context","volume":"5","author":"Harper","year":"2015","journal-title":"ACM Trans. Interact. Intell. Syst."},{"unstructured":"Fernandez, J.A.E., and Ver\u00f3n, M.\u00c1.H. (2020). Mild Differentiability Conditions for Newton\u2019s Method in Banach Spaces, Springer Nature.","key":"ref_38"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1400","DOI":"10.1214\/10-AAP728","article-title":"Random graphs with a given degree sequence","volume":"21","author":"Chatterjee","year":"2011","journal-title":"Ann. Appl. Probab."},{"unstructured":"Lo\u00e9ve, M. (1977). Probability Theory I, Springer. [4th ed.].","key":"ref_40"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1162\/rest_a_00805","article-title":"An Empirical Model of Dyadic Link Formation in a Network with Unobserved Heterogeneity","volume":"101","author":"Dzemski","year":"2019","journal-title":"Rev. Econ. Stat."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/2005\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T05:15:16Z","timestamp":1763702116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/2005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,19]]},"references-count":41,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17112005"],"URL":"https:\/\/doi.org\/10.3390\/sym17112005","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,11,19]]}}}