{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T06:09:09Z","timestamp":1770962949991,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000138","name":"U.S. Department of Education","doi-asserted-by":"crossref","award":["R305D210023"],"award-info":[{"award-number":["R305D210023"]}],"id":[{"id":"10.13039\/100000138","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>With the growing accessibility of tools such as online surveys and web scraping, longitudinal social network data are more commonly collected in social science research along with non-network survey data. Such data play a critical role in helping social scientists understand how relationships develop and evolve over time. Existing dynamic network models such as the Stochastic Actor-Oriented Model and the Temporal Exponential Random Graph Model provide frameworks to analyze traits of both the networks and the external non-network covariates. However, research on the dynamic latent space model (DLSM) has focused mainly on factors intrinsic to the networks themselves. Despite some discussion, the role of non-network data such as contextual or behavioral covariates remain a topic to be further explored in the context of DLSMs. In this study, one application of the DLSM to incorporate dynamic non-network covariates collected alongside friendship networks using autoregressive processes is presented. By analyzing two friendship network datasets with different time points and psychological covariates, it is shown how external factors can contribute to a deeper understanding of social interaction dynamics over time.<\/jats:p>","DOI":"10.3390\/computation14020034","type":"journal-article","created":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T12:49:44Z","timestamp":1770036584000},"page":"34","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Application of the Dynamic Latent Space Model to Social Networks with Time-Varying Covariates"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-9869-641X","authenticated-orcid":false,"given":"Ziqian","family":"Xu","sequence":"first","affiliation":[{"name":"Department of Psychology, University of Notre Dame, 390 Corbett Family Hall, Notre Dame, IN 46556, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0590-2196","authenticated-orcid":false,"given":"Zhiyong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Psychology, University of Notre Dame, 390 Corbett Family Hall, Notre Dame, IN 46556, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2134","DOI":"10.1080\/10494820.2021.1875001","article-title":"Trends in artificial intelligence-supported e-learning: A systematic review and co-citation network analysis (1998\u20132019)","volume":"31","author":"Tang","year":"2023","journal-title":"Interact. Learn. Environ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.jclinepi.2020.11.019","article-title":"Citation bias and other determinants of citation in biomedical research: Findings from six citation networks","volume":"132","author":"Urlings","year":"2021","journal-title":"J. Clin. Epidemiol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1007\/s11205-011-9861-2","article-title":"How friendship network characteristics influence subjective well-being","volume":"107","year":"2012","journal-title":"Soc. Indic. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15274","DOI":"10.1073\/pnas.0900282106","article-title":"Inferring friendship network structure by using mobile phone data","volume":"106","author":"Eagle","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1080\/00273171.2018.1479629","article-title":"Structural Equation Modeling of Social Networks: Specification, Estimation, and Application","volume":"53","author":"Liu","year":"2018","journal-title":"Multivar. Behav. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"34","DOI":"10.35566\/jbds\/v1n1\/p3","article-title":"Birds of a Feather Flock Together and Opposites Attract: The Nonlinear Relationship Between Personality and Friendship","volume":"1","author":"Liu","year":"2021","journal-title":"J. Behav. Data Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1080\/10705511.2025.2488030","article-title":"Structural Equation Models with Social Networks","volume":"33","author":"Xu","year":"2025","journal-title":"Struct. Equ. Model. Multidiscip. J."},{"key":"ref_8","first-page":"1","article-title":"Social network analysis and agent-based modeling in social epidemiology","volume":"9","author":"Scarborough","year":"2012","journal-title":"Epidemiol. Perspect. Innov."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.meegid.2016.05.042","article-title":"Integrating molecular epidemiology and social network analysis to study infectious diseases: Towards a socio-molecular era for public health","volume":"46","author":"Vasylyeva","year":"2016","journal-title":"Infect. Genet. Evol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1109\/MCS.2021.3092810","article-title":"Learning hidden influences in large-scale dynamical social networks: A data-driven sparsity-based approach, in memory of roberto tempo","volume":"41","author":"Ravazzi","year":"2021","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Veenstra, R., Bertogna, T., and Laninga-Wijnen, L. (2023). The growth of longitudinal social network analysis: A review of the key data sets and topics in research on child and adolescent development. Teen Friendsh. Netw. Dev. Risky Behav., 326\u2013352.","DOI":"10.1093\/oso\/9780197602317.003.0014"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1090","DOI":"10.1198\/016214502388618906","article-title":"Latent space approaches to social network analysis","volume":"97","author":"Hoff","year":"2002","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1646","DOI":"10.1080\/01621459.2014.988214","article-title":"Latent space models for dynamic networks","volume":"110","author":"Sewell","year":"2015","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v083.i06","article-title":"Temporal exponential random graph models with btergm: Estimation and bootstrap confidence intervals","volume":"83","author":"Leifeld","year":"2018","journal-title":"J. Stat. Softw."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1146\/annurev-statistics-060116-054035","article-title":"Stochastic actor-oriented models for network dynamics","volume":"4","author":"Snijders","year":"2017","journal-title":"Annu. Rev. Stat. Its Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1017\/nws.2018.26","article-title":"A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model","volume":"7","author":"Leifeld","year":"2019","journal-title":"Netw. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1111\/rssb.12200","article-title":"Statistical clustering of temporal networks through a dynamic stochastic block model","volume":"79","author":"Matias","year":"2017","journal-title":"J. R. Stat. Soc. Ser. Stat. Methodol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1111\/ajps.12263","article-title":"Navigating the range of statistical tools for inferential network analysis","volume":"61","author":"Cranmer","year":"2017","journal-title":"Am. J. Political Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.socnet.2021.07.003","article-title":"Friendship network formation in Chinese middle schools: Patterns of inequality and homophily","volume":"68","author":"An","year":"2022","journal-title":"Soc. Netw."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1017\/nws.2022.24","article-title":"Strong and weak tie homophily in adolescent friendship networks: An analysis of same-race and same-gender ties","volume":"10","author":"McMillan","year":"2022","journal-title":"Netw. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"100256","DOI":"10.1016\/j.jmh.2024.100256","article-title":"Does network homophily persist in multicultural volunteering programs? Results from an Exponential Random Graph Model","volume":"10","author":"Cao","year":"2024","journal-title":"J. Migr. Health"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.indmarman.2024.02.008","article-title":"Evolution of structural properties of the global strategic emerging industries\u2019 trade network and its determinants: An TERGM analysis","volume":"118","author":"Wang","year":"2024","journal-title":"Ind. Mark. Manag."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1111\/jopy.12585","article-title":"Making and maintaining relationships through the prism of the dark triad traits: A longitudinal social network study","volume":"89","author":"Rogoza","year":"2021","journal-title":"J. Personal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1177\/1094428118825300","article-title":"Stochastic actor-oriented models for the co-evolution of networks and behavior: An introduction and tutorial","volume":"23","author":"Kalish","year":"2020","journal-title":"Organ. Res. Methods"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"e408","DOI":"10.1002\/sta4.408","article-title":"Stochastic actor-oriented modelling of the impact of COVID-19 on financial network evolution","volume":"10","author":"Chu","year":"2021","journal-title":"Stat"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1068","DOI":"10.1080\/01621459.2021.2024436","article-title":"Dynamic stochastic blockmodel regression for network data: Application to international militarized conflicts","volume":"117","author":"Olivella","year":"2022","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1111\/rssa.12708","article-title":"Analysis of longitudinal advice-seeking networks following implementation of high stakes testing","volume":"184","author":"Adhikari","year":"2021","journal-title":"J. R. Stat. Soc. Ser. Stat. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Loyal, J.D. (2024). Fast variational inference of latent space models for dynamic networks using Bayesian P-splines. arXiv.","DOI":"10.1214\/25-BA1545"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1080\/10705511.2023.2230519","article-title":"Bayesian Inference of Dynamic Mediation Models for Longitudinal Data","volume":"31","author":"Zhao","year":"2024","journal-title":"Struct. Equ. Model. Multidiscip. J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1007\/s11336-020-09736-z","article-title":"Social Network Mediation Analysis: A Latent Space Approach","volume":"86","author":"Liu","year":"2021","journal-title":"Psychometrika"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1145\/1117454.1117459","article-title":"Dynamic social network analysis using latent space models","volume":"7","author":"Sarkar","year":"2005","journal-title":"ACM SIGKDD Explor. Newsl."},{"key":"ref_32","unstructured":"R Core Team (2021). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1080\/00031305.2017.1375990","article-title":"Extending R with C++: A brief introduction to Rcpp","volume":"72","author":"Eddelbuettel","year":"2018","journal-title":"Am. Stat."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.1016\/j.csda.2013.02.005","article-title":"RcppArmadillo: Accelerating R with high-performance C++ linear algebra","volume":"71","author":"Eddelbuettel","year":"2014","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1207\/s15327752jpa6601_2","article-title":"UCLA Loneliness Scale (Version 3): Reliability, validity, and factor structure","volume":"66","author":"Russell","year":"1996","journal-title":"J. Personal. Assess."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1037\/1040-3590.18.2.192","article-title":"The mini-IPIP scales: Tiny-yet-effective measures of the Big Five factors of personality","volume":"18","author":"Donnellan","year":"2006","journal-title":"Psychol. Assess."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Geweke, J. (1991). Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments, Federal Reserve Bank of Minneapolis. Technical report.","DOI":"10.21034\/sr.148"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"47","DOI":"10.35566\/jbds\/v2n2\/p3","article-title":"The Performances of Gelman-Rubin and Geweke\u2019s Convergence Diagnostics of Monte Carlo Markov Chains in Bayesian Analysis","volume":"2","author":"Du","year":"2022","journal-title":"J. Behav. Data Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.socnet.2015.07.005","article-title":"Latent space models for dynamic networks with weighted edges","volume":"44","author":"Sewell","year":"2016","journal-title":"Soc. Netw."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1007\/s11336-023-09926-5","article-title":"Joint Latent Space Model for Social Networks with Multivariate Attributes","volume":"88","author":"Wang","year":"2023","journal-title":"Psychometrika"},{"key":"ref_41","first-page":"599","article-title":"Comparison of Methods for Imputing Social Network Data","volume":"21","author":"Xu","year":"2022","journal-title":"J. Data Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1214\/10-BJPS132","article-title":"A note on the robustness of a full Bayesian method for nonignorable missing data analysis","volume":"26","author":"Zhang","year":"2012","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1080\/10705511.2020.1721292","article-title":"Network Mediation Analysis Using Model-Based Eigenvalue Decomposition","volume":"28","author":"Che","year":"2020","journal-title":"Struct. Equ. Model. Multidiscip. J."},{"key":"ref_44","unstructured":"Li, D., Tan, S., Zhang, Y., Jin, M., Pan, S., Okumura, M., and Jiang, R. (2024). Dyg-mamba: Continuous state space modeling on dynamic graphs. arXiv."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/14\/2\/34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T05:16:01Z","timestamp":1770959761000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/14\/2\/34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,1]]},"references-count":44,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["computation14020034"],"URL":"https:\/\/doi.org\/10.3390\/computation14020034","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,1]]}}}