{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,17]],"date-time":"2026-05-17T04:06:21Z","timestamp":1778990781996,"version":"3.51.4"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,5,17]],"date-time":"2026-05-17T00:00:00Z","timestamp":1778976000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,5,17]],"date-time":"2026-05-17T00:00:00Z","timestamp":1778976000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"UKRI Medical Research Council","award":["MC_UU_00002\/2"],"award-info":[{"award-number":["MC_UU_00002\/2"]}]},{"DOI":"10.13039\/501100018956","name":"NIHR Cambridge Biomedical Research Centre","doi-asserted-by":"publisher","award":["NIHR203312"],"award-info":[{"award-number":["NIHR203312"]}],"id":[{"id":"10.13039\/501100018956","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2026,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Exponential random graph models (ERGMs) are flexible probabilistic frameworks to model statistical networks through a variety of network summary statistics. Conventional Bayesian estimation for ERGMs involves iteratively exchanging with an auxiliary variable due to the intractability of the ERGM likelihood. However, this approach has limited scalability in large-scale implementations. Neural posterior estimation (NPE) is a recent advancement in simulation-based inference, using a neural network-based density estimator to infer the posterior for models with doubly intractable likelihoods for which simulations can be generated. While NPE has been successfully adopted in various fields such as cosmology, little research has investigated its use for ERGMs. Performing NPE on ERGMs not only provides a different approach to estimation for intractable ERGM likelihoods but also allows more efficient and scalable inference using the amortisation properties of NPE, and therefore we investigate how NPE can be effectively implemented in ERGMs. In this study, we present the first systematic implementation of NPE for ERGMs, rigorously evaluating potential biases, interpreting the bias magnitudes, and assessing computational costs. We compare NPE fits with conventional Bayesian ERGM fits as well as related neural simulation-based methods, namely neural likelihood estimation and neural ratio estimation. In our synthetic data analysis, we show that training a neural posterior estimator on 500,000 simulations circumvents the roughly 4,000,000,000 simulations required by conventional exchange-algorithm inference, enabling real-time posterior estimation. More importantly, our work highlights ERGM-specific areas that may pose particular challenges for the adoption of NPE.<\/jats:p>","DOI":"10.1007\/s11222-026-10896-8","type":"journal-article","created":{"date-parts":[[2026,5,17]],"date-time":"2026-05-17T03:14:31Z","timestamp":1778987671000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neural posterior estimation on exponential random graph models: evaluating bias and implementation challenges"],"prefix":"10.1007","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-6039-2994","authenticated-orcid":false,"given":"Yefeng","family":"Fan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8642-7037","authenticated-orcid":false,"given":"Simon Richard","family":"White","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,5,17]]},"reference":[{"issue":"5","key":"10896_CR1","doi-asserted-by":"publisher","first-page":"3001633","DOI":"10.1371\/journal.pbio.3001633","volume":"20","author":"G Avecilla","year":"2022","unstructured":"Avecilla, G., Chuong, J.N., Li, F., Sherlock, G., Gresham, D., Ram, Y.: Neural networks enable efficient and accurate simulation-based inference of evolutionary parameters from adaptation dynamics. PLoS Biol. 20(5), 3001633 (2022)","journal-title":"PLoS Biol."},{"issue":"1","key":"10896_CR2","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.socnet.2010.09.004","volume":"33","author":"A Caimo","year":"2011","unstructured":"Caimo, A., Friel, N.: Bayesian inference for exponential random graph models. Social Networks 33(1), 41\u201355 (2011)","journal-title":"Social Networks"},{"key":"10896_CR3","unstructured":"Caimo, A., Friel, N.: Bergm: Bayesian exponential random graph models in r. arXiv preprint arXiv:1703.05144 (2017)"},{"key":"10896_CR4","unstructured":"Durkan, C., Bekasov, A., Murray, I., Papamakarios, G.: Neural spline flows. Advances in Neural Information Processing Systems 32 (2019)"},{"issue":"24","key":"10896_CR5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.127.241103","volume":"127","author":"M Dax","year":"2021","unstructured":"Dax, M., Green, S.R., Gair, J., Macke, J.H., Buonanno, A., Sch\u00f6lkopf, B.: Real-time gravitational wave science with neural posterior estimation. Phys. Rev. Lett. 127(24), 241103 (2021)","journal-title":"Phys. Rev. Lett."},{"key":"10896_CR6","doi-asserted-by":"publisher","first-page":"23135","DOI":"10.52202\/068431-1681","volume":"35","author":"M Deistler","year":"2022","unstructured":"Deistler, M., Goncalves, P.J., Macke, J.H.: Truncated proposals for scalable and hassle-free simulation-based inference. Adv. Neural. Inf. Process. Syst. 35, 23135\u201323149 (2022)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"key":"10896_CR7","unstructured":"Duan, Y., Li, X., Avestruz, C., Regier, J., Collaboration, L.D.E.S.: Neural Posterior Estimation for Cataloging Astronomical Images from the Legacy Survey of Space and Time (2025). arxiv.org\/abs\/2510.15315"},{"key":"10896_CR8","unstructured":"Fen, C.: Simulation-Based Estimation of General Structural Network Models (2024)"},{"key":"10896_CR9","doi-asserted-by":"publisher","first-page":"65074","DOI":"10.7554\/eLife.65074","volume":"10","author":"A Fengler","year":"2021","unstructured":"Fengler, A., Govindarajan, L.N., Chen, T., Frank, M.J.: Likelihood approximation networks (lans) for fast inference of simulation models in cognitive neuroscience. Elife 10, 65074 (2021)","journal-title":"Elife"},{"key":"10896_CR10","doi-asserted-by":"publisher","first-page":"56261","DOI":"10.7554\/eLife.56261","volume":"9","author":"PJ Gon\u00e7alves","year":"2020","unstructured":"Gon\u00e7alves, P.J., Lueckmann, J.-M., Deistler, M., Nonnenmacher, M., \u00d6cal, K., Bassetto, G., Chintaluri, C., Podlaski, W.F., Haddad, S.A., Vogels, T.P., et al.: Training deep neural density estimators to identify mechanistic models of neural dynamics. ELife 9, 56261 (2020)","journal-title":"ELife"},{"key":"10896_CR11","unstructured":"Greenberg, D., Nonnenmacher, M., Macke, J.: Automatic posterior transformation for likelihood-free inference. In: International Conference on Machine Learning, pp. 2404\u20132414 (2019). PMLR"},{"key":"10896_CR12","doi-asserted-by":"crossref","unstructured":"Harris, J.K.: An Introduction to Exponential Random Graph Modeling, vol. 173. Sage Publications, Thousand Oaks, CA (2013)","DOI":"10.4135\/9781452270135"},{"key":"10896_CR13","unstructured":"Hermans, J., Begy, V., Louppe, G.: Likelihood-free mcmc with amortized approximate ratio estimators. In: International Conference on Machine Learning, pp. 4239\u20134248 (2020). PMLR"},{"issue":"481","key":"10896_CR14","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1198\/016214507000000446","volume":"103","author":"DR Hunter","year":"2008","unstructured":"Hunter, D.R., Goodreau, S.M., Handcock, M.S.: Goodness of fit of social network models. J. Am. Stat. Assoc. 103(481), 248\u2013258 (2008)","journal-title":"J. Am. Stat. Assoc."},{"issue":"3","key":"10896_CR15","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1198\/106186006X133069","volume":"15","author":"DR Hunter","year":"2006","unstructured":"Hunter, D.R., Handcock, M.S.: Inference in curved exponential family models for networks. J. Comput. Graph. Stat. 15(3), 565\u2013583 (2006)","journal-title":"J. Comput. Graph. Stat."},{"issue":"3","key":"10896_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v024.i03","volume":"24","author":"DR Hunter","year":"2008","unstructured":"Hunter, D.R., Handcock, M.S., Butts, C.T., Goodreau, S.M., Morris, M.: ergm: A package to fit, simulate and diagnose exponential-family models for networks. J. Stat. Softw. 24(3), 1\u201329 (2008). https:\/\/doi.org\/10.18637\/jss.v024.i03","journal-title":"J. Stat. Softw."},{"key":"10896_CR17","unstructured":"Handcock, M.S., Hunter, D.R., Butts, C.T., Goodreau, S.M., Krivitsky, P.N., Morris, M.: Ergm: Fit, Simulate and Diagnose Exponential-Family Models for Networks. (2023). The Statnet Project. https:\/\/CRAN.R-project.org\/package=ergm"},{"issue":"4","key":"10896_CR18","doi-asserted-by":"publisher","first-page":"920","DOI":"10.1080\/10618600.2012.679224","volume":"21","author":"RM Hummel","year":"2012","unstructured":"Hummel, R.M., Hunter, D.R., Handcock, M.S.: Improving simulation-based algorithms for fitting ergms. J. Comput. Graph. Stat. 21(4), 920\u2013939 (2012)","journal-title":"J. Comput. Graph. Stat."},{"key":"10896_CR19","unstructured":"Handcock, M.S., Robins, G., Snijders, T., Moody, J., Besag, J.: Assessing degeneracy in statistical models of social networks. Technical report, Working paper (2003)"},{"key":"10896_CR20","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/j.socnet.2006.08.005","volume":"29","author":"D Hunter","year":"2007","unstructured":"Hunter, D.: Curved exponential family models for social networks. Social Networks 29, 216\u2013230 (2007). https:\/\/doi.org\/10.1016\/j.socnet.2006.08.005","journal-title":"Social Networks"},{"key":"10896_CR21","doi-asserted-by":"crossref","unstructured":"Karabatsos, G.: Copula approximate bayesian computation using distribution random forests. arXiv preprint arXiv:2402.18450 (2024)","DOI":"10.3390\/stats7030061"},{"issue":"6","key":"10896_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v105.i06","volume":"105","author":"PN Krivitsky","year":"2023","unstructured":"Krivitsky, P.N., Hunter, D.R., Morris, M., Klumb, C.: ergm 4: New features for analyzing exponential-family random graph models. J. Stat. Softw. 105(6), 1\u201344 (2023). https:\/\/doi.org\/10.18637\/jss.v105.i06","journal-title":"J. Stat. Softw."},{"key":"10896_CR23","unstructured":"Lazega, E.: The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership. OUP Oxford, Oxford (2001)"},{"key":"10896_CR24","doi-asserted-by":"crossref","unstructured":"Labb\u00e9, B., H\u00e9rault, R., Chatelain, C.: Learning deep neural networks for high dimensional output problems. In: 2009 International Conference on Machine Learning and Applications, pp. 63\u201368 (2009). IEEE","DOI":"10.1109\/ICMLA.2009.48"},{"key":"10896_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2020.117480","volume":"225","author":"B Lehmann","year":"2021","unstructured":"Lehmann, B., Henson, R., Geerligs, L., White, S., et al.: Characterising group-level brain connectivity: a framework using bayesian exponential random graph models. Neuroimage 225, 117480 (2021)","journal-title":"Neuroimage"},{"key":"10896_CR26","doi-asserted-by":"crossref","unstructured":"Lusher, D., Koskinen, J., Robins, G.: Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications. Cambridge University Press, Cambridge (2013)","DOI":"10.1017\/CBO9780511894701"},{"issue":"3","key":"10896_CR27","doi-asserted-by":"publisher","first-page":"825","DOI":"10.3982\/ECTA10400","volume":"85","author":"A Mele","year":"2017","unstructured":"Mele, A.: A structural model of dense network formation. Econometrica 85(3), 825\u2013850 (2017)","journal-title":"Econometrica"},{"key":"10896_CR28","doi-asserted-by":"crossref","unstructured":"Mele, A.: Estimating network models using neural networks. arXiv preprint arXiv:2502.01810 (2025)","DOI":"10.2139\/ssrn.5123084"},{"key":"10896_CR29","doi-asserted-by":"crossref","unstructured":"Macias, O., Mason, Z., Ho, M., Ferri\u00e8re, A., Benoit-L\u00e9vy, A., Tueros, M.: Simulation-Based Inference for Direction Reconstruction of Ultra-High-Energy Cosmic Rays with Radio Arrays (2025). arxiv.org\/abs\/2508.15991","DOI":"10.1103\/j77n-1pl3"},{"key":"10896_CR30","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.jsc.2013.09.003","volume":"60","author":"BD McKay","year":"2014","unstructured":"McKay, B.D., Piperno, A.: Practical graph isomorphism, ii. J. Symb. Comput. 60, 94\u2013112 (2014)","journal-title":"J. Symb. Comput."},{"key":"10896_CR31","unstructured":"Papamakarios, G., Murray, I.: Fast $$\\varepsilon $$-free inference of simulation models with bayesian conditional density estimation. Advances in Neural Information Processing Systems 29 (2016)"},{"key":"10896_CR32","unstructured":"Papamakarios, G., Pavlakou, T., Murray, I.: Masked autoregressive flow for density estimation. Advances in Neural Information Processing Systems 30 (2017)"},{"issue":"2","key":"10896_CR33","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1111\/insr.12522","volume":"91","author":"H Pesonen","year":"2023","unstructured":"Pesonen, H., Simola, U., K\u00f6hn-Luque, A., Vuollekoski, H., Lai, X., Frigessi, A., Kaski, S., Frazier, D.T., Maneesoonthorn, W., Martin, G.M., et al.: Abc of the future. Int. Stat. Rev. 91(2), 243\u2013268 (2023)","journal-title":"Int. Stat. Rev."},{"key":"10896_CR34","unstructured":"Papamakarios, G., Sterratt, D., Murray, I.: Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows. In: The 22nd International Conference on Artificial Intelligence and Statistics, pp. 837\u2013848 (2019). PMLR"},{"key":"10896_CR35","doi-asserted-by":"crossref","unstructured":"Robins, G.: Exponential random graph models for social networks. The Sage handbook of social network analysis, 484\u2013500 (2011)","DOI":"10.4135\/9781446294413.n32"},{"key":"10896_CR36","doi-asserted-by":"crossref","unstructured":"Rubin, D.B.: Bayesianly justifiable and relevant frequency calculations for the applied statistician. The Annals of Statistics, 1151\u20131172 (1984)","DOI":"10.1214\/aos\/1176346785"},{"issue":"5","key":"10896_CR37","doi-asserted-by":"publisher","first-page":"20039","DOI":"10.1371\/journal.pone.0020039","volume":"6","author":"SL Simpson","year":"2011","unstructured":"Simpson, S.L., Hayasaka, S., Laurienti, P.J.: Exponential random graph modeling for complex brain networks. PLoS ONE 6(5), 20039 (2011)","journal-title":"PLoS ONE"},{"issue":"1","key":"10896_CR38","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1111\/j.1467-9531.2006.00176.x","volume":"36","author":"TA Snijders","year":"2006","unstructured":"Snijders, T.A., Pattison, P.E., Robins, G.L., Handcock, M.S.: New specifications for exponential random graph models. Sociol. Methodol. 36(1), 99\u2013153 (2006)","journal-title":"Sociol. Methodol."},{"key":"10896_CR39","doi-asserted-by":"crossref","unstructured":"Shafto, M.A., Tyler, L.K., Dixon, M., Taylor, J.R., Rowe, J.B., Cusack, R., Calder, A.J., Marslen-Wilson, W.D., Duncan, J., Dalgleish, T., Henson, R.N., Brayne, C., Cam-CAN, Matthews, F.E.: The cambridge centre for ageing and neuroscience (cam-can) study protocol:a cross-sectional, lifespan, multidisciplinary examination of healthy cognitive ageing. BMC Neurology 14(1), 204 (2014)","DOI":"10.1186\/s12883-014-0204-1"},{"key":"10896_CR40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.socnet.2023.05.003","volume":"76","author":"P Tolochko","year":"2024","unstructured":"Tolochko, P., Boomgaarden, H.G.: Same but different: A comparison of estimation approaches for exponential random graph models for multiple networks. Social Networks 76, 1\u201311 (2024)","journal-title":"Social Networks"},{"key":"10896_CR41","doi-asserted-by":"publisher","unstructured":"Tejero-Cantero, A., Boelts, J., Deistler, M., Lueckmann, J.-M., Durkan, C., Gon\u00e7alves, P.J., Greenberg, D.S., Macke, J.H.: sbi: A toolkit for simulation-based inference. Journal of Open Source Software 5(52), 2505 (2020) https:\/\/doi.org\/10.21105\/joss.02505","DOI":"10.21105\/joss.02505"},{"issue":"3","key":"10896_CR42","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1007\/s10614-018-9853-2","volume":"54","author":"J Van der Pol","year":"2019","unstructured":"Van der Pol, J.: Introduction to network modeling using exponential random graph models (ergm): theory and an application using r-project. Comput. Econ. 54(3), 845\u2013875 (2019)","journal-title":"Comput. Econ."},{"key":"10896_CR43","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1051\/0004-6361\/202245263","volume":"672","author":"M Vasist","year":"2023","unstructured":"Vasist, M., Rozet, F., Absil, O., Molli\u00e8re, P., Nasedkin, E., Louppe, G.: Neural posterior estimation for exoplanetary atmospheric retrieval. Astronomy & Astrophysics 672, 147 (2023)","journal-title":"Astronomy & Astrophysics"},{"key":"10896_CR44","first-page":"33845","volume":"35","author":"D Ward","year":"2022","unstructured":"Ward, D., Cannon, P., Beaumont, M., Fasiolo, M., Schmon, S.: Robust neural posterior estimation and statistical model criticism. Adv. Neural. Inf. Process. Syst. 35, 33845\u201333859 (2022)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"key":"10896_CR45","unstructured":"Yin, F., Butts, C.T.: Kernel-based approximate bayesian inference for exponential family random graph models. arXiv preprint arXiv:2004.08064 (2020)"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-026-10896-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-026-10896-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-026-10896-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,17]],"date-time":"2026-05-17T03:14:37Z","timestamp":1778987677000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-026-10896-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,17]]},"references-count":45,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["10896"],"URL":"https:\/\/doi.org\/10.1007\/s11222-026-10896-8","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,17]]},"assertion":[{"value":"11 May 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"141"}}