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arxiv: 2605.25873 · v1 · pith:HORLKMEDnew · submitted 2026-05-25 · 📊 stat.ME · stat.CO

Bayesian perspectives on exponential random graph models

Alberto Caimo , Isabella Gollini This is my paper

Pith reviewed 2026-06-29 20:36 UTC · model grok-4.3

classification 📊 stat.ME stat.CO
keywords exponential random graph modelsBayesian inferencenetwork data analysisdoubly intractable posteriorMCMC methodsvariational inferencemodel selectionnetwork extensions
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The pith

Bayesian ERGMs enable principled uncertainty quantification and prior incorporation for network data despite doubly intractable posteriors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews Bayesian approaches to exponential random graph models for analyzing network data and testing structural mechanisms. These models support full probabilistic inference that quantifies uncertainty and incorporates prior knowledge. The central computational obstacle is the doubly intractable posterior caused by a likelihood normalising constant that depends on unknown parameters. The review organises existing solutions into auxiliary variable MCMC methods, adjusted pseudo-likelihood approaches, and variational methods, while also addressing model selection and extensions such as missing data, longitudinal dynamics, weighted networks, and populations of networks.

Core claim

Bayesian ERGMs provide principled uncertainty quantification and enable the incorporation of prior knowledge through fully probabilistic modelling. Computation remains challenging because the posterior is doubly intractable, with a likelihood normalising constant that depends on unknown parameters. This paper reviews Bayesian approaches to ERGM inference, categorising inference methods into three broad classes: auxiliary variable MCMC methods, adjusted pseudo-likelihood approaches, and variational methods, alongside dedicated treatment of model selection. It also discusses modelling extensions for missing data, longitudinal dynamics, populations of networks, weighted networks, highlighting a

What carries the argument

Doubly intractable posterior of Bayesian ERGMs, addressed by categorising inference into auxiliary variable MCMC methods, adjusted pseudo-likelihood approaches, and variational methods.

If this is right

  • Enables hypothesis testing on structural mechanisms in observed networks with full uncertainty quantification
  • Supports incorporation of prior knowledge into network models
  • Facilitates extensions to missing data, longitudinal dynamics, weighted networks, and populations of networks
  • Applies across multiple scientific disciplines through the reviewed computational strategies

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The three-class taxonomy could serve as a template for classifying inference methods in other network models that share similar intractability issues.
  • Extensions to longitudinal dynamics open the possibility of using these Bayesian methods for predictive modeling of network evolution over time.
  • Treating populations of networks suggests room for hierarchical Bayesian extensions that share parameters across multiple observed networks.
  • Model selection techniques reviewed here could be adapted to compare ERGMs against non-exponential alternatives in applied network studies.

Load-bearing premise

The three broad classes of methods together with the dedicated model selection treatment adequately capture the main computational strategies for Bayesian ERGM inference.

What would settle it

Identification of an efficient Bayesian ERGM inference technique that fits none of the three categorised classes would falsify the completeness of the review's taxonomy.

read the original abstract

Exponential random graph models (ERGMs) are a widely used framework for network data, enabling hypothesis testing on the structural mechanisms underlying observed networks. Bayesian ERGMs provide principled uncertainty quantification and enable the incorporation of prior knowledge through fully probabilistic modelling. However, computation remains challenging because the posterior is doubly intractable, with a likelihood normalising constant that depends on unknown parameters. This paper reviews Bayesian approaches to ERGM inference, categorising inference methods into three broad classes: auxiliary variable MCMC methods, adjusted pseudo-likelihood approaches, and variational methods, alongside dedicated treatment of model selection. We also discuss modelling extensions for missing data, longitudinal dynamics, populations of networks, weighted networks, highlighting applications across various scientific disciplines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript reviews Bayesian approaches to exponential random graph models (ERGMs), emphasizing the doubly intractable posterior arising from the parameter-dependent normalizing constant. It categorizes inference methods into three classes: auxiliary variable MCMC methods, adjusted pseudo-likelihood approaches, and variational methods, and includes dedicated sections on model selection as well as extensions for missing data, longitudinal dynamics, populations of networks, and weighted networks, with applications in various disciplines.

Significance. This review organizes the literature on Bayesian ERGM inference, which is valuable for providing principled uncertainty quantification and prior incorporation in network analysis despite computational challenges. By cataloguing existing methods without advancing new theorems or algorithms, its contribution lies in synthesis and guidance for practitioners and researchers facing doubly intractable problems.

minor comments (1)
  1. [Abstract] The abstract mentions 'highlighting applications across various scientific disciplines' but does not specify which disciplines or provide concrete examples; including at least one or two in the abstract would better convey the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The referee's description accurately reflects the paper's scope as a review of Bayesian methods for ERGMs, including the categorization of inference approaches and the discussion of extensions. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; literature review only

full rationale

The manuscript is a review paper that organises existing literature on Bayesian ERGM inference into three computational classes plus model selection. No original derivations, theorems, algorithms, or empirical claims are advanced; the text describes the doubly-intractable posterior and catalogues published remedies without presenting equations or fitted quantities that could reduce to self-definition or self-citation by construction. The central content is descriptive organisation of prior work, which is self-contained against external benchmarks and carries no load-bearing derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a review paper, the work introduces no free parameters, axioms, or invented entities; all content summarizes prior literature on ERGMs.

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discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 4 canonical work pages

  1. [1]

    Journal of the American Statistical Association81(395), 832–842 (1986)

    Frank, O., Strauss, D.: Markov graphs. Journal of the American Statistical Association81(395), 832–842 (1986)

    1986

  2. [2]

    Department of Statistics, Stockholm University research report2004, 2 (2004)

    Koskinen, J.: Bayesian analysis of exponential random graphs-estimation of parameters and model selection. Department of Statistics, Stockholm University research report2004, 2 (2004)

    2004

  3. [3]

    Journal of the American Statistical Association113(523), 1372–1390 (2018)

    Park, J., Haran, M.: Bayesian inference in the presence of intractable normalizing functions. Journal of the American Statistical Association113(523), 1372–1390 (2018)

    2018

  4. [4]

    Social Networks33(1), 41–55 (2011)

    Caimo, A., Friel, N.: Bayesian inference for exponential random graph models. Social Networks33(1), 41–55 (2011)

    2011

  5. [5]

    In: Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence (UAI-06)

    Murray, I., Ghahramani, Z., MacKay, D.: MCMC for doubly-intractable distribu- tions. In: Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence (UAI-06). AUAI Press, Arlington, Virginia (2006)

    2006

  6. [6]

    Journal of Computational and Graphical Statistics21(4), 940–960 (2012)

    Everitt, R.G.: Bayesian parameter estimation for latent Markov random fields and social networks. Journal of Computational and Graphical Statistics21(4), 940–960 (2012)

    2012

  7. [7]

    Statistics and Computing 25, 113–125 (2015)

    Caimo, A., Mira, A.: Efficient computational strategies for doubly intractable problems with applications to Bayesian social networks. Statistics and Computing 25, 113–125 (2015)

    2015

  8. [8]

    Statistics and Computing 26(1), 29–47 (2016)

    Alquier, P., Friel, N., Everitt, R., Boland, A.: Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels. Statistics and Computing 26(1), 29–47 (2016)

    2016

  9. [9]

    Journal of the American Statistical Association111(513), 377–393 (2016)

    Liang, F., Jin, I.H., Song, Q., Liu, J.S.: An adaptive exchange algorithm for sampling from distributions with intractable normalizing constants. Journal of the American Statistical Association111(513), 377–393 (2016)

    2016

  10. [10]

    Neural Computation 25(8), 2199–2234 (2013)

    Liang, F., Jin, I.-H.: A Monte Carlo Metropolis-Hastings algorithm for sampling from distributions with intractable normalizing constants. Neural Computation 25(8), 2199–2234 (2013)

    2013

  11. [11]

    Bayesian Analysis19(2), 13 595 (2024)

    Zhang, Q., Liang, F.: Bayesian analysis of exponential random graph models using stochastic gradient Markov chain Monte Carlo. Bayesian Analysis19(2), 13 595 (2024)

    2024

  12. [12]

    In: Proceedings of the 28th International Conference on Machine Learning (ICML-11), pp

    Welling, M., Teh, Y.W.: Bayesian learning via stochastic gradient Langevin dynamics. In: Proceedings of the 28th International Conference on Machine Learning (ICML-11), pp. 681–688 (2011)

    2011

  13. [13]

    Social Networks50, 98–108 (2017)

    Bouranis, L., Friel, N., Maire, F.: Efficient Bayesian inference for exponential random graph models by correcting the pseudo-posterior distribution. Social Networks50, 98–108 (2017)

    2017

  14. [14]

    Journal of the Royal Statistical Society, Series B54, 657–699 (1992)

    Geyer, C.J., Thompson, E.A.: Constrained Monte Carlo maximum likelihood for dependent data (with discussion). Journal of the Royal Statistical Society, Series B54, 657–699 (1992)

    1992

  15. [15]

    Journal of Computational and Graphical Statistics29(4), 910–928 (2020)

    Tan, L.S., Friel, N.: Bayesian variational inference for exponential random graph models. Journal of Computational and Graphical Statistics29(4), 910–928 (2020)

    2020

  16. [16]

    Social Networks35(1), 11–24 (2013)

    Caimo, A., Friel, N.: Bayesian model selection for exponential random graph models. Social Networks35(1), 11–24 (2013)

    2013

  17. [17]

    Biometrika82, 711–732 (1995)

    Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika82, 711–732 (1995)

    1995

  18. [18]

    Journal of Computational and Graphical Statistics 1–13 (2018)

    Bouranis, L., Friel, N., Maire, F.: Bayesian model selection for exponential ran- dom graph models via adjusted pseudo-likelihoods. Journal of Computational and Graphical Statistics 1–13 (2018)

    2018

  19. [19]

    Journal of Computational and Graphical Statistics31(2), 360–377 (2022)

    Park, J., Jeon, Y., Shin, M., Jeon, M., Jin, I.H.: Bayesian shrinkage for functional network models, with applications to longitudinal item response data. Journal of Computational and Graphical Statistics31(2), 360–377 (2022)

    2022

  20. [20]

    Statistical Methodology7(3), 366–384 (2010)

    Koskinen, J.H., Robins, G.L., Pattison, P.E.: Analysing exponential random graph (p-star) models with missing data using Bayesian data augmentation. Statistical Methodology7(3), 366–384 (2010)

    2010

  21. [21]

    Biometrika63(3), 581–592 (1976)

    Rubin, D.B.: Inference and missing data. Biometrika63(3), 581–592 (1976)

    1976

  22. [22]

    Social Networks 35(4), 514–527 (2013)

    Koskinen, J.H., Robins, G.L., Wang, P., Pattison, P.E.: Bayesian analysis for par- tially observed network data, missing ties, attributes and actors. Social Networks 35(4), 514–527 (2013)

    2013

  23. [23]

    Social Networks46, 11–28 (2016)

    Thiemichen, S., Friel, N., Caimo, A., Kauermann, G.: Bayesian exponential random graph models with nodal random effects. Social Networks46, 11–28 (2016)

    2016

  24. [24]

    Statistica Neerlandica58, 234–254 (2004) 14

    Duijn, M.A., Snijders, T.A.B., Zijlstra, B.H.: p2: A random effects model with covariates for directed graphs. Statistica Neerlandica58, 234–254 (2004) 14

    2004

  25. [25]

    Political Analysis26(1), 3–19 (2018)

    Box-Steffensmeier, J.M., Christenson, D.P., Morgan, J.W.: Modeling unobserved heterogeneity in social networks with the frailty exponential random graph model. Political Analysis26(1), 3–19 (2018)

    2018

  26. [26]

    Journal of the Royal Statistical Society Series B: Statistical Methodology77(3), 647–676 (2015)

    Schweinberger, M., Handcock, M.S.: Local dependence in random graph mod- els: Characterization, properties and statistical inference. Journal of the Royal Statistical Society Series B: Statistical Methodology77(3), 647–676 (2015)

    2015

  27. [27]

    Social Networks44, 334–345 (2016)

    Slaughter, A.J., Koehly, L.M.: Multilevel models for social networks: Hierarchical bayesian approaches to exponential random graph modeling. Social Networks44, 334–345 (2016)

    2016

  28. [28]

    NeuroImage225, 117480 (2021)

    Lehmann, B., Henson, R., Geerligs, L., White, S.: Characterising group-level brain connectivity: A framework using bayesian exponential random graph models. NeuroImage225, 117480 (2021)

    2021

  29. [29]

    Statistics and Computing34(4), 136 (2024)

    Lehmann, B., White, S.: A Bayesian multilevel model for populations of networks using exponential-family random graphs. Statistics and Computing34(4), 136 (2024)

    2024

  30. [30]

    Bayesian Analysis17(4), 1153–1191 (2022)

    Yin, F., Shen, W., Butts, C.T.: Finite Mixtures of ERGMs for Modeling Ensembles of Networks. Bayesian Analysis17(4), 1153–1191 (2022)

    2022

  31. [31]

    Social Networks74, 156–165 (2023)

    Ren, S., Wang, X., Liu, P., Zhang, J.: Bayesian nonparametric mixtures of expo- nential random graph models for ensembles of networks. Social Networks74, 156–165 (2023)

    2023

  32. [32]

    Plos one17(8), 0273039 (2022)

    Yin, F., Butts, C.T.: Highly scalable maximum likelihood and conjugate Bayesian inference for ERGMs on graph sets with equivalent vertices. Plos one17(8), 0273039 (2022)

    2022

  33. [33]

    Computational Statistics & Data Analysis142, 106825 (2020)

    Caimo, A., Gollini, I.: A multilayer exponential random graph modelling approach for weighted networks. Computational Statistics & Data Analysis142, 106825 (2020)

    2020

  34. [34]

    Statistics in Medicine43(20), 3881–3898 (2024)

    Fan, Y., White, S.R.: Review of weighted exponential random graph models frameworks applied to neuroimaging. Statistics in Medicine43(20), 3881–3898 (2024)

    2024

  35. [35]

    Journal of statistical physics151, 523–548 (2013)

    Koskinen, J., Lomi, A.: The local structure of globalization: The network dynam- ics of foreign direct investments in the international electricity industry. Journal of statistical physics151, 523–548 (2013)

    2013

  36. [36]

    Network Science3(1), 58–77 (2015) 15

    Koskinen, J., Caimo, A., Lomi, A.: Simultaneous modeling of initial conditions and time heterogeneity in dynamic networks: An application to foreign direct investments. Network Science3(1), 58–77 (2015) 15

    2015

  37. [37]

    Journal of Statistical Software105(6), 1–44 (2023)

    Krivitsky, P.N., Hunter, D.R., Morris, M., Klumb, C.: ergm 4: New features for analyzing exponential-family random graph models. Journal of Statistical Software105(6), 1–44 (2023)

    2023

  38. [38]

    Journal of Statistical Software24(1), 1548 (2008)

    Handcock, M.S., Hunter, D.R., Butts, C.T., Goodreau, S.M., Morris, M.: statnet: Software tools for the representation, visualization, analysis and simulation of network data. Journal of Statistical Software24(1), 1548 (2008)

    2008

  39. [39]

    Journal of Statistical Software61(2), 1–25 (2014)

    Caimo, A., Friel, N.: Bergm: Bayesian exponential random graphs in R. Journal of Statistical Software61(2), 1–25 (2014)

    2014

  40. [40]

    Journal of Statistical Software85, 1–39 (2018)

    Schweinberger, M., Luna, P.: Hergm: Hierarchical exponential-family random graph models. Journal of Statistical Software85, 1–39 (2018)

    2018

  41. [41]

    Melbourne School of Psychological Sciences, The University of Melbourne (2009)

    Wang, P., Robins, G., Pattison, P., Koskinen, J.: Pnet: A program for the simu- lation and estimation of exponential random graph models. Melbourne School of Psychological Sciences, The University of Melbourne (2009)

    2009

  42. [42]

    https://arxiv

    Fan, Y., White, S.R.: Neural Posterior Estimation on Exponential Random Graph Models: Evaluating Bias and Implementation Challenges (2025). https://arxiv. org/abs/2504.09349

    work page arXiv 2025

  43. [43]

    https://arxiv.org/abs/2506.04558

    Fan, Y., White, S.R.: A Scalable Exponential Random Graph Model: Amor- tised Hierarchical Sequential Neural Posterior Estimation with Applications in Neuroscience (2025). https://arxiv.org/abs/2506.04558

    work page arXiv 2025

  44. [44]

    https:// arxiv.org/abs/2502.01810

    Mele, A.: Estimating Network Models using Neural Networks (2025). https:// arxiv.org/abs/2502.01810

    work page arXiv 2025

  45. [45]

    https://dx.doi.org/10.1093/jrsssa/ qnag053 16

    Caimo, A., Gollini, I.: Separable Models for Dynamic Signed Networks Journal of the Royal Statistical Society Series A (2026). https://dx.doi.org/10.1093/jrsssa/ qnag053 16

    work page doi:10.1093/jrsssa/ 2026