## Exponential Random Graph Models (ERGMs)

*14 Jul 2022 14:27*

See exponential families and network data analysis, naturally.

Around 2011, I set out to show that maximum likelihood estimation is
consistent for ERGMs, because they seemed like the best kind of network model
going. Doing so radically changed my perspective on these models; for
instance, I became convinced that maximum likelihood generally *isn't*
consistent, because these models aren't even *self*-consistent about
what happens as we see more data. But that story is a little long to tell
here; see here, or just read the
paper.

— Although
many of the relevant papers appear in the journal Social Networks,
published by Elsevier, a company known to also publish advertising disguised as
peer-reviewed scientific journals (e.g., The Australasian Journal of Bone
and Joint Medicine), I know of no *particular* reason to
believe that their findings are actually meretricious propaganda on behalf of a
paying client. It would, however, be better if the community would shift to a
journal whose publisher did not pollute the process of scientific communication
whenever it was profitable to do so.

- Recommended, over-views:
- Peter J. Carrington, John Scott and Stanley Wasserman (eds.), Models and Methods in Social Network Analysis [Many, but not all, of the papers are based on using ERGMs.]
- Steven M. Goodreau, James A. Kitts and Martina Morris,
"Birds of a Feather, Or Friend of a Friend?: Using Exponential Random Graph Models to Investigate Adolescent Social Networks", Demography
**46**(2009): 103--125 [In addition to the substantive findings, this is a great introduction to the approach.] - Mark S. Handcock, David R. Hunter, Carter T. Butts, Steven M. Goodreau, and Martina Morris (eds.), "Statistical Modeling of Social Networks with 'statnet'", special volume (24) of the Journal of Statistical Software (2008) [Introduction to a whole issue on the ERGM approach.]

- Recommended, close-ups:
- Shankar Bhamidi, Guy Bresler and Allan Sly, "Mixing time of exponential random graphs", Annals of Applied Probability
**21**(2011): 2146--2170, arxiv:0812.2265 [Extended abstract in FOCS 2008 conference proceedings] - Arun Chandrasekhar, Matthew O. Jackson, "Tractable and Consistent Random Graph Models", arxiv:1210.7375
- Sourav Chatterjee and Persi Diaconis, "Estimating and Understanding Exponential Random Graph Models", arxiv:1102.2650 [Results on conditions under which the mean field approximation for ERGMs becomes exact, and they consequently grow indistinguishable from simple Erdos-Renyi models.]
- Katherine Faust and John Skvoretz, "Comparing Networks Across Space
and Time, Size and
Species", Sociological
Methodology
**32**(2002): 267--299 [Though see my paper with Alessandro Rinaldo] - Stephen E. Fienberg, Alessandro Rinaldo and Yi Zhou, "On the Geometry of Discrete Exponential Families with Applications to Exponential Random Graph Models", arxiv:0901.0026
- Diego Garlaschelli and Maria I. Loffredo, "Maximum likelihood:
extracting unbiased information from complex
networks", cond-mat/0609015
[This is a much-needed corrective to the physics literature, but it makes it
sound as though exponential families of random graphs were invented in 2004,
and they're the first ones to apply maximum likelihood to network analysis. No
doubt these are inadvertent lapses. Definitely worth reading as a first
glimpse of how to do parameter estimation
*correctly*. Thanks to Dave Feldman for pointing it out to me.] - Krista Gile and Mark S. Handcock, "Model-based Assessment of the Impact of Missing Data on Inference for Networks" [Working Paper 66, Center for Statistics and the Social Sciences, University of Washington (2006). PDF preprint.]
- Mark S. Handcock, "Assessing degeneracy in statistical models of social networks", CSSS working paper 39 (2003)
- Mark S. Handcock and Krista J. Gile, "Modeling social networks from
sampled
data", Annals
of Applied Statistics
**4**(2010): 5--25, arxiv:1010.0891 - Steve Hanneke and Eric Xing, "Discrete Temporal Models for Social Networks", in Airoldi et al. (eds.) above [Extending exponential-family random graph models to dynamic networks. Makes me extra proud to have taught Steve stochastic processes. PDF preprint]
- Steve Hanneke, Wenjie Fu, and Eric P. Xing, "Discrete temporal models of social networks", Electronic Journal of Statistics
**4**(2010): 585--605 - David R. Hunter
and Mark S. Handcock, "Inference in curved exponential family models for
networks", Journal of Computational and Graphical Statistics
**15**(2006): 565--583 [PDF reprint] - Eric D. Kolaczyk and Pavel N. Krivitsky, "On the question of effective sample size in network modeling", arxiv:1112.0840
- Pavel N. Krivitsky, "Exponential-Family Random Graph Models for
Valued
Networks", Electronic
Journal of Statistics
**6**(2012): 1100--1128, arxiv:1101.1359 - Pavel N. Krivitsky, Mark S. Handcock and Martina Morris, "Adjusting
for Network Size and Composition Effects in Exponential-Family Random Graph
Models", Statistical
Methodology
**8**(2011): 319--339, arxiv:1004.5328 - John Levi Martin, "Comment on 'Geodesic Cycle Length Distributions in Delusional and Other Social Networks' ", Journal of Social Structure
**21**(2020): 77--93 - Juyong Park and M. E. J. Newman
- "The Statistical Mechanics of Networks", Physical
Review E
**70**(2004): 066117, arxiv:cond-mat/0405566 [I particularly like the way diagrammatic perturbation theory is introduced] - "Solution of the 2-star model of a network",
Physical Review E
**70**(2004): 066146, arxiv:cond-mat/0405457 - "Solution for the properties of a clustered network",
Physical Review E
**72**(2006): 026136, arxiv:cond-mat/0412579

- "The Statistical Mechanics of Networks", Physical
Review E
- Garry Robins, Tom Snijders, Peng Wang, Mark Handcock and Philippa
Pattison, "Recent developments in exponential random graph (p*) models for
social networks", Social Networks
**29**(2007): 192--215 [PDF reprint via Prof. Snijders] - Michael Schweinberberg, "Instability, Sensitivity, and Degeneracy
in Discrete Exponential
Families", Journal
of the American Statistical Association
**forthcoming**[Tech Report 10-07, Penn State, PDF (oddly, a scan of a LaTeX-produced paper] - Tom A. B. Snijders, "Conditional Marginalization for Exponential Random Graph Models", Journal of Mathematical Sociology
**34**(2010): 239--252 [Reprint] - Lin Yuan, Sergey Kirshner, Robert Givan, "Estimating Densities with Non-Parametric Exponential Families", arxiv:1206.5036

- Modesty forbids me to recommend:
- CRS and Alessandro
Rinaldo, "Consistency under Sampling of Exponential Random Graph
Models", Annals of
Statistics
**41**(2013): 508--535, arxiv:1111.3054 [More]

- To read:
- David Aristoff, Charles Radin, "Emergent structures in large networks", arxiv:1110.1912
- James Atwood, Don Towsley, Krista Gile, David Jensen, "Learning to Generate Networks", arxiv:1405.5868
- Ginestra Bianconi, "Information theory of spatial network ensembles", arxiv:2206.05614
- A. Caimo, N. Friel, "Bergm: Bayesian Exponential Random Graphs in R", arxiv:1201.2770
- Giulio Cimini, Tiziano Squartini, Fabio Saracco, Diego Garlaschelli, Andrea Gabrielli, Guido Caldarelli, "The Statistical Physics of Real-World Networks", arxiv:1810.05095
- Frank den Hollander, Maarten Markering, "Breaking of ensemble equivalence for dense random graphs under a single constraint", arxiv:2107.04351
- Alexander Engstrom, Patrik Noren, "Polytopes from Subgraph Statistics", arxiv:1011.3552
- Richard G. Everitt, "Bayesian Parameter Estimation for Latent Markov Random Fields and Social Networks", Journal of Computational and Graphical Statistic
**21**(2012): 940--960 - Andreas Flache and Tobias Stark, "Preference or opportunity? Why do we find more friendship segregation in more heterogeneous schools?", arxiv:0901.2825
- Agata Fronczak, "Exponential random graph models", arxiv:1210.7828
- Shweta Gaonkar, Angelo Mele, "A model of inter-organizational network formation", arxiv:2105.00458
- Neha Gondal, "The local and global structure of knowledge production in an emergent research field: An exponential random graph analysis", Social Networks forthcoming (2011)
- Ruth M. Hummel, David R. Hunter and Mark S. Handcock, "Improving
Simulation-Based Algorithms for Fitting ERGMs",
Journal of
Computational and Graphical Statistics
**21**(2012): 920--939 - David R. Hunter, Pavel N. Krivitsky and Michael Schweinberger,
"Computational Statistical Methods for Social Network Models",
Journal of Computational and Graphical Statistics
**21**(2012): 856--882 - Pavel N. Krivitsky and Mark S. Handcock, "A Separable Model for Dynamic Networks", arxiv:1011.1937
- Dean Lusher, Johan Koskinen, and Garry Robins (eds.), Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications
- Sumit Mukherjee, "Consistent estimation in the two star Exponential Random Graph Model", arxiv:1310.4526
- Saisuke Okabayashi and Charles J. Geyer, "Long range search for
maximum likelihood in exponential
families", Electronic
Journal of Statistics
**6**(2012): 123--147 - Philippa E. Pattison, Garry L. Robins, Tom A.B. Snijders, Peng Wang, "Conditional estimation of exponential random graph models from snowball sampling designs", Journal of Mathematical Psychology forthcoming (2013)
- Tiago P. Peixoto, "The entropy of stochastic blockmodel ensembles", Physical Review E
**85**(2012): 056122, arxiv:1112.6028 - Wen Pu, Jaesik Choi, Yunseong Hwang and Eyal Amir, "A Deterministic Partition Function Approximation for Exponential Random Graph Models" [PDF]
- Charles Radin, Mei Yin, "Phase transitions in exponential random graphs", arxiv:1108.0649
- Michael Schweinberger, "Statistical modelling of network panel data: Goodness of fit", British Journal of Mathematical and Statistical Psychology
**65**(2012): 263--281 - Sean L. Simpson, Satoru Hayasaka, Paul J. Laurienti, "Selecting an exponential random graph model for complex brain networks", arxiv:1007.3230
- Jeffrey A. Smith, "Macrostructure from Microstructure: Generating Whole Systems from Ego Networks", Sociological Methodology
**42**(2012): 155--205 - Thomas Suesse, "Marginalized Exponential Random Graph Models",
Journal of Computational and Graphical Statistics
**21**(2012): 883--900 - Mei Yin, "Critical phenomena in exponential random graphs", arxiv:1208.2992