Exponential random graph models

Exponential random graph models (ERGMs) are a family of statistical models for analyzing data about social and other networks.

Background

Many metrics exist to describe the structural features of an observed network such as the density, centrality, or assortativity.^[1][2] However, these metrics describe the observed network which is only one instance of a large number of possible alternative networks. This set of alternative networks may have similar or dissimilar structural features. To support statistical inference on the processes influencing the formation of network structure, a statistical model should consider the set of all possible alternative networks weighted on their similarity to an observed network. However because network data is inherently relational, it violates the assumptions of independence and identical distribution of standard statistical models like linear regression.^[3] Alternative statistical models should reflect the uncertainty associated with a given observation, permit inference about the relative frequency about network substructures of theoretical interest, disambiguating the influence of confounding processes, efficiently representing complex structures, and linking local-level processes to global-level properties.^[4] Degree Preserving Randomization, for example, is a specific way in which an observed network could be considered in terms of multiple alternative networks.

Definition

The Exponential family is a broad family of models for covering many types of data, not just networks. An ERGM is a model from this family which describes networks.

Formally a random graph $Y$ consists of a set of $n$ nodes and $m$ dyads (edges) $\{Y_{{ij}}:i=1,\dots ,n;j=1,\dots ,n\}$ where $Y_{{ij}}=1$ if the nodes $(i,j)$ are connected and $Y_{{ij}}=0$ otherwise.

The basic assumption of these models is that the structure in an observed graph $y$ can be explained by any statistics $s(y)$ depending on the observed network and nodal attributes. This way, it is possible to describe any kind of dependence between the dyadic variables:

$P(Y=y|\theta )={\frac {\exp(\theta ^{{T}}s(y))}{c(\theta )}}$

where $\theta$ is a vector of model parameters associated with $s(y)$ and $c(\theta )$ is a normalising constant.

These models represent a probability distribution on each possible network on $n$ nodes. However, the size of the set of possible networks for an undirected network (simple graph) of size $n$ is $2^{{n(n-1)/2}}$. Because the number of possible networks in the set vastly outnumbers the number of parameters which can constrain the model, the ideal probability distribution is the one which maximizes the Gibbs entropy.^[5]

References

1. Wasserman, Stanley; Faust, Katherine (1994). Social Network Analysis: Methods and Applications. ISBN 978-0-521-38707-1.
2. Newman, M.E.J. "The Structure and Function of Complex Networks". SIAM Review. 45 (2): 167–256. doi:10.1137/S003614450342480.
3. Contractor, Noshir; Wasserman, Stanley; Faust, Katherine. "Testing Multitheoretical, Multilevel Hypotheses About Organizational Networks: An Analytic Framework and Empirical Example". Academy of Management Review. 31 (3): 681–703. doi:10.5465/AMR.2006.21318925.
4. Robins, G.; Pattison, P.; Kalish, Y.; Lusher, D. (2007). "An introduction to exponential random graph models for social networks". Social Networks. 29: 173–191. doi:10.1016/j.socnet.2006.08.002.
5. Newman, M.E.J. "Other Network Models". Networks. pp. 565–585. ISBN 978-0-19-920665-0.

Further reading

• Caimo, A.; Friel, N (2011). "Bayesian inference for exponential random graph models". Social Networks. 33: 41–55. doi:10.1016/j.socnet.2010.09.004.
• Erdős, P.; Rényi, A (1959). "On random graphs". Publicationes Mathematicae. 6: 290–297.
• Fienberg, S. E.; Wasserman, S. (1981). "Discussion of An Exponential Family of Probability Distributions for Directed Graphs by Holland and Leinhardt". Journal of the American Statistical Association. 76: 54–57. doi:10.1080/01621459.1981.10477600.
• Frank, O.; Strauss, D (1986). "Markov Graphs". Journal of the American Statistical Association. 81: 832–842. doi:10.2307/2289017.
• Handcock, M. S.; Hunter, D. R.; Butts, C. T.; Goodreau, S. M.; Morris, M. (2008). "statnet: Software Tools for the Representation, Visualization, Analysis and Simulation of Network Data". Journal of Statistical Software. 24: 1–11.
• Hunter, D. R.; Goodreau, S. M.; Handcock, M. S. (2008). "Goodness of Fit of Social Network Models". Journal of the American Statistical Association. 103: 248–258. doi:10.1198/016214507000000446.
• Hunter, D. R; Handcock, M. S. (2006). "Inference in curved exponential family models for networks". Journal of Computational and Graphical Statistics. 15: 565–583. doi:10.1198/106186006X133069.
• Hunter, D. R.; Handcock, M. S.; Butts, C. T.; Goodreau, S. M.; Morris, M. (2008). "ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks". Journal of Statistical Software. 24: 1–29.
• Jin, I.H.; Liang, F. (2012). "Fitting social networks models using varying truncation stochastic approximation MCMC algorithm". Journal of Computational and Graphical Statistics. doi:10.1080/10618600.2012.680851.
• Koskinen, J. H.; Robins, G. L.; Pattison, P. E. (2010). "Analysing exponential random graph (p-star) models with missing data using Bayesian data augmentation". Statistical Methodology. 7: 366–384. doi:10.1016/j.stamet.2009.09.007.
• Morris, M.; Handcock, M. S.; Hunter, D. R. (2008). "Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects". Journal of Statistical Software. 24.
• Rinaldo, A.; Fienberg, S. E.; Zhou, Y. (2009). "On the geometry of descrete exponential random families with application to exponential random graph models". Electronic Journal of Statistics. 3: 446–484. doi:10.1214/08-EJS350.
• Robins, G.; Snijders, T.; Wang, P.; Handcock, M.; Pattison, P (2007). "Recent developments in exponential random graph (p*) models for social networks". Social Networks. 29: 192–215. doi:10.1016/j.socnet.2006.08.003.
• Schweinberger, Michael (2011). "Instability, sensitivity, and degeneracy of discrete exponential families". Journal of the American Statistical Association. 106 (496): 1361-1370.
• Schweinberger, Michael; Handcock, Mark (2015). "Local dependence in random graph models: characterization, properties and statistical inference". Journal of the Royal Statistical Society, Series B. 77 (3): 647-676.
• Snijders, T. A. B. (2002). "Markov chain Monte Carlo estimation of exponential random graph models" (PDF). Journal of Social Structure. 3.
• Snijders, T. A. B.; Pattison, P. E.; Robins, G. L. (2006). "New specifications for exponential random graph models". Sociological Methodology. 36: 99–153. doi:10.1111/j.1467-9531.2006.00176.x.
• Strauss, D; Ikeda, M (1990). "Pseudolikelihood estimation for social networks". Journal of the American Statistical Association. 5: 204–212. doi:10.2307/2289546.
• van Duijn, M. A.; Snijders, T. A. B.; Zijlstra, B. H. (2004). "p2: a random effects model with covariates for directed graphs". Statistica Neerlandica. 58: 234–254. doi:10.1046/j.0039-0402.2003.00258.x.
• van Duijn, M. A. J.; Gile, K. J.; Handcock, M. S. (2009). "A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models". Social Networks. 31: 52–62. doi:10.1016/j.socnet.2008.10.003.