## Exponential random graph models with R

This note documents the small but growing microverse of R packages on CRAN to produce various forms of exponential random graph models (ERGMs), which are a kind of modelling strategy akin to logistic regression for dyadic data.

## The starting point: `ergm`

The gravitational centre of the ERGM microverse is the `ergm`

package, by Handcock *et al.* The package is part of the `statnet`

suite of software packages, and is well documented through articles primarily published in *Social Networks* (for the theoretical explanation of how ERGMs operate) and in the *Journal of Statistical Software* (for the R implementation of the models).

Cosma Shalizi has compiled a nicely organised list of references on ERGMs, which includes the *JSS* special issue that introduced me to the topic. As Shalizi notes, another very recommended reading on the topic is the classic “Birds of a Feather” paper published in *Demography*, which introduces ERGMs through an excellent empirical example that clearly explains how homophily works.

As far as ERGM-related blog posts go, the best read I have stumbled upon so far is Alex Hanna's “Lessons on exponential random graph modeling”, which is based on an accessible and fun example. However, blogs are not the best knowledge source on ERGMs: to get precise answers to precise modelling questions, users should turn to the statnet mailing-list.

## Generalizing the models

The “dependent variable” of an ERGM is a binary network. When the network is weighted, the modelling strategy needs to be amended in several ways, since both the distribution of reference and the model terms are going to change to take edge values into account.

There are currently two packages to generalize ERGMs: the `ergm.count`

package, by Pavel N. Krivitsky and others, which is well documented in a Sunbelt tutorial, and the very recent `GERGM`

package, by Matthew D. Denny and others, which is well illustrated in the README of its GitHub repository.

## Extending the model terms

The `ergm`

package implements an amazingly long list of ERGM terms, but that is not the end of the story: additional terms can be added through the `ergm.userterms`

package, such as these two terms to model local cyclic and transitive triples, or these four terms to model graphlets.

The possibility to extend ERGM terms calls for an additional remark: the term/parameter space of ERGMs is absolutely *huge* and is quickly expanding. If you are used to building regression models from a limited number of variables and a few sensible interactions between them, get ready for a totally different modelling experience.

In particular, some of the terms that control for structural effects in ERGMs are highly sensitive to their internal parameters, such as the decay parameter of geometrically-weighted distribution terms. As a consequence, it takes ages to calibrate an ERGM, often by testing all possible values of the parameters over small increments. The literature contains many illustrations of that strategy (here's one; see the footnote at page 16).

## Extending the Bayesian logic

Another related remark about ERGMs is that their average estimation time has nothing in common with the average estimation time of (lowly hierarchical) regression models. If the network(s) that you want to feed to your model(s) contain(s) many nodes, think “days” instead of “seconds” when planning execution time.

The explanation is that ERGMs rely on MCMC estimation, which can take very long to converge, without the user being able to determine in advance exactly how long. This feature of ERGMs severely constrains their computational tractability, with runtimes of several hours or even days when estimating ERGMs on networks with, say, 200+ nodes.

For a complete Bayesian framework to use with ERGMs, users can turn to the `Bergm`

package, which Alberto Caimo and Nial Friel have carefully documented in three separate papers. Their work adds sampling from the posterior distribution (and much more) to the ERGM logic, in order to turn it into a fully Bayesian modelling strategy.

## Extending the model strategies

There are many more ways to extend ERGMs through R packages:

- Bootstrapped and temporal ERGMs can be modelled with the
`btergm`

,`tergm`

,`xergm`

and`xergm.common`

packages. - The
`blkergm`

package allows to fit “block ERGMs” that take the block structure of the networks into account, and the`hergm`

package implements “hierarchical” ERGMs.

I intend to better document these last strategies, as well as the other ones presented in this note, as soon as I find the time to learn more about them. For now, I will close this note by citing a forthcoming review article that will undoubtedly mention ERGMs, “Navigating the Range of Statistical Tools for Inferential Network Analysis”, by Skyler Cranmer and others, which is to be published in the *American Journal of Political Science*.

- First published on February 6th, 2016