R: MCMC Hints for Exponential-Family Random Graph Models

ergmHint {ergm} | R Documentation

## MCMC Hints for Exponential-Family Random Graph Models

### Description

This page describes how to provide to the ergm's MCMC algorithms information about the sample space.

### “Hints” for MCMC

In an exponential-family random graph model (ERGM), the probability or density of a given network, y in sample space Y, on a set of nodes is

h(y) exp{eta(theta).g(y)} / c(theta),

where h(y) is the reference distribution (particularly for valued network models), g(y) is a vector of network statistics for y, η(θ) is a natural parameter vector of the same length (with eta(theta)=theta for most terms), . is the dot product, and c(theta) is the normalizing constant for the distribution. A complete ERGM specification requires a list of network statistics g(y) and (if applicable) their η(θ) mappings provided by a formula of `ergmTerm`s; and, optionally, sample space Y and reference distribution h(y) information provided by `ergmConstraint`s and, for valued ERGMs, by `ergmReference`s.

It is often the case that there is additional information available about the distribution of networks being modelled. For example, you may be aware that the network is sparse or that there are strata among the dyads. “Hints”, typically passed on the right-hand side of `MCMC.prop` and `obs.MCMC.prop` arguments to `control.ergm()`, `control.simulate.ergm()`, and others, allow this information to be provided. By default, hint `sparse` is in effect.

Unlike constraints, model terms, and reference distributions, “hints” do not affect the specification of the model. That is, regardless of what “hints” may or may not be in effect, the sample space and the probabilities within it are the same. However, “hints” may affect the MCMC proposal distribution used by the samplers.

Note that not all proposals support all “hints”: and if the most suitable proposal available cannot incorporate a particular “hint”, a warning message will be printed.

“Hints” use the same underlying API as constraints, and, if present, `%ergmlhs%` attributes `constraints` and `constraints.obs` will be substituted in its place.

### Hints available to the package

The following hints are known to ergm at this time:

### References

- Goodreau SM, Handcock MS, Hunter DR, Butts CT, Morris M (2008a). A statnet Tutorial. Journal of Statistical Software, 24(8). doi: 10.18637/jss.v024.i08

- Hunter, D. R. and Handcock, M. S. (2006) Inference in curved exponential family models for networks, Journal of Computational and Graphical Statistics.

- Hunter DR, Handcock MS, Butts CT, Goodreau SM, Morris M (2008b). ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks. Journal of Statistical Software, 24(3). doi: 10.18637/jss.v024.i03

- Karwa V, Krivitsky PN, and Slavkovi\'c AB (2016). Sharing Social Network Data: Differentially Private Estimation of Exponential-Family Random Graph Models. Journal of the Royal Statistical Society, Series C, 66(3): 481-500. doi: 10.1111/rssc.12185

- Krivitsky PN (2012). Exponential-Family Random Graph Models for Valued Networks. Electronic Journal of Statistics, 6, 1100-1128. doi: 10.1214/12-EJS696

- Morris M, Handcock MS, Hunter DR (2008). Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects. Journal of Statistical Software, 24(4). doi: 10.18637/jss.v024.i04

[Package ergm version 4.2.1 Index]