R: Sample Space Constraints for Exponential-Family Random Graph...

ergmConstraint {ergm} | R Documentation

## Sample Space Constraints for Exponential-Family Random Graph Models

### Description

This page describes how to specify the constraints on the network sample space (the set of possible networks Y, the set of networks y for which h(y)>0) and sometimes the baseline weights h(y) to functions in the `ergm` package. It also provides an indexed list of the constraints visible to the ergm's API.

### Specifying constraints

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. Constraints typically affect Y, or, equivalently, set h(y)=0 for some y, but some (“soft” constraints) set h(y) to values other than 0 and 1.

A constraints formula is a one- or two-sided formula whose left-hand side is an optional direct selection of the `InitErgmProposal` function and whose right-hand side is a series of one or more terms separated by `"+"` and `"-"` operators, specifying the constraint.

The sample space (over and above the reference distribution) is determined by iterating over the constraints terms from left to right, each term updating it as follows:

- If the constraint introduces complex dependence structure (e.g., constrains degree or number of edges in the network), then this constraint always restricts the sample space. It may only have a `"+"` sign.

- If the constraint only restricts the set of dyads that may vary in the sample space (e.g., block-diagonal structure or fixing specific dyads at specific values) and has a `"+"` sign, the set of dyads that may vary is restricted to those that may vary according to this constraint and all the constraints to date.

- If the constraint only restricts the set of dyads that may vary in the sample space but has a `"-"` sign, the set of dyads that may vary is expanded to those that may vary according to this constraint or all the constraints up to date.

For example, a constraints formula `~a-b+c-d` with all constraints dyadic will allow dyads permitted by either `a` or `b` but only if they are also permitted by `c`; as well as all dyads permitted by `d`. If `A`, `B`, `C`, and `D` were logical matrices, the matrix of variable dyads would be equal to `((A|B)&C)|D`.

Terms with a positive sign can be viewed as "adding" a constraint while those with a negative sign can be viewed as "relaxing" a constraint.

The constraints formula can also contain a `.`. If present, `%ergmlhs%` attributes `constraints` and `constraints.obs` will be substituted in its place.

### Constraints visible to the package

#### All constraints

#### Constraints by keywords

### 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 4.2.2 Index]