ergmConstraint: Sample Space Constraints for Exponential-Family Random Graph...
In ergm: Fit, Simulate and Diagnose Exponential-Family Models for Networks

ergmConstraint

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. Constraints can also be searched via search.ergmConstraints, and help for an individual constraint can be obtained with ⁠ergmConstraint?<constraint>⁠ or help("<constraint>-ergmConstraint").

Specifying constraints

\ERGMspec

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.

Inheriting constraints from LHS `network`

By default, %ergmlhs% attributes constraints or constraints.obs (depending on which constraint) attached to the LHS of the model formula or the ⁠basis=⁠ argument will be added in front of the specified constraints formula. This is the desired behaviour most of the time, since those constraints are usually determined by how the network was constructed (e.g., structural zeros in a block-diagonal network).

For those situations in which this is not the desired behavior, a . term (with a positive sign or no sign at all) can be used to manually set the position of the inherited constraints in the formula, and a -. (minus-dot) term anywhere in the constraints formula will suppress the inherited formula altogether.

Constraints visible to the package

\ergmCSS \Sexpr[results=rd,stage=render]{ergm:::.formatIndexHtml(ergm:::.buildTermsDataframe("ergmConstraint"))}

All constraints

\Sexpr[results=rd,stage=render]{ergm:::.formatMatrixHtml(ergm:::.termMatrix("ergmConstraint"))}

Constraints by keywords

\Sexpr[results=rd,stage=render]{ergm:::.formatTocHtml(ergm:::.termToc("ergmConstraint"))}

References

Goodreau SM, Handcock MS, Hunter DR, Butts CT, Morris M (2008a). A statnet Tutorial. Journal of Statistical Software, 24(8). \Sexpr[results=rd]{tools:::Rd_expr_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). \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/rssc.12185")}
Krivitsky PN (2012). Exponential-Family Random Graph Models for Valued Networks. Electronic Journal of Statistics, 6, 1100-1128. \Sexpr[results=rd]{tools:::Rd_expr_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). \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v024.i04")}