| ergmConstraint {ergm} | R Documentation |
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.
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 ergmTerms; and, optionally, sample space Y and reference distribution h(y) information provided by ergmConstraints and, for valued ERGMs, by ergmReferences.
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.
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