hergm-terms {hergm}R Documentation

Hierarchical Exponential-Family Random Graph Models: Terms

Description

Hierarchical Exponential-Family Random Graph Models can be specified by calling the function hergm(formula), where formula is a formula of the form network ~ terms.

By using suitable terms, it is possible to specify

- the p_1 model for directed networks of Holland and Leinhardt (1981) and its extension to undirected random graph models with Dirichlet process priors (see arcs_i, arcs_j, mutual_i, and edges_i). While the p_1 model for undirected and directed networks with parametric priors contains O(n) parameters (n = number of nodes) and therefore is not parsimonious, the non-parametric Dirichlet process prior encourages a small number of unique parameters and therefore represents an elegant alternative to parametric priors.

- the stochastic block model of Snijders and Nowicki (1997) and Nowicki and Snijders (2001) with natural parameterization (restricted between-block parameters) and Dirichlet process priors (see edges_ij).

- the hierarchical exponential-family models of Schweinberger and Handcock (2009) with stick-breaking priors (see mutual_ij, twostar_ijk, triangle_ijk, ttriple_ijk, ctriple_ijk). Hierarchical exponential-family models replace the strong dependence of simple exponential-family models by weak dependence with an eye to solving the near-degeneracy problem of simple exponential-family model.

hergm.terms can be found here. Additional terms, e.g. covariate-dependent terms, can be found in ergm.terms.

Arguments

edges_i(k) (undirected network)

adding the term edges_i to the model adds node-dependent edge terms to the model; the optional argument k is the maximum number of blocks (default: the number of nodes).

arcs_i(k) (directed network)

adding the term arcs_i to the model adds node-dependent outdegree terms to the model; the optional argument k is the maximum number of blocks (default: the number of nodes).

arcs_j(k) (directed network)

adding the term arcs_j to the model adds node-dependent indegree terms to the model; the optional argument k is the maximum number of blocks (default: the number of nodes).

edges_ij(k) (undirected, directed network)

adding the term edges_ij to the model adds block-dependent edge terms to the model. the optional argument k is the maximum number of blocks (default: the number of nodes).

mutual_i(k) (directed network)

adding the term mutual_i to the model adds additive, block-dependent mutual edge terms to the model. the optional argument k is the maximum number of blocks (default: the number of nodes).

mutual_ij(k) (directed network)

adding the term mutual_ij to the model adds block-dependent mutual edge terms to the model. the optional argument k is the maximum number of blocks (default: the number of nodes).

twostar_ijk(k) (undirected network)

adding the term twostar_ijk to the model adds block-dependent two-star terms to the model. the optional argument k is the maximum number of blocks (default: the number of nodes).

triangle_ijk(k) (undirected, directed network)

adding the term triangle_ijk to the model adds block-dependent triangle terms to the model. the optional argument k is the maximum number of blocks (default: the number of nodes).

ttriple_ijk(k) (directed network)

adding the term ttriple_ijk to the model adds block-dependent transitive triple terms to the model. the optional argument k is the maximum number of blocks (default: the number of nodes).

ctriple_ijk(k) (directed network)

adding the term ctriple_ijk to the model adds block-dependent cyclic triple terms to the model. the optional argument k is the maximum number of blocks (default: the number of nodes).

References

Holland, P. W. and S. Leinhardt (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association 76 (373), 33–65.

Nowicki, K. and T. A. B. Snijders (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association 96 (455), 1077–1087.

Schweinberger, M. and M. S. Handcock (2009). Hierarchical exponential-family random graph models. Technical report, Pennsylvania State University. Submitted.

Snijders, T. A. B. and K. Nowicki (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. Journal of Classification 14, 75–100.

See Also

network, hergm, ergm.terms, hergm.postprocess

Examples

## Not run: # Load undirected network with 15 nodes (see ?example)
data(example)
# p_1 model for undirected network with Dirichlet process prior
hergm(d ~ edges_i)

# Load directed network with 18 nodes (see ?sampson)
data(sampson) 
# p_1 model for directed network with Dirichlet process prior
hergm(samplike ~ arcs_i + arcs_j + mutual)

# Load undirected network with 15 nodes (see ?example)
data(example)
# Stochastic block model for undirected network 
# with natural parameterization and Dirichlet process prior
hergm(d ~ edges_ij)

# Load directed network with 18 nodes (see ?sampson)
data(sampson) 
# Stochastic block model for directed network 
# with natural parameterization and Dirichlet process prior
hergm(samplike ~ edges_ij + mutual)

# Load undirected network with 15 nodes (see ?example)
data(example)
# Hierarchical exponential-family model with stick-breaking prior
hergm(d ~ edges + mutual + ttriple_ijk)

# Load directed network with 18 nodes (see ?sampson)
data(sampson) 
# Hierarchical exponential-family model with stick-breaking prior
hergm(samplike ~ edges + mutual + ttriple_ijk)

## End(Not run)

[Package hergm version 1.3-5 Index]