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R/bigergm.R
In bigergm: Fit, Simulate, and Diagnose Hierarchical Exponential-Family Models for Big Networks
Defines functions
bigergm
Documented in
bigergm
#' bigergm: Exponential-family random graph models for large networks with local dependence
#' @description
#' The function \code{bigergm} estimates and simulates three classes of exponential-family
#' random graph models for large networks under local dependence:
#'\enumerate{
#' \item{The p_1 model of Holland and Leinhardt (1981) in exponential-family form and extensions by Vu, Hunter, and Schweinberger (2013), Schweinberger, Petrescu-Prahova, and Vu (2014), Dahbura et al. (2021), and Fritz et al. (2024) to both directed and undirected random graphs with additional model terms, with and without covariates.}
#' \item{The stochastic block model of Snijders and Nowicki (1997) and Nowicki and Snijders (2001) in exponential-family form.}
#' \item{The exponential-family random graph models with local dependence of Schweinberger and Handcock (2015), with and without covariates. The exponential-family random graph models with local dependence replace the long-range dependence of conventional exponential-family random graph models by short-range dependence. Therefore, exponential-family random graph models with local dependence replace the strong dependence of conventional exponential-family random graph models by weak dependence, reducing the problem of model degeneracy (Handcock, 2003; Schweinberger, 2011) and improving goodness-of-fit (Schweinberger and Handcock, 2015). In addition, exponential-family random graph models with local dependence satisfy a weak form of self-consistency in the sense that these models are self-consistent under neighborhood sampling (Schweinberger and Handcock, 2015), which enables consistent estimation of neighborhood-dependent parameters (Schweinberger and Stewart, 2017; Schweinberger, 2017).}
#' }
#' @useDynLib bigergm
#' @importFrom utils tail
#' @importFrom cachem cache_disk
#' @importFrom tidyr expand
#' @importFrom Rcpp sourceCpp
#' @importFrom stats simulate terms update
#' @importFrom ergm ergm.getnetwork gof control.ergm ergmMPLE
#' @importFrom graphics abline boxplot lines
#' @importFrom network `%v%` list.vertex.attributes get.vertex.attribute as.network delete.edges as.network
#' @importFrom network `%v%<-` delete.vertex.attribute network.initialize set.vertex.attribute
#' @param object An \R \code{\link{formula}} object or \code{\link{bigergm}} class object.
#' If a formula is given, the function estimates a new model specified by it.
#' It needs to be of the form
#' \code{y ~ <model terms>}, where \code{y} is a
#' \code{\link[network]{network}} object.
#' For the details on the possible \code{<model terms>}, see
#' \code{\link[ergm]{ergmTerm}} and Morris, Handcock and Hunter (2008).
#' All terms that induce dependence are excluded from the between block model, while the within block model includes all terms.
#' When you pass a \code{\link{bigergm}} class object to the function, you continue from the previous MM step.
#' Note that the block allocation (which is either provided by parameter \code{blocks} or estimated in the first step) is saved as the vertex.attribute `block` of the network.
#' This attribute can also be used in the specified formula.
#' The \code{\link[ergm.multi]{L-ergmTerm}} is supported to enable size-dependent coefficients for the within-blocks model.
#' Note, however, that for size-dependent parameters of terms that are included in the between-blocks model,
#' the intercept in the linear model provided to \code{\link[ergm.multi]{L-ergmTerm}} should not include the intercept.
#' See the second example below for a demonstration.
#' @param add_intercepts Boolean value to indicate whether adequate intercepts
#' should be added to the provided formula so that the model in the first stage
#' of the estimation is a nested model of the estimated model in the second stage of the estimation.
#' @param n_blocks The number of blocks. This must be specified by the user.
#' When you pass a \code{\link{bigergm}} class object to the function, you don't have to specify this argument.
#' @param n_cores The number of CPU cores to use.
#' @param blocks The pre-specified block memberships for each node.
#' If `NULL`, the latent community structure is estimated, assuming that the number of communities is `n_blocks`.
#' @param estimate_parameters If `TRUE`, both clustering and parameter estimation are implemented.
#' If `FALSE`, only clustering is executed.
#' @param verbose A logical or an integer: if this is TRUE/1,
#' the program will print out additional information about the progress of estimation and simulation.
#' A higher value yields lower level information.
#' @param n_MM_step_max The maximum number of MM iterations.
#' Currently, no early stopping criteria is introduced. Thus `n_MM_step_max` MM iterations are exactly implemented.
#' @param tol_MM_step Tolerance regarding the relative change of the lower bound of the likelihood
#' used to decide on the convergence of the clustering step
#' @param initialization How the blocks should be initialized.
#' If `infomap` (the default), `igraph`' or Python's infomap is implemented.
#' If `random`, the initial clusters are randomly uniformally selected.
#' If `spectral`, spectral clustering is conducted.
#' If `walktrap`, the walktrap clustering algorithm as implemented in \code{\link[igraph]{cluster_walktrap}} is conducted.
#' If \code{initialization}
#' is a vector of integers of the same length as the number of nodes in the provided network (in \code{object}),
#' then the provided vector is used as the initial cluster assignment.
#' If \code{initialization} is a string relating to a file path, \code{\link{bigergm}} will interpret it as block allocations saved in Python's infomap .clu format under that path.
#' @param use_infomap_python If `TRUE`, the cluster initialization is implemented using Pythons' infomap.
#' @param virtualenv_python Which virtual environment should be used for the infomap algorithm?
#' @param seed_infomap seed value (integer) for the infomap algorithm, which can be used to initialize the estimation of the blocks.
#' @param weight_for_initialization weight value used for cluster initialization. The higher this value, the more weight is put on the initialized block allocation.
#' @param seed seed value (integer) for the random number generator.
#' @param method_within If "MPLE" (the default), then the maximum pseudolikelihood estimator is implemented when estimating the within-block network model.
#' If "MLE", then an approximate maximum likelihood estimator is conducted. If "CD" (EXPERIMENTAL), the Monte-Carlo contrastive divergence estimate is returned.
#' @param control_within A list of control parameters for the \code{\link[ergm]{ergm}} function used to estimate the parameters of the within model. See \code{\link[ergm]{control.ergm}} for details.
#' @param only_use_preprocessed If `TRUE`, the function only uses the preprocessed data from a previous fit but does not continue the estimation from its final iteration, instead the estimation is started again from the provided initialization.
#' @param clustering_with_features If `TRUE`, clustering is implemented using the discrete covariates specified in the formula. At the moment, we only support the \code{nodematch} term and disregard diff = TRUE if provided.
#' @param compute_pi If `TRUE`, this function keeps track of pi matrices at each MM iteration.
#' If the network is large, we strongly recommend to set to be `FALSE`.
#' @param check_alpha_update If `TRUE`, this function keeps track of alpha matrices at each MM iteration.
#' If the network is large, we strongly recommend to set to be `FALSE`.
#' @param check_blocks If TRUE, this function keeps track of estimated block memberships at each MM iteration.
#' @param cache a `cachem` cache object used to store intermediate calculations such as eigenvector decomposition results.
#' @param return_checkpoint If `TRUE`, the function returns the checkpoint list. For most applications, this should be set to `TRUE` but if memory space needed by the output is an issue, set to `FALSE`.
#' @param ... Additional arguments, to be passed to lower-level functions (mainly to the \code{\link[ergm]{ergm}} function used for the estimation of within-block connections).
#' @references
#' Babkin, S., Stewart, J., Long, X., and M. Schweinberger (2020). Large-scale estimation of random graph models with local dependence. Computational Statistics and Data Analysis, 152, 1--19.
#'
#' Dahbura, J. N. M., Komatsu, S., Nishida, T. and Mele, A. (2021), ‘A structural model of business cards exchange networks’.
#' https://arxiv.org/abs/2105.12704
#'
#' Fritz C., Georg C., Mele A., and Schweinberger M. (2024). A strategic model of software dependency networks.
#' https://arxiv.org/abs/2402.13375
#'
#' Handcock, M. S. (2003). Assessing degeneracy in statistical models of social networks. Technical report, Center for Statistics and the Social Sciences, University of Washington, Seattle. \cr
#' https://csss.uw.edu/Papers/wp39.pdf
#'
#' Holland, P. W. and S. Leinhardt (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, Theory & Methods, 76, 33--65.
#'
#' Morris M, Handcock MS, Hunter DR (2008). Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects.
#' Journal of Statistical Software, 24.
#'
#' Nowicki, K. and T. A. B. Snijders (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, Theory & Methods, 96, 1077--1087.
#'
#' Schweinberger, M. (2011). Instability, sensitivity, and degeneracy of discrete exponential families. Journal of the American Statistical Association, Theory & Methods, 106, 1361--1370.
#'
#' Schweinberger, M. (2020). Consistent structure estimation of exponential-family random graph models with block structure. Bernoulli, 26, 1205--1233.
#'
#' Schweinberger, M. and M. S. Handcock (2015). Local dependence in random graph models: characterization, properties, and statistical inference. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 7, 647-676.
#'
#' Schweinberger, M., Krivitsky, P. N., Butts, C.T. and J. Stewart (2020). Exponential-family models of random graphs: Inference in finite, super, and infinite population scenarios. Statistical Science, 35, 627-662.
#'
#' Schweinberger, M. and P. Luna (2018). HERGM: Hierarchical exponential-family random graph models. Journal of Statistical Software, 85, 1--39.
#'
#' Schweinberger, M., Petrescu-Prahova, M. and D. Q. Vu (2014). Disaster response on September 11, 2001 through the lens of statistical network analysis. Social Networks, 37, 42--55.
#'
#' Schweinberger, M. and J. Stewart (2020). Concentration and consistency results for canonical and curved exponential-family random graphs. The Annals of Statistics, 48, 374--396.
#'
#' 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.
#'
#' Stewart, J., Schweinberger, M., Bojanowski, M., and M. Morris (2019). Multilevel network data facilitate statistical inference for curved {ERGM}s with geometrically weighted terms. Social Networks, 59, 98--119.
#'
#' Vu, D. Q., Hunter, D. R. and M. Schweinberger (2013). Model-based clustering of large networks. Annals of Applied Statistics, 7, 1010--1039.
#' @return An object of class 'bigergm' including the results of the fitted model.
#' These include:
#'\describe{
#' \item{call:}{call of the mode}
#' \item{block:}{vector of the found block of the nodes into cluster}
#' \item{initial_block:}{vector of the initial block of the nodes into cluster}
#' \item{sbm_pi:}{Connection probabilities represented as a \code{n_blocks x n_blocks} matrix from the first stage of the estimation between all clusters}
#' \item{MM_list_z:}{list of cluster allocation for each node and each iteration}
#' \item{MM_list_alpha:}{list of posterior distributions of cluster allocations for all nodes for each iteration}
#' \item{MM_change_in_alpha:}{change in 'alpha' for each iteration}
#' \item{MM_lower_bound:}{ vector of the evidence lower bounds from the MM algorithm}
#' \item{alpha: }{matrix representing the converged posterior distributions of cluster allocations for all nodes}
#' \item{counter_e_step:}{ integer number indicating the number of iterations carried out}
#' \item{adjacency_matrix:}{sparse matrix representing the adjacency matrix used for the estimation}
#' \item{estimation_status:}{character stating the status of the estimation}
#' \item{est_within:}{\code{\link[ergm]{ergm}} object of the model for within cluster connections }
#' \item{est_between:}{\code{\link[ergm]{ergm}} object of the model for between cluster connections}
#' \item{checkpoint:}{list of information to continue the estimation (only returned if \code{return_checkpoint = TRUE})}
#' \item{membership_before_kmeans:}{vector of the found blocks of the nodes into cluster before the final check for bad clusters}
#' \item{estimate_parameters:}{binary value if the parameters in the second step of the algorithm should be estimated or not}
#' }
#' @examples
#' # Load an embedded network object.
#' data(toyNet)
#'
#' # Specify the model that you would like to estimate.
#' model_formula <- toyNet ~ edges + nodematch("x") + nodematch("y") + triangle
#' # Estimate the model
#' bigergm_res <- bigergm(
#' object = model_formula,
#' # The model you would like to estimate
#' n_blocks = 4,
#' # The number of blocks
#' n_MM_step_max = 10,
#' # The maximum number of MM algorithm steps
#' estimate_parameters = TRUE,
#' # Perform parameter estimation after the block recovery step
#' clustering_with_features = TRUE,
#' # Indicate that clustering must take into account nodematch on characteristics
#' check_blocks = FALSE)
#'
#' # Example with N() operator
#'
#' \dontrun{
#' set.seed(1)
#' # Prepare ingredients for simulating a network
#' N <- 500
#' K <- 10
#'
#' list_within_params <- c(1, 2, 2,-0.5)
#' list_between_params <- c(-8, 0.5, -0.5)
#' formula <- g ~ edges + nodematch("x") + nodematch("y") + N(~edges,~log(n)-1)
#'
#' memb <- sample(1:K,prob = c(0.1,0.2,0.05,0.05,0.10,0.1,0.1,0.1,0.1,0.1),
#' size = N, replace = TRUE)
#' vertex_id <- as.character(11:(11 + N - 1))
#'
#' x <- sample(1:2, size = N, replace = TRUE)
#' y <- sample(1:2, size = N, replace = TRUE)
#'
#'
#' df <- tibble::tibble(
#' id = vertex_id,
#' memb = memb,
#' x = x,
#' y = y
#' )
#' g <- network::network.initialize(n = N, directed = FALSE)
#' g %v% "vertex.names" <- df$id
#' g %v% "block" <- df$memb
#' g %v% "x" <- df$x
#' g %v% "y" <- df$y
#'
#' # Simulate a network
#' g_sim <-
#' simulate_bigergm(
#' formula = formula,
#' coef_within = list_within_params,
#' coef_between = list_between_params,
#' nsim = 1,
#' control_within = control.simulate.formula(MCMC.burnin = 200000))
#'
#' estimation <- bigergm(update(formula,new = g_sim~.), n_blocks = 10,
#' verbose = T)
#' summary(estimation)
#' }
#' @export
bigergm <- function(object,
add_intercepts = FALSE,
n_blocks = NULL,
n_cores = 1,
blocks = NULL,
estimate_parameters = TRUE,
verbose = 0,
n_MM_step_max = 100,
tol_MM_step = 0.0001,
initialization = "infomap",
use_infomap_python = FALSE,
virtualenv_python = "r-bigergm",
seed_infomap = NULL,
weight_for_initialization = 1000,
seed = NULL,
method_within = "MPLE",
control_within = ergm::control.ergm(),
clustering_with_features = TRUE,
compute_pi = FALSE,
check_alpha_update = FALSE,
check_blocks = FALSE,
cache = NULL,
return_checkpoint = TRUE,
only_use_preprocessed = FALSE,
...) {
###################################################################################
###### Preparations for estimation ################################################
###################################################################################
# Add a check to look for the indicator whether the estimation should not be continued but
# only the preprocessed data be used
if ("bigergm" %in% class(object)) {
if(only_use_preprocessed){
preprocessed_data <- object$checkpoint$preprocessed_data
object <- object$checkpoint$formula
}
}
# When the given object is formula:
if ("formula" %in% class(object)) {
# Check if initialization is a vector of block memberships, a .clu file to be read or whether blocks are known
check <- all(stringr::str_detect(initialization, ".clu")) + (length(initialization) >1) + !is.null(blocks)
# If n_cluster is missing and blocks and initialized_cluster_data are NULL, stop the process.
if (!check && is.null(n_blocks)){
stop("\nThe argument 'n_blocks' is missing. Please specify the number of clusters.")
}
if(length(initialization) >1){
# If initialization is a vector of block memberships, use it as the initial block memberships but leave the n_blocks argument if provided as is.
if (is.null(n_blocks)){
n_blocks <- length(unique(initialization))
}
} else if (all(stringr::str_detect(initialization, ".clu"))){
initialized_cluster_data <- readr::read_delim(initialization, delim = " ", skip = 9, col_names = c("node_id", "block", "flow"), col_types = "iid")
initialization <- as.numeric(dplyr::arrange(initialized_cluster_data, by_group = .data$node_id)$block)
n_blocks <- length(unique(initialization))
}
if (!is.null(blocks)){
if (!is.null(n_blocks)){
warning("The argument 'blocks' is given. The argument 'n_blocks' will be ignored.")
}
n_blocks <- length(unique(blocks))
# Have the blocks be a numeric vector
blocks <- as.numeric(factor(blocks))
}
# Get the formula
formula <- object
# Get network object from formula
network <- ergm::ergm.getnetwork(formula)
if(clustering_with_features) {
# Check if clustering_with_features is TRUE without any features
list_varname <- as.character(formula)[3]
list_varname <- unlist(stringr::str_extract_all(string = list_varname, pattern = '"[^"]*"'))
list_varname <- stringr::str_remove_all(string = list_varname, pattern = '\"')
if(length(list_varname) == 0){
warning("Parameter clustering_with_features was set to TRUE although no covariates are needed for the given formula")
clustering_with_features <- FALSE
}
}
if(!exists("preprocessed_data")){
preprocessed_data <- NULL
}
# If preprocessed_data is not NULL, check if the order of features names is the same with that of formula.
if (!is.null(preprocessed_data)) {
# If clustering_with_features = FALSE, stop the process.
if (!clustering_with_features) {
stop("\nSet clustering_with_features = TRUE since you are going to use vertex features for cluster estimation.")
}
}
}
MM_restart_object <- NULL
# When the given object has a class of "bigergm":
if ("bigergm" %in% class(object)) {
# Inherit the following arguments from the previous estimation.
# These arguments are so important that they won't be replaced by the arguments given by the user.
network <- object$checkpoint$network
formula <- object$checkpoint$formula
n_blocks <- object$checkpoint$n_blocks
clustering_with_features <- object$checkpoint$clustering_with_features
add_intercepts <- object$checkpoint$add_intercepts
preprocessed_data <- object$checkpoint$preprocessed_data
# Inherit the following arguments from the previous estimation if not given by the user.
# If given by the user, discard the inherited argument and use the given one.
vec_arguments <-
c(
"n_cores",
"estimate_parameters",
"verbose",
"n_MM_step_max",
"seed",
"method_within",
"compute_pi",
"check_alpha_update",
"check_blocks"
)
message("Arguments:")
message(glue::glue("Number of clusters = {n_blocks}"))
for (i in 1:length(vec_arguments)) {
if (do.call(missing, list(vec_arguments[[i]])) == TRUE) {
assign(vec_arguments[[i]], object$checkpoint[[vec_arguments[[i]]]])
}
message(glue::glue("{vec_arguments[[i]]} = {get(vec_arguments[[i]])}"))
}
# Prepare an object to restart the MM with.
MM_restart_object <-
list(
alpha = object$alpha,
list_alpha = object$MM_list_alpha,
list_z = object$MM_list_z,
z_memb_init = object$z_memb_final_before_kmeans,
change_in_alpha = object$MM_change_in_alpha,
lower_bound = object$MM_lower_bound,
counter_e_step = object$counter_e_step,
adjacency_matrix = object$adjacency_matrix
)
}
# For the entire estimation process the vertex.names are changed to be integers running from 1 to n
old_vertex_names <- network::network.vertex.names(network)
network::network.vertex.names(network) <- 1:length(old_vertex_names)
# If the blockss of each node are specified in the variable 'blocks' as integers between 1 and n_blocks,
# the specified block memberships are fixed.
all_blockss_fixed <- ifelse(is.null(blocks), FALSE, TRUE)
# Make sure that if all_blockss_fixed == TRUE, estimate_parameters must also be TRUE.
estimate_parameters <- ifelse(all_blockss_fixed, TRUE, estimate_parameters)
###################################################################################
###### First step: Estimating block memberships ###################################
###################################################################################
# Estimate the memberships if they are not specified.
if (!all_blockss_fixed) {
set.seed(seed)
# Estimate the block memberships
answer <- MM_wrapper(
network = network,
formula = formula,
n_blocks = n_blocks,
n_MM_step_max = n_MM_step_max,
tol_MM_step = tol_MM_step,
min_size = 2,
initialization = initialization,
use_infomap_python = use_infomap_python,
virtualenv_python = virtualenv_python,
seed_infomap = seed_infomap,
clustering_with_features = clustering_with_features,
list_multiplied_feature_matrices = preprocessed_data,
verbose = verbose,
weight_for_initialization = weight_for_initialization,
compute_pi = compute_pi,
check_alpha_update = check_alpha_update,
check_blocks = check_blocks,
MM_restart_object = MM_restart_object,
cache = cache
)
blocks <- answer$z_memb_final
initial_block <- answer$z_memb_init
membership_before_kmeans <- answer$z_memb_final_before_kmeans
sbm_pi <- answer$Pi
MM_list_alpha <- answer$list_alpha
MM_list_z <- answer$list_z
MM_change_in_alpha <- answer$change_in_alpha
MM_lower_bound <- answer$lower_bound
alpha <- answer$alpha
counter_e_step <- answer$counter_e_step
adjacency_matrix <- as(answer$adjacency_matrix, "nMatrix")
preprocessed_data <- answer$list_multiplied_feature_matrices
}
else {
if (verbose > 0) {
message("\nSkipping Steps 1 and 2: z specified")
}
param_MM_wrapper <- NULL
initial_block <- NULL
membership_before_kmeans <- NULL
sbm_pi <- NULL
MM_list_z <- NULL
MM_list_alpha <- NULL
MM_change_in_alpha <- NULL
MM_lower_bound <- NULL
alpha <- NULL
counter_e_step <- NULL
adjacency_matrix <- NULL
preprocessed_data <- NULL
}
# Relabel the block to be in canonical form
blocks <- relabel(block = blocks, n_blocks = n_blocks)
####################################################################################################
###### Second step: Estimating between-block parameters ############################################
####################################################################################################
network %v% "block" <- as.numeric(blocks)
if (estimate_parameters) {
if (verbose > 0) {
message("\nEstimate between-block parameters")
}
## Estimate between-block parameters
est_between <- est_between(
formula = formula,
network = network,
add_intercepts = add_intercepts,
clustering_with_features = clustering_with_features
)
} else {
# If you don't estimate any parameters...
est_between <- NULL
}
####################################################################################################
###### Third step: Estimating within-block parameters ##############################################
####################################################################################################
# When you estimate with-block parameters, then:
if (estimate_parameters) {
if (verbose > 0) {
message("\n\nStep 3: Estimate parameters conditional on z")
}
# Estimate within-block parameters
if (verbose == 0) {
suppressMessages(est_within <- est_within(formula =formula,network = network,
block = blocks,number_cores = n_cores,
seed = NULL,method = method_within,
add_intercepts = add_intercepts,
clustering_with_features = clustering_with_features,
control = control_within,
...))
} else {
est_within <- est_within(formula =formula,network = network,
block = blocks,number_cores = n_cores,
seed = NULL,method = method_within,
add_intercepts = add_intercepts,
clustering_with_features = clustering_with_features,
control = control_within,
...)
}
####################################################################################################
###### Store the results ###########################################################################
####################################################################################################
estimation_status <- ifelse(est_within$failure, "Estimation failed", "Estimated")
} else {
# estimate_parameters = FALSE:
message("\n")
estimation_status <- "Not estimated"
est_within <- NULL
}
if(return_checkpoint){
# Change the names back to the old ones
network%v% "vertex.names" <- old_vertex_names
# Store the given arguments. These will be inherited for the next MM.
checkpoint <- list(
network = network,
formula = formula,
n_blocks = n_blocks,
clustering_with_features = clustering_with_features,
preprocessed_data = preprocessed_data,
n_cores = n_cores,
estimate_parameters = estimate_parameters,
verbose = verbose,
n_MM_step_max = n_MM_step_max,
seed = seed,
method_within = method_within,
compute_pi = compute_pi,
check_alpha_update = check_alpha_update,
check_blocks = check_blocks,
add_intercepts = add_intercepts
)
} else{
checkpoint <- NULL
}
# Store the results in a list
output <- list(
call = sys.calls()[[1]],
formula = formula,
block = blocks,
initial_block = initial_block,
sbm_pi = sbm_pi,
MM_list_z = MM_list_z,
MM_list_alpha = MM_list_alpha,
MM_change_in_alpha = MM_change_in_alpha,
MM_lower_bound = MM_lower_bound,
alpha = alpha,
counter_e_step = counter_e_step,
adjacency_matrix = adjacency_matrix,
directed = network$gal$directed,
estimation_status = estimation_status,
est_within = est_within,
est_between = est_between,
checkpoint = checkpoint,
membership_before_kmeans = membership_before_kmeans,
estimate_parameters = estimate_parameters
)
# Return the output
return(structure(output, class = "bigergm"))
}
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Defines functions bigergm
Documented in bigergm
#' bigergm: Exponential-family random graph models for large networks with local dependence
#' @description
#' The function \code{bigergm} estimates and simulates three classes of exponential-family
#' random graph models for large networks under local dependence:
#'\enumerate{
#' \item{The p_1 model of Holland and Leinhardt (1981) in exponential-family form and extensions by Vu, Hunter, and Schweinberger (2013), Schweinberger, Petrescu-Prahova, and Vu (2014), Dahbura et al. (2021), and Fritz et al. (2024) to both directed and undirected random graphs with additional model terms, with and without covariates.}
#' \item{The stochastic block model of Snijders and Nowicki (1997) and Nowicki and Snijders (2001) in exponential-family form.}
#' \item{The exponential-family random graph models with local dependence of Schweinberger and Handcock (2015), with and without covariates. The exponential-family random graph models with local dependence replace the long-range dependence of conventional exponential-family random graph models by short-range dependence. Therefore, exponential-family random graph models with local dependence replace the strong dependence of conventional exponential-family random graph models by weak dependence, reducing the problem of model degeneracy (Handcock, 2003; Schweinberger, 2011) and improving goodness-of-fit (Schweinberger and Handcock, 2015). In addition, exponential-family random graph models with local dependence satisfy a weak form of self-consistency in the sense that these models are self-consistent under neighborhood sampling (Schweinberger and Handcock, 2015), which enables consistent estimation of neighborhood-dependent parameters (Schweinberger and Stewart, 2017; Schweinberger, 2017).}
#' }
#' @useDynLib bigergm
#' @importFrom utils tail
#' @importFrom cachem cache_disk
#' @importFrom tidyr expand
#' @importFrom Rcpp sourceCpp
#' @importFrom stats simulate terms update
#' @importFrom ergm ergm.getnetwork gof control.ergm ergmMPLE
#' @importFrom graphics abline boxplot lines
#' @importFrom network `%v%` list.vertex.attributes get.vertex.attribute as.network delete.edges as.network
#' @importFrom network `%v%<-` delete.vertex.attribute network.initialize set.vertex.attribute
#' @param object An \R \code{\link{formula}} object or \code{\link{bigergm}} class object.
#' If a formula is given, the function estimates a new model specified by it.
#' It needs to be of the form
#' \code{y ~ <model terms>}, where \code{y} is a
#' \code{\link[network]{network}} object.
#' For the details on the possible \code{<model terms>}, see
#' \code{\link[ergm]{ergmTerm}} and Morris, Handcock and Hunter (2008).
#' All terms that induce dependence are excluded from the between block model, while the within block model includes all terms.
#' When you pass a \code{\link{bigergm}} class object to the function, you continue from the previous MM step.
#' Note that the block allocation (which is either provided by parameter \code{blocks} or estimated in the first step) is saved as the vertex.attribute `block` of the network.
#' This attribute can also be used in the specified formula.
#' The \code{\link[ergm.multi]{L-ergmTerm}} is supported to enable size-dependent coefficients for the within-blocks model.
#' Note, however, that for size-dependent parameters of terms that are included in the between-blocks model,
#' the intercept in the linear model provided to \code{\link[ergm.multi]{L-ergmTerm}} should not include the intercept.
#' See the second example below for a demonstration.
#' @param add_intercepts Boolean value to indicate whether adequate intercepts
#' should be added to the provided formula so that the model in the first stage
#' of the estimation is a nested model of the estimated model in the second stage of the estimation.
#' @param n_blocks The number of blocks. This must be specified by the user.
#' When you pass a \code{\link{bigergm}} class object to the function, you don't have to specify this argument.
#' @param n_cores The number of CPU cores to use.
#' @param blocks The pre-specified block memberships for each node.
#' If `NULL`, the latent community structure is estimated, assuming that the number of communities is `n_blocks`.
#' @param estimate_parameters If `TRUE`, both clustering and parameter estimation are implemented.
#' If `FALSE`, only clustering is executed.
#' @param verbose A logical or an integer: if this is TRUE/1,
#' the program will print out additional information about the progress of estimation and simulation.
#' A higher value yields lower level information.
#' @param n_MM_step_max The maximum number of MM iterations.
#' Currently, no early stopping criteria is introduced. Thus `n_MM_step_max` MM iterations are exactly implemented.
#' @param tol_MM_step Tolerance regarding the relative change of the lower bound of the likelihood
#' used to decide on the convergence of the clustering step
#' @param initialization How the blocks should be initialized.
#' If `infomap` (the default), `igraph`' or Python's infomap is implemented.
#' If `random`, the initial clusters are randomly uniformally selected.
#' If `spectral`, spectral clustering is conducted.
#' If `walktrap`, the walktrap clustering algorithm as implemented in \code{\link[igraph]{cluster_walktrap}} is conducted.
#' If \code{initialization}
#' is a vector of integers of the same length as the number of nodes in the provided network (in \code{object}),
#' then the provided vector is used as the initial cluster assignment.
#' If \code{initialization} is a string relating to a file path, \code{\link{bigergm}} will interpret it as block allocations saved in Python's infomap .clu format under that path.
#' @param use_infomap_python If `TRUE`, the cluster initialization is implemented using Pythons' infomap.
#' @param virtualenv_python Which virtual environment should be used for the infomap algorithm?
#' @param seed_infomap seed value (integer) for the infomap algorithm, which can be used to initialize the estimation of the blocks.
#' @param weight_for_initialization weight value used for cluster initialization. The higher this value, the more weight is put on the initialized block allocation.
#' @param seed seed value (integer) for the random number generator.
#' @param method_within If "MPLE" (the default), then the maximum pseudolikelihood estimator is implemented when estimating the within-block network model.
#' If "MLE", then an approximate maximum likelihood estimator is conducted. If "CD" (EXPERIMENTAL), the Monte-Carlo contrastive divergence estimate is returned.
#' @param control_within A list of control parameters for the \code{\link[ergm]{ergm}} function used to estimate the parameters of the within model. See \code{\link[ergm]{control.ergm}} for details.
#' @param only_use_preprocessed If `TRUE`, the function only uses the preprocessed data from a previous fit but does not continue the estimation from its final iteration, instead the estimation is started again from the provided initialization.
#' @param clustering_with_features If `TRUE`, clustering is implemented using the discrete covariates specified in the formula. At the moment, we only support the \code{nodematch} term and disregard diff = TRUE if provided.
#' @param compute_pi If `TRUE`, this function keeps track of pi matrices at each MM iteration.
#' If the network is large, we strongly recommend to set to be `FALSE`.
#' @param check_alpha_update If `TRUE`, this function keeps track of alpha matrices at each MM iteration.
#' If the network is large, we strongly recommend to set to be `FALSE`.
#' @param check_blocks If TRUE, this function keeps track of estimated block memberships at each MM iteration.
#' @param cache a `cachem` cache object used to store intermediate calculations such as eigenvector decomposition results.
#' @param return_checkpoint If `TRUE`, the function returns the checkpoint list. For most applications, this should be set to `TRUE` but if memory space needed by the output is an issue, set to `FALSE`.
#' @param ... Additional arguments, to be passed to lower-level functions (mainly to the \code{\link[ergm]{ergm}} function used for the estimation of within-block connections).
#' @references
#' Babkin, S., Stewart, J., Long, X., and M. Schweinberger (2020). Large-scale estimation of random graph models with local dependence. Computational Statistics and Data Analysis, 152, 1--19.
#'
#' Dahbura, J. N. M., Komatsu, S., Nishida, T. and Mele, A. (2021), ‘A structural model of business cards exchange networks’.
#' https://arxiv.org/abs/2105.12704
#'
#' Fritz C., Georg C., Mele A., and Schweinberger M. (2024). A strategic model of software dependency networks.
#' https://arxiv.org/abs/2402.13375
#'
#' Handcock, M. S. (2003). Assessing degeneracy in statistical models of social networks. Technical report, Center for Statistics and the Social Sciences, University of Washington, Seattle. \cr
#' https://csss.uw.edu/Papers/wp39.pdf
#'
#' Holland, P. W. and S. Leinhardt (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, Theory & Methods, 76, 33--65.
#'
#' Morris M, Handcock MS, Hunter DR (2008). Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects.
#' Journal of Statistical Software, 24.
#'
#' Nowicki, K. and T. A. B. Snijders (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, Theory & Methods, 96, 1077--1087.
#'
#' Schweinberger, M. (2011). Instability, sensitivity, and degeneracy of discrete exponential families. Journal of the American Statistical Association, Theory & Methods, 106, 1361--1370.
#'
#' Schweinberger, M. (2020). Consistent structure estimation of exponential-family random graph models with block structure. Bernoulli, 26, 1205--1233.
#'
#' Schweinberger, M. and M. S. Handcock (2015). Local dependence in random graph models: characterization, properties, and statistical inference. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 7, 647-676.
#'
#' Schweinberger, M., Krivitsky, P. N., Butts, C.T. and J. Stewart (2020). Exponential-family models of random graphs: Inference in finite, super, and infinite population scenarios. Statistical Science, 35, 627-662.
#'
#' Schweinberger, M. and P. Luna (2018). HERGM: Hierarchical exponential-family random graph models. Journal of Statistical Software, 85, 1--39.
#'
#' Schweinberger, M., Petrescu-Prahova, M. and D. Q. Vu (2014). Disaster response on September 11, 2001 through the lens of statistical network analysis. Social Networks, 37, 42--55.
#'
#' Schweinberger, M. and J. Stewart (2020). Concentration and consistency results for canonical and curved exponential-family random graphs. The Annals of Statistics, 48, 374--396.
#'
#' 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.
#'
#' Stewart, J., Schweinberger, M., Bojanowski, M., and M. Morris (2019). Multilevel network data facilitate statistical inference for curved {ERGM}s with geometrically weighted terms. Social Networks, 59, 98--119.
#'
#' Vu, D. Q., Hunter, D. R. and M. Schweinberger (2013). Model-based clustering of large networks. Annals of Applied Statistics, 7, 1010--1039.
#' @return An object of class 'bigergm' including the results of the fitted model.
#' These include:
#'\describe{
#' \item{call:}{call of the mode}
#' \item{block:}{vector of the found block of the nodes into cluster}
#' \item{initial_block:}{vector of the initial block of the nodes into cluster}
#' \item{sbm_pi:}{Connection probabilities represented as a \code{n_blocks x n_blocks} matrix from the first stage of the estimation between all clusters}
#' \item{MM_list_z:}{list of cluster allocation for each node and each iteration}
#' \item{MM_list_alpha:}{list of posterior distributions of cluster allocations for all nodes for each iteration}
#' \item{MM_change_in_alpha:}{change in 'alpha' for each iteration}
#' \item{MM_lower_bound:}{ vector of the evidence lower bounds from the MM algorithm}
#' \item{alpha: }{matrix representing the converged posterior distributions of cluster allocations for all nodes}
#' \item{counter_e_step:}{ integer number indicating the number of iterations carried out}
#' \item{adjacency_matrix:}{sparse matrix representing the adjacency matrix used for the estimation}
#' \item{estimation_status:}{character stating the status of the estimation}
#' \item{est_within:}{\code{\link[ergm]{ergm}} object of the model for within cluster connections }
#' \item{est_between:}{\code{\link[ergm]{ergm}} object of the model for between cluster connections}
#' \item{checkpoint:}{list of information to continue the estimation (only returned if \code{return_checkpoint = TRUE})}
#' \item{membership_before_kmeans:}{vector of the found blocks of the nodes into cluster before the final check for bad clusters}
#' \item{estimate_parameters:}{binary value if the parameters in the second step of the algorithm should be estimated or not}
#' }
#' @examples
#' # Load an embedded network object.
#' data(toyNet)
#'
#' # Specify the model that you would like to estimate.
#' model_formula <- toyNet ~ edges + nodematch("x") + nodematch("y") + triangle
#' # Estimate the model
#' bigergm_res <- bigergm(
#' object = model_formula,
#' # The model you would like to estimate
#' n_blocks = 4,
#' # The number of blocks
#' n_MM_step_max = 10,
#' # The maximum number of MM algorithm steps
#' estimate_parameters = TRUE,
#' # Perform parameter estimation after the block recovery step
#' clustering_with_features = TRUE,
#' # Indicate that clustering must take into account nodematch on characteristics
#' check_blocks = FALSE)
#'
#' # Example with N() operator
#'
#' \dontrun{
#' set.seed(1)
#' # Prepare ingredients for simulating a network
#' N <- 500
#' K <- 10
#'
#' list_within_params <- c(1, 2, 2,-0.5)
#' list_between_params <- c(-8, 0.5, -0.5)
#' formula <- g ~ edges + nodematch("x") + nodematch("y") + N(~edges,~log(n)-1)
#'
#' memb <- sample(1:K,prob = c(0.1,0.2,0.05,0.05,0.10,0.1,0.1,0.1,0.1,0.1),
#' size = N, replace = TRUE)
#' vertex_id <- as.character(11:(11 + N - 1))
#'
#' x <- sample(1:2, size = N, replace = TRUE)
#' y <- sample(1:2, size = N, replace = TRUE)
#'
#'
#' df <- tibble::tibble(
#' id = vertex_id,
#' memb = memb,
#' x = x,
#' y = y
#' )
#' g <- network::network.initialize(n = N, directed = FALSE)
#' g %v% "vertex.names" <- df$id
#' g %v% "block" <- df$memb
#' g %v% "x" <- df$x
#' g %v% "y" <- df$y
#'
#' # Simulate a network
#' g_sim <-
#' simulate_bigergm(
#' formula = formula,
#' coef_within = list_within_params,
#' coef_between = list_between_params,
#' nsim = 1,
#' control_within = control.simulate.formula(MCMC.burnin = 200000))
#'
#' estimation <- bigergm(update(formula,new = g_sim~.), n_blocks = 10,
#' verbose = T)
#' summary(estimation)
#' }
#' @export
bigergm <- function(object,
add_intercepts = FALSE,
n_blocks = NULL,
n_cores = 1,
blocks = NULL,
estimate_parameters = TRUE,
verbose = 0,
n_MM_step_max = 100,
tol_MM_step = 0.0001,
initialization = "infomap",
use_infomap_python = FALSE,
virtualenv_python = "r-bigergm",
seed_infomap = NULL,
weight_for_initialization = 1000,
seed = NULL,
method_within = "MPLE",
control_within = ergm::control.ergm(),
clustering_with_features = TRUE,
compute_pi = FALSE,
check_alpha_update = FALSE,
check_blocks = FALSE,
cache = NULL,
return_checkpoint = TRUE,
only_use_preprocessed = FALSE,
...) {
###################################################################################
###### Preparations for estimation ################################################
###################################################################################
# Add a check to look for the indicator whether the estimation should not be continued but
# only the preprocessed data be used
if ("bigergm" %in% class(object)) {
if(only_use_preprocessed){
preprocessed_data <- object$checkpoint$preprocessed_data
object <- object$checkpoint$formula
}
}
# When the given object is formula:
if ("formula" %in% class(object)) {
# Check if initialization is a vector of block memberships, a .clu file to be read or whether blocks are known
check <- all(stringr::str_detect(initialization, ".clu")) + (length(initialization) >1) + !is.null(blocks)
# If n_cluster is missing and blocks and initialized_cluster_data are NULL, stop the process.
if (!check && is.null(n_blocks)){
stop("\nThe argument 'n_blocks' is missing. Please specify the number of clusters.")
}
if(length(initialization) >1){
# If initialization is a vector of block memberships, use it as the initial block memberships but leave the n_blocks argument if provided as is.
if (is.null(n_blocks)){
n_blocks <- length(unique(initialization))
}
} else if (all(stringr::str_detect(initialization, ".clu"))){
initialized_cluster_data <- readr::read_delim(initialization, delim = " ", skip = 9, col_names = c("node_id", "block", "flow"), col_types = "iid")
initialization <- as.numeric(dplyr::arrange(initialized_cluster_data, by_group = .data$node_id)$block)
n_blocks <- length(unique(initialization))
}
if (!is.null(blocks)){
if (!is.null(n_blocks)){
warning("The argument 'blocks' is given. The argument 'n_blocks' will be ignored.")
}
n_blocks <- length(unique(blocks))
# Have the blocks be a numeric vector
blocks <- as.numeric(factor(blocks))
}
# Get the formula
formula <- object
# Get network object from formula
network <- ergm::ergm.getnetwork(formula)
if(clustering_with_features) {
# Check if clustering_with_features is TRUE without any features
list_varname <- as.character(formula)[3]
list_varname <- unlist(stringr::str_extract_all(string = list_varname, pattern = '"[^"]*"'))
list_varname <- stringr::str_remove_all(string = list_varname, pattern = '\"')
if(length(list_varname) == 0){
warning("Parameter clustering_with_features was set to TRUE although no covariates are needed for the given formula")
clustering_with_features <- FALSE
}
}
if(!exists("preprocessed_data")){
preprocessed_data <- NULL
}
# If preprocessed_data is not NULL, check if the order of features names is the same with that of formula.
if (!is.null(preprocessed_data)) {
# If clustering_with_features = FALSE, stop the process.
if (!clustering_with_features) {
stop("\nSet clustering_with_features = TRUE since you are going to use vertex features for cluster estimation.")
}
}
}
MM_restart_object <- NULL
# When the given object has a class of "bigergm":
if ("bigergm" %in% class(object)) {
# Inherit the following arguments from the previous estimation.
# These arguments are so important that they won't be replaced by the arguments given by the user.
network <- object$checkpoint$network
formula <- object$checkpoint$formula
n_blocks <- object$checkpoint$n_blocks
clustering_with_features <- object$checkpoint$clustering_with_features
add_intercepts <- object$checkpoint$add_intercepts
preprocessed_data <- object$checkpoint$preprocessed_data
# Inherit the following arguments from the previous estimation if not given by the user.
# If given by the user, discard the inherited argument and use the given one.
vec_arguments <-
c(
"n_cores",
"estimate_parameters",
"verbose",
"n_MM_step_max",
"seed",
"method_within",
"compute_pi",
"check_alpha_update",
"check_blocks"
)
message("Arguments:")
message(glue::glue("Number of clusters = {n_blocks}"))
for (i in 1:length(vec_arguments)) {
if (do.call(missing, list(vec_arguments[[i]])) == TRUE) {
assign(vec_arguments[[i]], object$checkpoint[[vec_arguments[[i]]]])
}
message(glue::glue("{vec_arguments[[i]]} = {get(vec_arguments[[i]])}"))
}
# Prepare an object to restart the MM with.
MM_restart_object <-
list(
alpha = object$alpha,
list_alpha = object$MM_list_alpha,
list_z = object$MM_list_z,
z_memb_init = object$z_memb_final_before_kmeans,
change_in_alpha = object$MM_change_in_alpha,
lower_bound = object$MM_lower_bound,
counter_e_step = object$counter_e_step,
adjacency_matrix = object$adjacency_matrix
)
}
# For the entire estimation process the vertex.names are changed to be integers running from 1 to n
old_vertex_names <- network::network.vertex.names(network)
network::network.vertex.names(network) <- 1:length(old_vertex_names)
# If the blockss of each node are specified in the variable 'blocks' as integers between 1 and n_blocks,
# the specified block memberships are fixed.
all_blockss_fixed <- ifelse(is.null(blocks), FALSE, TRUE)
# Make sure that if all_blockss_fixed == TRUE, estimate_parameters must also be TRUE.
estimate_parameters <- ifelse(all_blockss_fixed, TRUE, estimate_parameters)
###################################################################################
###### First step: Estimating block memberships ###################################
###################################################################################
# Estimate the memberships if they are not specified.
if (!all_blockss_fixed) {
set.seed(seed)
# Estimate the block memberships
answer <- MM_wrapper(
network = network,
formula = formula,
n_blocks = n_blocks,
n_MM_step_max = n_MM_step_max,
tol_MM_step = tol_MM_step,
min_size = 2,
initialization = initialization,
use_infomap_python = use_infomap_python,
virtualenv_python = virtualenv_python,
seed_infomap = seed_infomap,
clustering_with_features = clustering_with_features,
list_multiplied_feature_matrices = preprocessed_data,
verbose = verbose,
weight_for_initialization = weight_for_initialization,
compute_pi = compute_pi,
check_alpha_update = check_alpha_update,
check_blocks = check_blocks,
MM_restart_object = MM_restart_object,
cache = cache
)
blocks <- answer$z_memb_final
initial_block <- answer$z_memb_init
membership_before_kmeans <- answer$z_memb_final_before_kmeans
sbm_pi <- answer$Pi
MM_list_alpha <- answer$list_alpha
MM_list_z <- answer$list_z
MM_change_in_alpha <- answer$change_in_alpha
MM_lower_bound <- answer$lower_bound
alpha <- answer$alpha
counter_e_step <- answer$counter_e_step
adjacency_matrix <- as(answer$adjacency_matrix, "nMatrix")
preprocessed_data <- answer$list_multiplied_feature_matrices
}
else {
if (verbose > 0) {
message("\nSkipping Steps 1 and 2: z specified")
}
param_MM_wrapper <- NULL
initial_block <- NULL
membership_before_kmeans <- NULL
sbm_pi <- NULL
MM_list_z <- NULL
MM_list_alpha <- NULL
MM_change_in_alpha <- NULL
MM_lower_bound <- NULL
alpha <- NULL
counter_e_step <- NULL
adjacency_matrix <- NULL
preprocessed_data <- NULL
}
# Relabel the block to be in canonical form
blocks <- relabel(block = blocks, n_blocks = n_blocks)
####################################################################################################
###### Second step: Estimating between-block parameters ############################################
####################################################################################################
network %v% "block" <- as.numeric(blocks)
if (estimate_parameters) {
if (verbose > 0) {
message("\nEstimate between-block parameters")
}
## Estimate between-block parameters
est_between <- est_between(
formula = formula,
network = network,
add_intercepts = add_intercepts,
clustering_with_features = clustering_with_features
)
} else {
# If you don't estimate any parameters...
est_between <- NULL
}
####################################################################################################
###### Third step: Estimating within-block parameters ##############################################
####################################################################################################
# When you estimate with-block parameters, then:
if (estimate_parameters) {
if (verbose > 0) {
message("\n\nStep 3: Estimate parameters conditional on z")
}
# Estimate within-block parameters
if (verbose == 0) {
suppressMessages(est_within <- est_within(formula =formula,network = network,
block = blocks,number_cores = n_cores,
seed = NULL,method = method_within,
add_intercepts = add_intercepts,
clustering_with_features = clustering_with_features,
control = control_within,
...))
} else {
est_within <- est_within(formula =formula,network = network,
block = blocks,number_cores = n_cores,
seed = NULL,method = method_within,
add_intercepts = add_intercepts,
clustering_with_features = clustering_with_features,
control = control_within,
...)
}
####################################################################################################
###### Store the results ###########################################################################
####################################################################################################
estimation_status <- ifelse(est_within$failure, "Estimation failed", "Estimated")
} else {
# estimate_parameters = FALSE:
message("\n")
estimation_status <- "Not estimated"
est_within <- NULL
}
if(return_checkpoint){
# Change the names back to the old ones
network%v% "vertex.names" <- old_vertex_names
# Store the given arguments. These will be inherited for the next MM.
checkpoint <- list(
network = network,
formula = formula,
n_blocks = n_blocks,
clustering_with_features = clustering_with_features,
preprocessed_data = preprocessed_data,
n_cores = n_cores,
estimate_parameters = estimate_parameters,
verbose = verbose,
n_MM_step_max = n_MM_step_max,
seed = seed,
method_within = method_within,
compute_pi = compute_pi,
check_alpha_update = check_alpha_update,
check_blocks = check_blocks,
add_intercepts = add_intercepts
)
} else{
checkpoint <- NULL
}
# Store the results in a list
output <- list(
call = sys.calls()[[1]],
formula = formula,
block = blocks,
initial_block = initial_block,
sbm_pi = sbm_pi,
MM_list_z = MM_list_z,
MM_list_alpha = MM_list_alpha,
MM_change_in_alpha = MM_change_in_alpha,
MM_lower_bound = MM_lower_bound,
alpha = alpha,
counter_e_step = counter_e_step,
adjacency_matrix = adjacency_matrix,
directed = network$gal$directed,
estimation_status = estimation_status,
est_within = est_within,
est_between = est_between,
checkpoint = checkpoint,
membership_before_kmeans = membership_before_kmeans,
estimate_parameters = estimate_parameters
)
# Return the output
return(structure(output, class = "bigergm"))
}
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