An Introduction to Estimating Joint Probability Models with iglm

Overview

This vignette provides an introduction to the iglm package, which is designed for estimating joint probability models that incorporate network structures. The package allows users to analyze how individual attributes and network connections jointly influence outcomes of interest.

Basic Usage

To use the iglm package, you first need to load it into your R session

library(iglm)

Next, you can create a iglm object by specifying the network structure and the attributes of interest. Here is a simple example:

n_actor <- 100

attribute_info <- rnorm(n_actor)
attribute_cov <- diag(attribute_info)
edge_cov <- outer(attribute_info, attribute_info, FUN = function(x, y) {
  abs(x - y)
})
set.seed(123)

alpha <- 0.3
block <- matrix(nrow = 50, ncol = 50, data = 1)
neighborhood <- as.matrix(Matrix::bdiag(replicate(n_actor / 50, block, simplify = FALSE)))

overlapping_degree <- 0.5
neighborhood <- matrix(nrow = n_actor, ncol = n_actor, data = 0)
block <- matrix(nrow = 5, ncol = 5, data = 0)
size_neighborhood <- 5
size_overlap <- ceiling(size_neighborhood * overlapping_degree)

end <- floor((n_actor - size_neighborhood) / size_overlap)
for (i in 0:end) {
  neighborhood[(1 + size_overlap * i):(size_neighborhood + size_overlap * i), (1 + size_overlap * i):(size_neighborhood + size_overlap * i)] <- 1
}
neighborhood[(n_actor - size_neighborhood + 1):(n_actor), (n_actor - size_neighborhood + 1):(n_actor)] <- 1

type_x <- "binomial"
type_y <- "binomial"
formula_beg <- as.formula("xyz_obj ~ 1 ")
formula_model <- as.formula("xyz_object ~ 1 ")

object <- iglm.data(neighborhood = neighborhood, directed = F, type_x = type_x, type_y = type_y)

Model Specification

You can specify a model formula that includes various network statistics and attribute effects. For example:

formula <- object ~ edges + attribute_y + attribute_x + degrees

To fully define the model, you need to set up a sampler for the MCMC estimation and set all necessary parameters:

# Parameters of edges(mode = "local"), attribute_y, and attribute_x
gt_coef <- c(3, -1, -1)
# Parameters for popularity effect
gt_coef_pop <- c(rnorm(n = n_actor, -2, 1))
# Define the sampler
sampler_tmp <- sampler.iglm(
  n_burn_in = 100, n_simulation = 10,
  sampler_x = sampler.net.attr(n_proposals = n_actor * 10, seed = 13),
  sampler_y = sampler.net.attr(n_proposals = n_actor * 10, seed = 32),
  sampler_z = sampler.net.attr(n_proposals = sum(neighborhood > 0) * 10, seed = 134),
  init_empty = F
)

model_tmp_new <- iglm(
  formula = formula,
  coef = gt_coef, coef_degrees = gt_coef_pop, sampler = sampler_tmp,
  control = control.iglm(accelerated = F, max_it = 200, display_progress = F)
)

Model Simulation

Once you have specified a model, you can simulate new data based on the fitted parameters:

# Simulate new networks
model_tmp_new$simulate()
# Get the samples
tmp <- model_tmp_new$get_samples()

Model Estimation

You can estimate the model parameters using the estimate method:

# First set the first simulated network as the target for estimation
model_tmp_new$set_target(tmp[[1]])
model_tmp_new$estimate()
model_tmp_new$iglm.data$degree_distribution(plot = TRUE)

Model Assessment

After estimation, you can assess the model fit using various diagnostics:

model_tmp_new$assess(formula = ~ degree_distribution +
  geodesic_distances_distribution + edgewise_shared_partner_distribution + mcmc_diagnostics)

model_tmp_new$results$plot(model_assessment = T)

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