# Joint Modeling of Networks and Attributes 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.

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

## 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 degree effect
    gt_coef_degrees <- 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),
      sampler_y = sampler.net.attr(n_proposals = n_actor * 10),
      sampler_z = sampler.net.attr(n_proposals = sum(neighborhood > 0) * 10),
      init_empty = F
    )

    model_tmp_new <- iglm(
      formula = formula,
      coef = gt_coef, coef_degrees = gt_coef_degrees, 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)