R/get_edge_assignment_probs.R

In multigraphr: Probability Models and Statistical Analysis of Random Multigraphs

#' @title Edge assignment probabilities under IEA model given fixed degrees
#' @description Calculates the edge assignment probabilities
#' given specified degree sequence under the two ways in which the RSM
#' model can be approximated by the IEA model: \cr
#'  - the IEAS (independent edge assignment of stubs) model, \cr
#'  - the ISA (independent stub assignment) model.
#' @param m integer giving number of edges in multigraph.
#' @param deg.seq vector of integers with the sum equal to 2\code{m} representing
#' the degree sequence of the multigraph.
#' @param model character string, either \code{'IEAS'} or \code{'ISA'}.
#' @return  A numeric vector representing the edge assignment probabilities
#' to all possible vertex pair sites. The number of vertex pair sites is given by \eqn{n(n+1)/2}.
#' @details The IEAS and ISA edge assignment probabilities to
#' possible vertex pairs are calculated given a fixed degree sequence \code{deq.seq}
#' under the IEAS model, and \code{deg.seq}/2\code{m} under the ISA model.
#'
#' Number of possible vertex pair sites (and thus the length of the edge assignment sequence) is given by
#' \eqn{(n+1)n/2} where \emph{n} is number of vertices.
#' @author Termeh Shafie
#' @seealso \code{\link{get_degree_seq}}, \code{\link{iea_model}}
#' @references Shafie, T. (2015). A Multigraph Approach to Social Network Analysis. \emph{Journal of Social Structure}, 16.
#' \cr
#'
#' Shafie, T. (2016). Analyzing Local and Global Properties of Multigraphs. \emph{The Journal of Mathematical Sociology}, 40(4), 239-264.
#' \cr
#'
#' #' Shafie, T., Schoch, D. (2021). Multiplexity analysis of networks using multigraph representations.
#' \emph{Statistical Methods & Applications} 30, 1425–1444.
#' @examples
#' # Under the IEAS model with 10 possible vertex pair sites (4 vertices)
#' get_edge_assignment_probs(m = 8, deg.seq = c(4, 4, 4, 4), model = "IEAS")
#'
#' # Under the ISA model with 21 possible vertex pair sites (6 vertices)
#' get_edge_assignment_probs(m = 10, deg.seq = c(8, 4, 2, 2, 2, 2), model = "ISA")
#' @export
#'
get_edge_assignment_probs <- function(m, deg.seq, model) {
    if (sum(deg.seq) %% 2 == 1) {
        stop("not a graphical degree sequence, the sum is not equal to 2m.")
    }
    n <- length(deg.seq)
    if (model == "IEAS") {
        Q.mat <- matrix(0, n, n)
        for (i in 1:n) {
            for (j in 1:n) {
                if (i == j) {
                    Q.mat[i, j] <- deg.seq[i] * (deg.seq[i] - 1) / (2 * m * (2 * m - 1))
                } else {
                    Q.mat[i, j] <- 2 * deg.seq[i] * (deg.seq[j]) / (2 * m * (2 * m - 1))
                }
            }
        }
        Q.seq <- t(Q.mat)[lower.tri(Q.mat, diag = TRUE)]
        return(Q.seq)
    } else if (model == "ISA") {
        p.seq <- deg.seq / (2 * m)
        Q.mat <- matrix(0, n, n)
        for (i in 1:n) {
            for (j in 1:n) {
                if (i == j) {
                    Q.mat[i, j] <- p.seq[i]^2
                } else {
                    Q.mat[i, j] <- 2 * p.seq[i] * p.seq[j]
                }
            }
        }
        Q.seq <- t(Q.mat)[lower.tri(Q.mat, diag = TRUE)]
        return(Q.seq)
    } else {
        stop("model must be one of 'IEAS' or 'ISA'.")
    }
}