install.packages("sna")
install.packages("igraph")
library(sna)
library(igraph)

# This is the Log-likelihood function in Meneses et al.'s article (2017).
# This function receives as inputs:
#    p: parameter of randomness, based on Watts and Strogatz's model, and
#    AdjMat: adjacency matrix of the social network.

loglike <- function(p, AdjMat){
    n <- nrow(AdjMat)                 # n: number of nodes
    Dens <- sna::gden(AdjMat)         # Dens: network density of the social network.
    K <- round(Dens * (n - 1)/2, 0)   # K: number of neighbors a node has to its rights side in the regular lattice before rewiring.

    # logProbs: vector of the logarithm of the probabilities of observing the degree centrality of each node in social network, based on Menezes et al.'s paper.
    logProbs <- log(apply(X = matrix(seq(n), ncol = 1),
        MARGIN = 1,
        FUN = function(j){
            m <- sum(AdjMat[, j])
            LowLim <- max(c(2 * K - m, 0))
            UppLim <- min(c(n - 1 - m, 2 * K))
            a <- 1/2 * p * (n - 1 - 2 * K)/n
            b <- p/n
            Factor1 <- dbinom(x = LowLim:UppLim, size = 2 * K, prob = a)
            Factor2 <- dbinom(x = m - 2 * K + LowLim:UppLim, size = (n - 2) * K, prob = b)
            sum(Factor1 * Factor2)
            }
        ))
    sum(logProbs)      # output: loglikelihood of observed data.
    }

# This part of the program attemps to build the sampling distribution of the maximum likelihood estimator (MLE) of the randomness parameter p, with 0 <= p <= 1.

p <- 0.2     # True value of the randomness parameter.

# This part of the program simulates 1000 times a social network of N nodes with density d and parameter of randomness p. For each social network, I obtain the MLE of p. Then, I build the sampling distribution of the MLE of p.

d <- 0.1
N <- 2000
MLE_p <- NULL
for(i in 1:100){
    K <- round(d * (N - 1)/2, 0)
    AdjMat <- as.matrix(igraph::get.adjacency(igraph::watts.strogatz.game(dim = 1,
    size = N, nei = K, p = p)))
    OptProblem <- nlminb(start = runif(1),
        objective = function(x) -loglike(p = x, AdjMat = AdjMat),
        lower = 0, upper = 1,
        control = list(trace = TRUE))
    MLE_p <- c(MLE_p, OptProblem$par)
    }

hist(MLE_p, col = "gray60")