# Estimating Network Formation Models with Degree Heterogeneity: the Bayesian Random Effect Approach `homophily.re` implements a Bayesian Probit estimator for network formation model with homophily. The model includes degree heterogeneity using random effects (see details). ```r homophily.re( network, formula, data, symmetry = FALSE, group.fe = FALSE, re.way = 1, init = list(), iteration = 1000, print = TRUE ) ``` ## Arguments - `network`: matrix or list of sub-matrix of social interactions containing 0 and 1, where links are represented by 1. - `formula`: an object of class formula : a symbolic description of the model. The `formula` should be as for example `~ x1 + x2` where `x1`, `x2` are explanatory variables for links formation. - `data`: an optional data frame, list, or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from `environment(formula)`, typically the environment from which `homophily` is called. - `symmetry`: indicates whether the network model is symmetric (see details). - `group.fe`: indicates whether the model includes group fixed effects. - `re.way`: indicates whether it is a one-way or two-way random effect model. The expected value is 1 or 2 (see details). - `init`: (optional) list of starting values containing `beta`, a K-dimensional vector of the explanatory variables parameter, `mu`, an n-dimensional vector, and `nu`, an n-dimensional vector, `smu2` the variance of `mu`, and `snu2` the variance of `nu`, where K is the number of explanatory variables and n is the number of individuals. - `iteration`: the number of iterations to be performed. - `print`: boolean indicating if the estimation progression should be printed. ## Returns A list consisting of: - **model.info**: list of model information, such as the type of random effects, whether the model is symmetric, number of observations, etc. - **posterior**: list of simulations from the posterior distribution. - **init**: returned list of starting values. ## Details Let $Pij$ be a probability for a link to go from the individual $i$ to the individual $j$. This probability is specified for two-way effect models (`re.way = 2`) as $$ p_{ij} = F(\mathbf{x}_{ij}'\beta + \mu_i + \nu_j), $$ where $F$ is the cumulative of the standard normal distribution. Unobserved degree heterogeneity is captured by $\mu_i$ and $\nu_j$. The latter are treated as random effects (see `homophily.fe` for fixed effect models). For one-way random effect models (`re.way = 1`), $\nu_j = \mu_j$. For symmetric models, the network is not directed and the random effects need to be one way. ## Examples ```r set.seed(1234) library(MASS) M <- 4 # Number of sub-groups nvec <- round(runif(M, 100, 500)) beta <- c(.1, -.1) Glist <- list() dX <- matrix(0, 0, 2) mu <- list() nu <- list() cst <- runif(M, -1.5, 0) smu2 <- 0.2 snu2 <- 0.2 rho <- 0.8 Smunu <- matrix(c(smu2, rho*sqrt(smu2*snu2), rho*sqrt(smu2*snu2), snu2), 2) for (m in 1:M) { n <- nvec[m] tmp <- mvrnorm(n, c(0, 0), Smunu) mum <- tmp[,1] - mean(tmp[,1]) num <- tmp[,2] - mean(tmp[,2]) X1 <- rnorm(n, 0, 1) X2 <- rbinom(n, 1, 0.2) Z1 <- matrix(0, n, n) Z2 <- matrix(0, n, n) for (i in 1:n) { for (j in 1:n) { Z1[i, j] <- abs(X1[i] - X1[j]) Z2[i, j] <- 1*(X2[i] == X2[j]) } } Gm <- 1*((cst[m] + Z1*beta[1] + Z2*beta[2] + kronecker(mum, t(num), "+") + rnorm(n^2)) > 0) diag(Gm) <- 0 diag(Z1) <- NA diag(Z2) <- NA Z1 <- Z1[!is.na(Z1)] Z2 <- Z2[!is.na(Z2)] dX <- rbind(dX, cbind(Z1, Z2)) Glist[[m]] <- Gm mu[[m]] <- mum nu[[m]] <- num } mu <- unlist(mu) nu <- unlist(nu) out <- homophily.re(network = Glist, formula = ~ dX, group.fe = TRUE, re.way = 2, iteration = 1e3) # plot simulations plot(out$posterior$beta[,1], type = "l") abline(h = cst[1], col = "red") plot(out$posterior$beta[,2], type = "l") abline(h = cst[2], col = "red") plot(out$posterior$beta[,3], type = "l") abline(h = cst[3], col = "red") plot(out$posterior$beta[,4], type = "l") abline(h = cst[4], col = "red") plot(out$posterior$beta[,5], type = "l") abline(h = beta[1], col = "red") plot(out$posterior$beta[,6], type = "l") abline(h = beta[2], col = "red") plot(out$posterior$sigma2_mu, type = "l") abline(h = smu2, col = "red") plot(out$posterior$sigma2_nu, type = "l") abline(h = snu2, col = "red") plot(out$posterior$rho, type = "l") abline(h = rho, col = "red") i <- 10 plot(out$posterior$mu[,i], type = "l") abline(h = mu[i], col = "red") plot(out$posterior$nu[,i], type = "l") abline(h = nu[i], col = "red") ``` ## See Also `homophily.fe`.

homophily.re function