# Estimating Network Formation Models with Degree Heterogeneity: the Fixed Effect Approach `homophily.fe` implements a Logit estimator for a network formation model with homophily. The model includes degree heterogeneity using fixed effects (see details). ```r homophily.fe( network, formula, data, symmetry = FALSE, fe.way = 1, init = NULL, method = c("L-BFGS", "Block-NRaphson", "Mix"), ctr = list(maxit.opt = 10000, maxit.nr = 50, eps_f = 1e-09, eps_g = 1e-09, tol = 1e-04), print = TRUE ) ``` ## Arguments - `network`: A matrix or list of sub-matrices 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, for example, `~ x1 + x2`, where `x1` and `x2` are explanatory variables for link formation. If missing, the model is estimated with fixed effects only. - `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). - `fe.way`: Indicates whether it is a one-way or two-way fixed effect model. The expected value is 1 or 2 (see details). - `init`: (optional) Either a list of starting values containing `beta`, a K-dimensional vector of the explanatory variables' parameters, `mu`, an n-dimensional vector, and `nu`, an n-dimensional vector, where K is the number of explanatory variables and n is the number of individuals; or a vector of starting values for `c(beta, mu, nu)`. - `method`: A character string specifying the optimization method. Expected values are `"L-BFGS"`, `"Block-NRaphson"`, or `"Mix"`. `"Block-NRaphson"` refers to the `Newton-Raphson` method applied to each subnetwork, and `"Mix"` combines the `Newton-Raphson` method for `beta` with the `L-BFGS` method for the fixed effects. - `ctr`: (optional) A list containing control parameters for the solver. For the `optim_lbfgs` method from the `RcppNumerical` package, the list should include `maxit.opt` (corresponding to `maxit` for the `L-BFGS` method), `eps_f`, and `eps_g`. For the `Block-NRaphson` method, the list should include `maxit.nr` (corresponding to `maxit` for the `Newton-Raphson` method) and `tol`. - `print`: A boolean indicating if the estimation progression should be printed. ## Returns A list consisting of: - **model.info**: A list of model information, such as the type of fixed effects, whether the model is symmetric, the number of observations, etc. - **estimate**: The maximizer of the log-likelihood. - **loglike**: The maximized log-likelihood. - **optim**: The returned value from the optimization solver, which contains details of the optimization. The solver used is `optim_lbfgs` from the `RcppNumerical` package. - **init**: The returned list of starting values. - **loglike.init**: The log-likelihood at the starting values. ## Details Let $p_{ij}$ be the probability for a link to go from individual $i$ to individual $j$. This probability is specified for two-way effect models (`fe.way = 2`) as $$ p_{ij} = F(\mathbf{x}_{ij}'\beta + \mu_i + \nu_j), $$ where $F$ is the cumulative distribution function of the standard logistic distribution. Unobserved degree heterogeneity is captured by $\mu_i$ and $\nu_j$. These are treated as fixed effects (see `homophily.re` for random effect models). As shown by Yan et al. (2019), the estimator of the parameter $\beta$ is biased. A bias correction is necessary but not implemented in this version. However, the estimators of $\mu_i$ and $\nu_j$ are consistent. For one-way fixed effect models (`fe.way = 1`), $\nu_j = \mu_j$. For symmetric models, the network is not directed, and the fixed effects need to be one-way. ## Examples ```r set.seed(1234) M <- 2 # Number of sub-groups nvec <- round(runif(M, 20, 50)) beta <- c(.1, -.1) Glist <- list() dX <- matrix(0, 0, 2) mu <- list() nu <- list() Emunu <- runif(M, -1.5, 0) # Expectation of mu + nu smu2 <- 0.2 snu2 <- 0.2 for (m in 1:M) { n <- nvec[m] mum <- rnorm(n, 0.7*Emunu[m], smu2) num <- rnorm(n, 0.3*Emunu[m], snu2) 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*((Z1*beta[1] + Z2*beta[2] + kronecker(mum, t(num), "+") + rlogis(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.fe(network = Glist, formula = ~ -1 + dX, fe.way = 2) muhat <- out$estimate$mu nuhat <- out$estimate$nu plot(mu, muhat) plot(nu, nuhat) ``` ## References Yan, T., Jiang, B., Fienberg, S. E., & Leng, C. (2019). Statistical inference in a directed network model with covariates. **Journal of the American Statistical Association**, 114(526), 857-868, tools:::Rd_expr_doi("https://doi.org/10.1080/01621459.2018.1448829") . ## See Also `homophily.re`.

homophily.fe function