# Fit a Generalized Stochastic Block Model Given an adjacency matrix with missing observations, the function `gsbm_mgcd` robustly estimates the probabilities of connections between nodes. ```r gsbm_mcgd( A, lambda1, lambda2, epsilon = 0.1, U = NULL, maxit = 100, thresh = 1e-06, S0 = NULL, L0 = NULL, R0 = NULL, trace.it = FALSE ) ``` ## Arguments - `A`: nxn adjacency matrix - `lambda1`: regularization parameter for nuclear norm penalty (positive number) - `lambda2`: regularization parameter for 2,1-norm penalty (positive number) - `epsilon`: regularization parameter for the L2-norm penalty (positive number, if NULL, default method is applied) - `U`: lower bound on nuclear norm (positive number, if NULL, default method is applied) - `maxit`: maximum number of iterations (positive integer, if NULL, default method is applied) - `thresh`: convergence tolerance (positive number, if NULL, default method is applied) - `S0`: initial value for the sparse component (if NULL, default method is applied)) - `L0`: initial value for the low-rank component (if NULL, default method is applied) - `R0`: lower bound on nuclear norm of L0 (positive number, if NULL, default method is applied) - `trace.it`: whether messages about convergence should be printed (boolean, if NULL, default is FALSE) ## Returns The estimate for the nxn matrix of probabilities of connections between nodes. It is given as the sum of a low-rank nxn matrix L, corresponding to connections between inlier nodes, and a column sparse nxn matrix S, corresponding to connections between outlier nodes and the rest of the network. The matrices L and S are such that E(A) = L - diag(L) + S + S' where E(A) is the expectation of the adjacency matrix, diag(L) is a nxn diagonal matrix with diagonal entries equal to those of L, and S' means S transposed. The return value is a list of components * `A`the adjacency matrix. * `L`estimate for the low-rank component. * `S`estimate for the column-sparse component. * `objective`the value of the objective function. * `R`a bound on the nuclear norm of the low-rank component. * `iter`number of iterations between convergence of the objective function. ## Examples ```r # Draw a 50x50 adjacency matrix # Generalized SBM with 2 communities and 2 outliers # Create low-rank matrix L L <- matrix(0,50,50) # low-rank component L[1:25, 1:25] <- 0.6 # connection probabilities within community 1 L[1:25, 26:48] <- 0.1 # connection probabilities between communities 1 and 2 L[26:48, 1:25] <- 0.1 # connection probabilities between communities 1 and 2 L[26:48, 26:48] <- 0.6 # connection probabilities within community 2 # Create column-sparse matrix S S <- matrix(0,50,50) # column sparse component S[49:50,1:48] <- 0.6 # connection probabilties between outliers and inliers # Draw connections and create the adjacency matrix undir <- rbinom(n=50*(50-1)/2, size=1, prob=(L+S+t(S))[upper.tri(L+S+t(S))]) # draw edges A <- matrix(0,50,50) A[upper.tri(A)] <- undir A <- (A+t(A)) # Estimate the probabilities of connection lambda1 <- 7 lambda2 <- 7 res <- gsbm_mcgd(A, lambda1, lambda2) ```

gsbm_mcgd function