  This paper proposes a logistic undirected network formation model which
allows for assortative matching on observed individual characteristics and the
presence of edge-wise fixed effects. We model the coefficients of observed
characteristics to have a latent community structure and the edge-wise fixed
effects to be of low rank. We propose a multi-step estimation procedure
involving nuclear norm regularization, sample splitting, iterative logistic
regression and spectral clustering to detect the latent communities. We show
that the latent communities can be exactly recovered when the expected degree
of the network is of order log n or higher, where n is the number of nodes in
the network. The finite sample performance of the new estimation and inference
methods is illustrated through both simulated and real datasets.
