  Factor analysis is a widely used statistical tool in many scientific
disciplines, such as psychology, economics, and sociology. As observations
linked by networks become increasingly common, incorporating network structures
into factor analysis remains an open problem. In this paper, we focus on
high-dimensional factor analysis involving network-connected observations, and
propose a generalized factor model with latent factors that account for both
the network structure and the dependence structure among high-dimensional
variables. These latent factors can be shared by the high-dimensional variables
and the network, or exclusively applied to either of them. We develop a
computationally efficient estimation procedure and establish asymptotic
inferential theories. Notably, we show that by borrowing information from the
network, the proposed estimator of the factor loading matrix achieves optimal
asymptotic variance under much milder identifiability constraints than existing
literature. Furthermore, we develop a hypothesis testing procedure to tackle
the challenge of discerning the shared and individual latent factors'
structure. The finite sample performance of the proposed method is demonstrated
through simulation studies and a real-world dataset involving a statistician
co-authorship network.
