  Network structures play important roles in social, technological and
biological systems. However, the observable nodes and connections in real cases
are often incomplete or unavailable due to measurement errors, private
protection issues, or other problems. Therefore, inferring the complete network
structure is useful for understanding human interactions and complex dynamics.
The existing studies have not fully solved the problem of inferring network
structure with partial information about connections or nodes. In this paper,
we tackle the problem by utilizing time-series data generated by network
dynamics. We regard the network inference problem based on dynamical time
series data as a problem of minimizing errors for predicting states of
observable nodes and proposed a novel data-driven deep learning model called
Gumbel-softmax Inference for Network (GIN) to solve the problem under
incomplete information. The GIN framework includes three modules: a dynamics
learner, a network generator, and an initial state generator to infer the
unobservable parts of the network. We implement experiments on artificial and
empirical social networks with discrete and continuous dynamics. The
experiments show that our method can infer the unknown parts of the structure
and the initial states of the observable nodes with up to 90\% accuracy. The
accuracy declines linearly with the increase of the fractions of unobservable
nodes. Our framework may have wide applications where the network structure is
hard to obtain and the time series data is rich.
