  We study nonlinear dynamics on complex networks. Each vertex $i$ has a state
$x_i$ which evolves according to a networked dynamics to a steady-state
$x_i^*$. We develop fundamental tools to learn the true steady-state of a small
part of the network, without knowing the full network. A naive approach and the
current state-of-the-art is to follow the dynamics of the observed partial
network to local equilibrium. This dramatically fails to extract the true
steady state. We use a mean-field approach to map the dynamics of the unseen
part of the network to a single node, which allows us to recover accurate
estimates of steady-state on as few as 5 observed vertices in domains ranging
from ecology to social networks to gene regulation. Incomplete networks are the
norm in practice, and we offer new ways to think about nonlinear dynamics when
only sparse information is available.
