  We investigate the reconstruction of time series from dynamical networks that
are partially observed. In particular, we address the extent to which the time
series at a node of the network can be successfully reconstructed when
measuring from another node, or subset of nodes, corrupted by observational
noise. We will assume the dynamical equations of the network are known, and
that the dynamics are not necessarily low-dimensional. The case of linear
dynamics is treated first, and leads to a definition of observation error
magnification factor (OEMF) that measures the magnification of noise in the
reconstruction process. Subsequently, the definition is applied to nonlinear
and chaotic dynamics. Comparison of OEMF for different target/observer
combinations can lead to better understanding of how to optimally observe a
network. As part of the study, a computational method for reconstructing time
series from partial observations is presented and analyzed.
