  Network reconstruction, i.e., obtaining network structure from data, is a
central theme in systems biology, economics and engineering. In some previous
work, we introduced dynamical structure functions as a tool for posing and
solving the problem of network reconstruction between measured states. While
recovering the network structure between hidden states is not possible since
they are not measured, in many situations it is important to estimate the
minimal number of hidden states in order to understand the complexity of the
network under investigation and help identify potential targets for
measurements. Estimating the minimal number of hidden states is also crucial to
obtain the simplest state-space model that captures the network structure and
is coherent with the measured data. This paper characterizes minimal order
state-space realizations that are consistent with a given dynamical structure
function by exploring properties of dynamical structure functions and
developing an algorithm to explicitly obtain such a minimal realization.
