  Understanding the dynamics of complex systems is a central task in many
different areas ranging from biology via epidemics to economics and
engineering. Unexpected behaviour of dynamic systems or even system failure is
sometimes difficult to comprehend. Such a data-mismatch can be caused by
endogenous model errors including misspecified interactions and inaccurate
parameter values. These are often difficult to distinguish from unmodelled
process influencing the real system like unknown inputs or faults. Localizing
the root cause of these errors or faults and reconstructing their dynamics is
only possible if the measured outputs of the system are sufficiently
informative.
  Here, we present criteria for the measurements required to localize the
position of error sources in large dynamic networks. We assume that faults or
errors occur at a limited number of positions in the network. This invariable
sparsity differs from previous sparsity definitions for inputs to dynamic
systems. We provide an exact criterion for the recovery of invariable sparse
inputs to nonlinear systems and formulate an optimization criterion for
invariable sparse input reconstruction. For linear systems we can provide exact
error bounds for this reconstruction method.
