  This paper deals with dynamical networks for which the relations between node
signals are described by proper transfer functions and external signals can
influence each of the node signals. We are interested in graph-theoretic
conditions for identifiability of such dynamical networks, where we assume that
only a subset of nodes is measured but the underlying graph structure of the
network is known. This problem has recently been investigated from a generic
viewpoint. Roughly speaking, generic identifiability means that the transfer
functions in the network can be identified for "almost all" network matrices
associated with the graph. In this paper, we investigate the stronger notion of
identifiability for all network matrices. To this end, we introduce a new
graph-theoretic concept called the graph simplification process. Based on this
process, we provide necessary and sufficient topological conditions for
identifiability. Notably, we also show that these conditions can be verified by
polynomial time algorithms. Finally, we explain how our results generalize
existing sufficient conditions for identifiability.
