This paper studies linear time-varying (LTV) network systems affected by multiple unknown inputs. The goal is to reconstruct both the initial state and the unknown input. The main result is a characterization of strong structural input and state observability, i.e., the conditions under which both the whole network state and the unknown input can be reconstructed for all system matrices that share a common zero/non-zero pattern. This characterization is in terms of strong structural observability of a suitably-defined linear time-invariant (LTI) subsystem.