  We analyze in detail the subtle yet critical differences between the
structural controllability and observability of the triplet $(A,B,C)$ in the
two cases that this is viewed as a linear dynamical network of interconnected
nodes or as a a single complex system. Investigating the controllability and
observability properties of each single node when the network is not completely
controllable and/or observable, we show that the first point of view requires
the development of novel tools leading, ultimately, to a state space
decomposition that is different from the one proposed in 1963 by R.E. Kalman
for linear systems.
