  Observing and controlling complex networks are of paramount interest for
understanding complex physical, biological and technological systems. Recent
studies have made important advances in identifying sensor or driver nodes,
through which we can observe or control a complex system. Yet, the
observational uncertainty induced by measurement noise and the energy required
for control continue to be significant challenges in practical applications.
Here we show that the variability of control energy and observational
uncertainty for different directions of the state space depend strongly on the
number of driver nodes. In particular, we find that if all nodes are directly
driven, control is energetically feasible, as the maximum energy increases
sublinearly with the system size. If, however, we aim to control a system
through a single node, control in some directions is energetically prohibitive,
increasing exponentially with the system size. For the cases in between, the
maximum energy decays exponentially when the number of driver nodes increases.
We validate our findings in several model and real networks, arriving to a
series of fundamental laws to describe the control energy that together deepen
our understanding of complex systems.
