  It is notoriously difficult to predict the behaviour of a complex
self-organizing system, where the interactions among dynamical units form a
heterogeneous topology. Even if the dynamics of each microscopic unit is known,
a real understanding of their contributions to the macroscopic system behaviour
is still lacking. Here we develop information-theoretical methods to
distinguish the contribution of each individual unit to the collective
out-of-equilibrium dynamics. We show that for a system of units connected by a
network of interaction potentials with an arbitrary degree distribution, highly
connected units have less impact on the system dynamics as compared to
intermediately connected units. In an equilibrium setting, the hubs are often
found to dictate the long-term behaviour. However, we find both analytically
and experimentally that the instantaneous states of these units have a
short-lasting effect on the state trajectory of the entire system. We present
qualitative evidence of this phenomenon from empirical findings about a social
network of product recommendations, a protein-protein interaction network, and
a neural network, suggesting that it might indeed be a widespread property in
nature.
