  Synchronization has been the subject of intense research during decades
mainly focused on determining the structural and dynamical conditions driving a
set of interacting units to a coherent state globally stable. However, little
attention has been paid to the description of the dynamical development of each
individual networked unit in the process towards the synchronization of the
whole ensemble. In this paper, we show how in a network of identical dynamical
systems, nodes belonging to the same degree class differentiate in the same
manner visiting a sequence of states of diverse complexity along the route to
synchronization independently on the global network structure. In particular,
we observe, just after interaction starts pulling orbits from the initially
uncoupled attractor, a general reduction of the complexity of the dynamics of
all units being more pronounced in those with higher connectivity. In the weak
coupling regime, when synchronization starts to build up, there is an increase
in the dynamical complexity whose maximum is achieved, in general, first in the
hubs due to their earlier synchronization with the mean field. For very strong
coupling, just before complete synchronization, we found a hierarchical
dynamical differentiation with lower degree nodes being the ones exhibiting the
largest complexity departure. We unveil how this differentiation route holds
for several models of nonlinear dynamics including toroidal chaos and how it
depends on the coupling function. This study provides new insights to
understand better strategies for network identification and control or to
devise effective methods for network inference.
