  Understanding the sensitivity of a system's behavior with respect to
parameter changes is essential for many applications. This sensitivity may be
desired - for instance in the brain, where a large repertoire of different
dynamics, particularly different synchronization patterns, is crucial - or may
be undesired - for instance in power grids, where disruptions to
synchronization may lead to blackouts. In this work, we show that the dynamics
of networks of phase oscillators can acquire a very large and complex
sensitivity to changes made in either their units' parameters or in their
connections - even modifications made to a parameter of a single unit can
radically alter the global dynamics of the network in an unpredictable manner.
As a consequence, each modification leads to a different path to phase
synchronization manifested as large fluctuations along that path. This
dynamical malleability occurs over a wide parameter region, around the
network's two transitions to phase synchronization. One transition is induced
by increasing the coupling strength between the units, and another is induced
by increasing the prevalence of long-range connections. Specifically, we study
Kuramoto phase oscillators connected under either Watts-Strogatz or
distance-dependent topologies to analyze the statistical properties of the
fluctuations along the paths to phase synchrony. We argue that this increase in
the dynamical malleability is a general phenomenon, as suggested by both
previous studies and the theory of phase transitions.
