  Phase transitions are crucial in shaping the collective dynamics of a broad
spectrum of natural systems across disciplines. Here, we report two distinct
heterogeneous nucleation facilitating single-step and multi-step phase
transitions to global synchronization in a finite-size adaptive network due to
the trade-off between time scale adaptation and coupling strength disparities.
Specifically, small intracluster nucleations coalesce either at the population
interface or within the populations resulting in the two distinct phase
transitions depending on the degree of the disparities. We find that the
coupling strength disparity largely controls the nature of phase transition in
the phase diagram irrespective of the adaptation disparity. We provide a
mesoscopic description for the cluster dynamics using the collective
coordinates approach that brilliantly captures the multicluster dynamics among
the populations leading to distinct phase transitions. Further, we also deduce
the upper bound for the coupling strength for the existence of two
intraclusters explicitly in terms of adaptation and coupling strength
disparities. These insights may have implications across domains ranging from
neurological disorders to segregation dynamics in social networks.
