  Identifying nucleation pathway is important for understanding the kinetics of
first-order phase transitions in natural systems. In the present work, we study
nucleation pathway of the Ising model in homogeneous and heterogeneous networks
using the forward flux sampling method, and find that the nucleation processes
represent distinct features along pathways for different network topologies.
For homogeneous networks, there always exists a dominant nucleating cluster to
which relatively small clusters are attached gradually to form the critical
nucleus. For heterogeneous ones, many small isolated nucleating clusters emerge
at the early stage of the nucleation process, until suddenly they form the
critical nucleus through a sharp merging process. By analyzing the properties
of the nucleating clusters along the pathway, we show that the main reason
behind the different routes is the heterogeneous character of the underlying
networks.
