Nucleation in scale-free networks.

We have studied nucleation dynamics of the Ising model in scale-free networks whose degree distribution follows a power law with the exponent γ by using the forward flux sampling method and focusing on how the network topology would influence the nucleation rate and pathway. For homogeneous nucleation, the new phase clusters grow from those nodes with smaller degree, while the cluster sizes follow a power-law distribution. Interestingly, we find that the nucleation rate R{Hom} decays exponentially with network size and, accordingly, the critical nucleus size increases linearly with network size, implying that homogeneous nucleation is not relevant in the thermodynamic limit. These observations are robust to the change of γ and are also present in random networks. In addition, we have also studied the dynamics of heterogeneous nucleation, wherein w impurities are initially added either to randomly selected nodes or to targeted ones with the largest degrees. We find that targeted impurities can enhance the nucleation rate R{Het} much more sharply than random ones. Moreover, ln(R{Het}/R{Hom}) scales as w{(γ-2)/(γ-1)} and w for targeted and random impurities, respectively. A simple mean-field analysis is also present to qualitatively illustrate the above simulation results.