Nucleation in scale-free networks

TL;DR

This paper investigates how the topology of scale-free networks influences the nucleation process of the Ising model, revealing that homogeneous nucleation becomes negligible in large networks and that targeted impurities significantly enhance heterogeneous nucleation rates.

Contribution

The study provides new insights into nucleation dynamics on scale-free networks, including the exponential decay of homogeneous nucleation rate with network size and the impact of targeted impurities on heterogeneous nucleation.

Findings

Homogeneous nucleation rate decays exponentially with network size

Cluster sizes follow a power-law distribution in homogeneous nucleation

Targeted impurities greatly increase heterogeneous nucleation rate

Abstract

We have studied nucleation dynamics of the Ising model in scale-free networks with degree distribution $P(k)\sim k^{-\gamma}$ by using forward flux sampling method, 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 the network size $N$, and accordingly the critical nucleus size increases linearly with $N$, implying that homogeneous nucleation is not relevant in the thermodynamic limit. These observations are robust to the change of $\gamma$ and 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 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 $w^{\gamma-2/\gamma-1}$ and $w$ for targeted and random impurities, respectively. A simple mean field analysis is also present to qualitatively illustrate above simulation results.