Preferential attachment during the evolution of a potential energy landscape

TL;DR

This study investigates how the network of minima on a potential energy landscape evolves with potential range, revealing a preferential attachment process that leads to scale-free network properties and basin size distributions.

Contribution

It demonstrates that the growth of the energy landscape network involves preferential attachment, explaining the emergence of scale-free properties in the minima network.

Findings

Network becomes scale-free as potential range decreases.

New edges preferentially attach to highly-connected minima.

Basin sizes follow a power-law distribution.

Abstract

It has previously been shown that the network of connected minima on a potential energy landscape is scale-free, and that this reflects a power-law distribution for the areas of the basins of attraction surrounding the minima. Here, we set out to understand more about the physical origins of these puzzling properties by examining how the potential energy landscape of a 13-atom cluster evolves with the range of the potential. In particular, on decreasing the range of the potential the number of stationary points increases and thus the landscape becomes rougher and the network gets larger. Thus, we are able to follow the evolution of the potential energy landscape from one with just a single minimum to a complex landscape with many minima and a scale-free pattern of connections. We find that during this growth process, new edges in the network of connected minima preferentially attach to more highly-connected minima, thus leading to the scale-free character. Furthermore, minima that appear when the range of the potential is shorter and the network is larger have smaller basins of attraction. As there are many of these smaller basins because the network grows exponentially, the observed growth process thus also gives rise to a power-law distribution for the hyperareas of the basins.