Preferential attachment during the evolution of a potential energy landscape

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.

10 pages, 10 figures

References in corpus (31)

• Emergence of scaling in random networks
• The structure and function of complex networks
• Power laws, Pareto distributions and Zipf's law
• Lethality and centrality in protein networks
• The diameter of the world wide web
• The large-scale organization of metabolic networks
• Evolution of the social network of scientific collaborations
• Clustering and preferential attachment in growing networks
• Topology of evolving networks: local events and universality
• Connectivity of Growing Random Networks
• Scale-Free Networks are Ultrasmall
• Measuring preferential attachment for evolving networks
• Inherent Structure Entropy of Supercooled Liquids
• Pattern Detection in Complex Networks: Correlation Profile of the Internet
• Crossover to Potential Energy Landscape Dominated Dynamics in a Model Glass-forming Liquid
• Preferential attachment in the protein network evolution
• Effect of the accelerating growth of communications networks on their structure
• The network topology of a potential energy landscape: A static scale-free network
• Saddle Points and Dynamics of Lennard-Jones Clusters, Solids and Supercooled Liquids
• The potential energy landscape of a model glass former: thermodynamics, anharmonicities, and finite size effects
• Thermodynamics and the Global Optimization of Lennard-Jones clusters
• Identifying "communities" within energy landscapes
• Self-similar disk packings as model spatial scale-free networks
• Structural relaxation in Morse clusters: Energy landscapes
• Structural relaxation in atomic clusters: Master equation dynamics
• Characterizing the network topology of the energy landscapes of atomic clusters
• Power-law distributions for the areas of the basins of attraction on a potential energy landscape
• Accelerating, hyper-accelerating, and decelerating probabilistic networks
• The effect of compression on the global optimization of atomic clusters
• Exploring the origins of the power-law properties of energy landscapes: An egg-box model
• A self-consistent approach to measure preferential attachment in networks and its application to an inherent structure network

Cited by in corpus (6)

• A Preferential Attachment Paradox: How Preferential Attachment Combines with Growth to Produce Networks with Log-normal In-degree Distributions
• Edwards thermodynamics of the jamming transition for frictionless packings: ergodicity test and role of angoricity and compactivity
• Transition from fractal to non-fractal scalings in growing scale-free networks
• Glass transition and random walks on complex energy landscapes
• Identifying time dependence in network growth
• Complex systems: features, similarity and connectivity