# Researchers at the University of Notre Dame, the University of Durham and the University of Cambridge Make Connections between an Unresolved Problem in Mathematics and Properties of Social Networks Using Energy Landscape Theory

Nadia Casas • DATE: June 24, 2016 |

Kinetic transition networks for the Thomoson problem and Smale's seventh problem.

In a paper published in Physical Review Letters, a group of researchers from the Department of Chemical and Biomolecular Engineering (CBE), the Department of Computer Science and Engineering (CSE), and the Department of Applied and Computational Mathematics and Statistics (ACMS) at the University of Notre Dame have collaborated with UK scientists at Durham and Cambridge to suggest a new angle for a famous mathematical problem. Their contribution provides a connection between Smale's 7th problem and J.J. Thomson's early model of atomic structure, which they establish using an approach based on energy landscapes. The team comprises Dr. Dhagash Mehta, a visiting research assistant professor in ACMS also collaborating with Assistant Professor Jonathan Whitmer (CBE), Prof. Danny Z. Chen (CSE) and his graduate student Mr Jianxu Chen (CSE), Prof. David J. Wales FRS (Department of Chemistry, University of Cambridge) and Dr Halim Kusumaatmaja (Department of Physics, University of Durham).

Steve Smale, a renowned Fields medalist, listed 18 problems in mathematics and computer science for the 21st century, along the lines of Hilbert's 23 problems for the 20th century. The 7th problem in Smale's list is to find a configuration nearby the most stable configuration (the global minimum) for particles with equal charges that are confined to the surface of a sphere. In chemical physics this model is known as the Thomson problem. Although easily stated, finding the global minimum becomes progressively harder as the number of particles increases. It also turns out that the solutions provide useful insight into a wide range of phenomena where systems are confined to spherical topology, and hence the Thomson problem has attracted considerable attention from mathematicians and scientists.

In their paper, Mehta et al. have applied methods from the computational side of energy landscape theory to locate most of the metastable configurations, corresponding to local minima. They also analyzed how the local minima can interconvert, identifying the corresponding pathways and energy barriers. They then constructed a network for the local minima, analogous to a social or friendship network, where structures are connected if they can interconvert in a single step. Computational techniques adopted from social network analysis were then applied, revealing that the networks of local minima for the Thomson problem exhibit 'small-world' properties. In such networks, every individual is, on average, only a few friends away from any other randomly picked individual.

In summary, the results show that the most stable configuration is on average only a few hops away from any other local minimum. It is this observation that may be relevant for Smale's 7th Problem in mathematics.

This highly interdisciplinary team of researchers is now investigating where the corresponding networks exhibit any other properties that have been highlighted for social networks.

Their paper has been selected as an 'Editor's Suggestion' by Physical Review Letters.