Spectral analysis and clustering of large stochastic networks. Application to the Lennard-Jones-75 cluster

Abstract

We consider stochastic networks with pairwise transition rates of the form where the temperature T is a small parameter. Such networks arise in physics and chemistry and serve as mathematically tractable models of complex systems. Typically, such networks contain large numbers of states and widely varying pairwise transition rates. We present a methodology for spectral analysis and clustering of such networks that takes advance of the small parameter T and consists of two steps: (1) computing zero-temperature asymptotics for eigenvalues and the collection of quasi-invariant sets, and (2) finite temperature continuation. Step (1) is reducible to a sequence of optimisation problems on graphs. A novel single-sweep algorithm for solving them is introduced. Its mathematical justification is provided. This algorithm is valid for both time-reversible and time-irreversible networks. For time-reversible networks, a finite temperature continuation technique combining lumping and truncation with Rayleigh quotient iteration is developed. The proposed methodology is applied to the network representing the energy landscape of the Lennard-Jones-75 cluster containing 169523 states and 226377 edges. The transition process between its two major funnels is analysed. The corresponding eigenvalue is shown to have a kink at the solid–solid phase transition temperature.

Keywords:

• Eigenvalues
• asymptotic estimates
• optimal W-graphs
• Lennard-Jones clusters
• eigencurrents

Acknowledgements

We thank Professor David Wales for providing us with the data for the network and valuable discussion. We are grateful to Mr Weilin Li for valuable discussions at the early stages of the development of Algorithm 1.

Notes

No potential conflict of interest was reported by the authors.

1 The data for the network were kindly provided by Professor D. Wales, Cambridge University, UK.

2 The value corresponds to Wales’s data-set containing 593320 local minima.

Additional information

Funding

This work was partially supported by NSF [grant number 1217118].