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

TL;DR

This paper introduces a spectral analysis and clustering methodology for large stochastic networks with applications to complex physical systems, specifically analyzing the energy landscape of the Lennard-Jones-75 cluster.

Contribution

It presents a novel two-step approach leveraging small parameter T for spectral analysis, including a new algorithm for zero-temperature asymptotics and finite temperature continuation techniques.

Findings

Applied to Lennard-Jones-75 cluster network with over 169,000 states.

Identified transition processes and eigenvalue behavior at phase transition.

Validated the methodology for large, complex stochastic networks.

Abstract

We consider stochastic networks with pairwise transition rates of the exponential 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 re- ducible to a sequence of optimization 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 169,523 states and 226,377 edges. The transition process between its two major funnels, is analyzed. The corresponding eigenvalue is shown to have a kink at the solid-solid phase transition temperature.