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We consider stochastic networks with pairwise transition rates of theexponential form where the temperature T is a small parameter. Such networksarise in physics and chemistry and serve as mathematically tractable models ofcomplex systems. Typically, such networks contain large numbers of states andwidely varying pairwise transition rates. We present a methodology for spectralanalysis and clustering of such networks that takes advance of the smallparameter T and consists of two steps: (1) computing zero-temperatureasymptotics for eigenvalues and the collection of quasi-invariant sets, and (2)finite temperature continuation. Step (1) is re- ducible to a sequence ofoptimization problems on graphs. A novel single-sweep algorithm for solvingthem is introduced. Its mathematical justification is provided. This algorithmis valid for both time-reversible and time-irreversible networks. Fortime-reversible networks, a finite temperature continuation technique combininglumping and truncation with Rayleigh quotient iteration is developed. Theproposed methodology is applied to the network representing the energylandscape of the Lennard-Jones-75 cluster containing 169,523 states and 226,377edges. The transition process between its two major funnels, is analyzed. Thecorresponding eigenvalue is shown to have a kink at the solid-solid phasetransition temperature.

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