  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.
