  Large continuous-time Markov chains with exponentially small transition rates
arise in modeling complex systems in physics, chemistry and biology. We propose
a constructive graph-algorithmic approach to determine the sequence of critical
timescales at which the qualitative behavior of a given Markov chain changes,
and give an effective description of the dynamics on each of them. This
approach is valid for both time-reversible and time-irreversible Markov
processes, with or without symmetry. Central to this approach are two graph
algorithms, Algorithm 1 and Algorithm 2, for obtaining the sequences of the
critical timescales and the hierarchies of Typical Transition Graphs or
T-graphs indicating the most likely transitions in the system {without and
with} symmetry respectively. The sequence of {critical} timescales includes the
subsequence of the reciprocals { of the real parts } of eigenvalues. Under a
certain assumption, we prove sharp asymptotic estimates for eigenvalues
(including prefactors) and show how one can extract them from the output of
Algorithm 1. We discuss the relationship between Algorithms 1 and 2, and
explain how one needs to interpret the output of Algorithm 1 if it is applied
in the case with symmetry instead of Algorithm 2. Finally, we analyze an
example motivated by R. D. Astumian's model of the dynamics of kinesin, a
molecular motor, by means of Algorithm 2.
