  We compute spectra of large stochastic matrices $W$, defined on sparse random
graphs, where edges $(i,j)$ of the graph are given positive random weights
$W_{ij}>0$ in such a fashion that column sums are normalized to one. We compute
spectra of such matrices both in the thermodynamic limit, and for single large
instances. The structure of the graphs and the distribution of the non-zero
edge weights $W_{ij}$ are largely arbitrary, as long as the mean vertex degree
remains finite in the thermodynamic limit and the $W_{ij}$ satisfy a detailed
balance condition. Knowing the spectra of stochastic matrices is tantamount to
knowing the complete spectrum of relaxation times of stochastic processes
described by them, so our results should have many interesting applications for
the description of relaxation in complex systems. Our approach allows to
disentangle contributions to the spectral density related to extended and
localized states, respectively, allowing to differentiate between time-scales
associated with transport processes and those associated with the dynamics of
local rearrangements.
