  We investigate the spectrum of the infinitesimal generator of the continuous
time random walk on a randomly weighted oriented graph. This is the
non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and
L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild
assumptions on the law of the weights, we establish convergence as n tends to
infinity of the empirical spectral distribution of L after centering and
rescaling. In particular, our assumptions include sparse random graphs such as
the oriented Erd\"os-R\'enyi graph where each edge is present independently
with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting
distribution is characterized as an additive Gaussian deformation of the
standard circular law. In free probability terms, this coincides with the Brown
measure of the free sum of the circular element and a normal operator with
Gaussian spectral measure. The density of the limiting distribution is analyzed
using a subordination formula. Furthermore, we study the convergence of the
invariant measure of L to the uniform distribution and establish estimates on
the extremal eigenvalues of L.
