  We compute analytically the distribution and moments of the largest
eigenvalues/singular values and resolvent statistics for random matrices with
(i) non-negative entries, (ii) small rank, and (iii) prescribed sums of
rows/columns. Applications are discussed in the context of Mean First Passage
Time of random walkers on networks, and the calculation of network "influence"
metrics. The analytical results are corroborated by numerical simulations.
