Approximating the statistics of various properties in randomly weighted graphs

Consider the setting of randomly weighted graphs, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the non-negative reals. Under this setting, weighted graph properties such as the diameter, the radius (with respect to a designated vertex), and the weight of a minimum spanning tree become random variables and we are interested in computing their expectation. Unfortunately, this turns out to be #P-hard. In this paper, we define a family of weighted graph properties (that includes the above three) and show that for each property in this family, the problem of computing the k^th moment (and in particular, the expectation) of the corresponding random variable admits a fully polynomial-time randomized approximation scheme (FPRAS) for every fixed k.