  We consider the following problem arising from the study of human problem
solving: Let $G$ be a vertex-weighted graph with marked "in" and "out"
vertices. Suppose a random walker begins at the in-vertex, steps to neighbors
of vertices with probability proportional to their weights, and stops upon
reaching the out-vertex. Could one deduce the weights from the paths that many
such walkers take? We analyze an iterative numerical solution to this
reconstruction problem, in particular, given the empirical mean occupation
times of the walkers. In the process, a result concerning the differentiation
of a matrix pseudoinverse is given, which may be of independent interest. We
then consider the existence of a choice of weights for the given occupation
times, formulating a natural conjecture to the effect that -- barring obvious
obstructions -- a solution always exists. It is shown that the conjecture holds
for a class of graphs that includes all trees and complete graphs. Several open
problems are discussed.
