Random flows through fixed network A flow network is a directed graph in which each edge has a capacity. A flow through this network is an assignment of a value to each edge that is less or equal to the edge capacity, and such that the net incoming flow to every node balances with the net outcoming flux at that node. Two special nodes are exempted from this last restriction: the source (which can output a net non-zero flux) and the sink (which can receive a net non-zero flux).

There are algorithms to find the maximum net flow from source to sink in such a network (for example, Ford-Fulkerson algorithm).

I am looking for algorithms that generate pseudo-random admissible flows through such a network. Hopefully the space of admissible flows should be sampled as uniformly as possible. What methods are available here?
Not very practical, but you can sample from a the polytope of feasible flows using the well-known random walk techniques, see for example this classical paper by Kannan, Lovasz and Simonovits. These algorithms allow you to sample in polynomial time (in the dimension) from a distribution which is arbitrarily close to uniform in $L_1$ distance.