This paper is concerned with minisum location-allocation problems on undirected networks in which demands can occur on links with uniform probability distributions. Two types of networks are considered. The first type considered is simply a chain graph. It is shown that except for the 1-median case, the problem is generally non-convex. However, for the p-median case, a discrete set of potential optimal facility locations may be identified, and hence it is shown that all local and global minima to the problem may be discovered by solving a series of trivial linear programming problems. This analysis is then extended to prescribe an algorithm for the 2-median location-allocation problem on a tree network involving uniform continuous demands on links. Some localization theorems are presented in the spirit of the work done on discrete nodal demand problems.