A shortest-route algorithm for finite networks is modified for application to path-dependent finite networks and to stochastic networks. The type of stochastic network considered has a "capture" probability associated with each node;The problem of finding a path of minimum expected value on a countable infinite stochastic network is discussed. Conditions are presented under which countable paths have the same minimum expected value as permutation paths;A continuous analog of the countable network, consisting of a hazard function on the plane and curves in the plane, is developed. Two optimality criteria are investigated: stochastic ordering and minimum expected value. Necessary conditions are given for paths to be optimal under the two criteria.