The work of Korman and others, based on investigations of navigational techniques of ‘crazy ants’, leads to the following problem: Let Q be a network with given arc lengths, and let H be a ‘home’ node. At each node a permanent direction is given (one of the adjacent arcs) which leads to a shortest path path from that node to H with probability p less than one. Starting from a node S, a searcher chooses the given direction at each reached node with probability q; otherwise he chooses randomly. He arrives home at node H in expected time T=T(S,H,p,q). How should q be chosen to minimize T? Perhaps when reaching a node the searcher sees its degree k so chooses the given direction with probability q(k). Related problems have been studied for tree from a graph theoretical perspective.