Search for an immobile entity on a network

We consider the problem of searching for immobile friendly entities on an undirected network. The focus is on finding the first such entity. The search time is a random variable, whose probability density function (pdf) depends upon the path and also upon information about entity location. We seek a path choice that minimizes the expected search time. This problem differs from arc-covering problems, e.g. the Chinese Postman Problem (CPP), in the way that here the objective is not to find the minimum length tour that covers all the links at least once, but instead to minimize the expected time to find the entity. We introduce a heuristic algorithm to deal with the search process given that there is exactly one entity on the network and it is equally likely to be at any point on the network. We later prove that this path is also optimal for the situation when a number of entities are present and these entities are uniformly distributed.