Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focuses on finding optimal distributions of weights which, for a given connection topology, maximize some kind of flow or minimize a given cost function. We show that both sets of problems can be approached through an analysis of the large-deviation functions of random walks. Specifically, a study of ensembles of trajectories allows us to find optimal paths, or design optimal weighted networks, by means of an auxiliary stochastic process (the generalized Doob transform). The paths are not limited to shortest paths, and the weights must not necessarily optimize a given function. Paths and weights can in fact be tailored to a given statistics of a time-integrated observable, which may be an activity or current, or local functions marking the passing of the random walker through a given node or link. We illustrate this idea with an exploration of optimal paths in the presence of obstacles, and networks that optimize flows under constraints on local observables.