Levy Walks in Non-Euclidean Spaces

Many animals display characteristic foraging patterns in their behavior when searching for food. Previous studies on foraging have shown that, in many cases, animals follow a Levy flight pattern with a power law distribution of step sizes that might be tuned for optimal search efficiency. While all of biology is constrained to live in Euclidean space, natural search processes may take place in effectively more complex spaces with a network topology such as networks of caves or other ecological niches.  Motivated by the recent equivalency that has been shown to exist between complex scale-free networks and hyperbolic space, we consider the question of optimal foraging in the case when searching occurs in a negatively curved space.  We study the search process in an appropriate projection of the hyperbolic space and make use of the equivalency to infer connections between optimal Levy walk searching in hyperbolic space and searching on a scale-free network.