Lévy Walk Navigation in Complex Networks: A Distinct Relation between Optimal Transport Exponent and Network Dimension

We investigate, for the first time, navigation on networks with a Lévy walk strategy such that the step probability scales as p(ij) ~ d(ij)(–α), where d(ij) is the Manhattan distance between nodes i and j, and α is the transport exponent. We find that the optimal transport exponent α(opt) of such a diffusion process is determined by the fractal dimension d(f) of the underlying network. Specially, we theoretically derive the relation α(opt) = d(f) + 2 for synthetic networks and we demonstrate that this holds for a number of real-world networks. Interestingly, the relationship we derive is different from previous results for Kleinberg navigation without or with a cost constraint, where the optimal conditions are α = d(f) and α = d(f) + 1, respectively. Our results uncover another general mechanism for how network dimension can precisely govern the efficient diffusion behavior on diverse networks.