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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.

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