Lévy walk in complex networks: An efficient way of mobility

• We use entropy rate to characterize the diffusion process of Lévy walk on networks.
• The entropy rate demonstrates the efficiency of mobility navigated by Lévy walk.
• The entropy rate of Lévy walk is sensitive to the distribution of the shortest distance.
• The Lévy walk shows distinct responses to irregularity compared with the random walk.

To obtain an efficient diffusion process is an intriguing and important issue in the study of dynamical behaviors on real networks. Most previous studies are mainly focused on the analysis based on the random walk strategy, whose entropy rate is bounded by the logarithm of the largest node degree of a given graph. In this paper, we take into account a novel strategy named Lévy walk and derive the general expression of entropy rate of Lévy walk on networks. We present numerical evidences for how the Lévy walk strategy delivers an efficient diffusion process on networks and significantly increases the entropy rate compared with the random walk strategy. It is further demonstrated that the capability of Lévy walk heavily relies on the network topology as well as the amount of information available regarding the network structure. Specifically, the behavior of Lévy walk is highly sensitive to the distribution of shortest distances of the network and its variation. To address this finding, we thereby give a theoretical explanation of the relationship between the variation of shortest distances and the entropy rate of Lévy walk. This work may help to enrich our understanding of the behavior of Lévy walk and further guide us to find an efficient diffusion process on networks.