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

Tongfeng Weng,Jie Zhang,Pan Hui,Michael Small

doi:10.1038/srep17309

Abstract

We investigate, for the first time, navigation on networks with a Lévy walk strategy such that the step probability scales as pij ~ dij–α, where dij is the Manhattan distance between nodes i and j, and α is the transport exponent. We find that the optimal transport exponent αopt of such a diffusion process is determined by the fractal dimension df of the underlying network. Specially, we theoretically derive the relation αopt = df + 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 = df and α = df + 1, respectively. Our results uncover another general mechanism for how network dimension can precisely govern the efficient diffusion behavior on diverse networks.

Highlights

Networks are ubiquitous in a vast range of natural and man-made systems ranging from the Internet through human society to the oil-water flow[1,2,3,4]
Roberson et al claims that when only local information is available, the optimal condition is the addition of long-range links taken from the distribution pij ∼ di−j d f, where df is the fractal dimension of the underlying network[8]
We investigate the Lévy diffusion processes on networks and find that the optimal exponent of such diffusion process occurs at α = df + 2, where df is the fractal dimension of the underlying network, in contrast to the previous findings, where α = df[7,8] and α = df + 110,11, respectively

Summary

Introduction

Networks are ubiquitous in a vast range of natural and man-made systems ranging from the Internet through human society to the oil-water flow[1,2,3,4]. Unlike the previous unconstrained situation, Li et al provide the design principles for optimal transport networks under imposition of a cost constraint of long-range links, where the best condition is obtained with the long-range links taken from pij ∼ di−j (d f +1), regardless of the strategy used based on local or global information of the whole network[10,11]. All these strategies have some common characteristics that the efficient mobility is achieved by choosing one of the available links of a site to follow (based on local or global knowledge of the network structure) that potentially optimizes the path. Our results indicate that this efficient global approach of mobility only depends on the dimension of the underlying network, sometimes that is impossible to obtain merely based on limited and local information

Methods

Results

Conclusion