Navigation by anomalous random walks on complex networks

TL;DR

This paper introduces the mean first traverse distance (MFTD) as a new measure for characterizing anomalous random walks on complex networks, providing insights into search and transport processes.

Contribution

It proposes the MFTD concept, offers a calculation procedure, and applies it to Levy walks and PageRank, revealing new understanding of anomalous diffusion dynamics.

Findings

MFTD effectively characterizes anomalous random walks.

Application to Levy walks shows interplay between diffusion and network structure.

Analysis of PageRank explains the empirically chosen damping factor.

Abstract

Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Levy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Levy walks and the underlying network structure. Interestingly, applying our framework to the famous PageRank search, we can explain why its damping factor empirically chosen to be around 0.85. The framework for analyzing anomalous random walks on complex networks offers a new useful paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks.