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 Lévy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Lévy walks and the underlying network structure. Moreover, applying our framework to the famous PageRank search, we show how to inform the optimality of the PageRank search. The framework for analyzing anomalous random walks on complex networks offers a useful new paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks.
Highlights
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
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
To characterize anomalous random walks, we propose the concept of a MFTD lij, which is the expected distance taken by a walker to first reach node j starting from node i
Summary
Tongfeng Weng[1], Jie Zhang[2], Moein Khajehnejad[1], Michael Small[3,4], Rui Zheng1 & Pan Hui[1] received: 23 February 2016 accepted: 01 November 2016 Published: 23 November 2016. A variety of measurements including mean first passage time (MFPT)[2], first passage time[4], and average trapping time[6] have been proposed, providing a comprehensive characterization of random walks on networks These studies facilitate our understanding of diverse dynamical processes on networks including epidemic spreading[7], synchronization[8], and transportation[9]. Traditional measurements like the mean first passage time neglect the difference between the cost associated with the nearest-neighbor jump and the long-range hopping, cannot properly characterize anomalous random walks on networks. We propose the mean first traverse distance that represents the expected traverse distance required by a walker moving from a source node to a target node This allows the cost associated with hopping to be taken into account in the characterization of anomalous random walks; this overcomes the problems of traditional measurements adopted in general random walks. Metric enables effective characterization of dynamics of anomalous random walks on networks, which promises more efficient search and transport processes on networks
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