Navigation by anomalous random walks on complex networks

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.

Mean First-Passage Time on Scale-Free Networks Based on Rectangle Operation

“…For example, Watts and Strogatz discovered and explained the ubiquitous small-world property of real social networks by building small-world network models with a relatively small diameter [1]; Barabasi and Albert constructed a scale-free network model to reveal the fact that degree distribution obeys the power-law distribution in real networks [2,3]. These two classical works have inspired many scholars to devote themselves to researching about complex networks, especially random walks in complex networks [4][5][6][7][8][9].…”

High-Order Degree and Combined Degree in Complex Networks

“…is called thedegree sequence of [54]. As we all know, the mean value and the distribution of degree sequence guide the definition of mean degree and degree distribution.…”

“…Same as 2-order degree, suppose matrix is an adjacency matrix of a network , we denote = ( ∈ * ) as the -order adjacency matrix of . The matrix was used to calculate the number of walk with length between nodes [54][55][56]. Denote { } and { } as the sum of the -th row and -th column respectively in matrix ; then the -order degree of node V in undirected network is { } and { } = { } (if is a directed network, then the -order out-degree is { } , and the -order in-degree is { } ).…”

Universal principles governing multiple random searchers on complex networks: The logarithmic growth pattern and the harmonic law

“…It has long been recognized that diverse dynamics of natural and artificial systems ranging from the stochastic motion of molecules [1], animals foraging [2], transportation [3], to information and disease spreading [4] can be described well and accurately modeled by random search processes on a network. The broad range of relevances have turned random search processes on complex networks into a long-standing topic of scientific interest [5][6][7][8].…”