Random walks on weighted networks

Zhang, Zhongzhi; Shan, Tong; Chen, Guanrong

doi:10.1103/physreve.87.012112

Cited by 116 publications

(86 citation statements)

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“…Particular attention has been devoted to the study of the characteristic times associated to random walks, such as the mean return times, or the mean first passage times, respectively the average time the walker takes to come back to the starting node or to hit a given node [12]. Such characteristic times can be determined analytically for random walks on regular lattices [13], but their calculation for graphs with heterogeneous structures is still the object of active research [14,15]. Recent results include the derivation of analytic expressions for the characteristic times of unbiased random walks on Erdös-Rényi random graphs [16], on fractal networks [17][18][19][20] and on particular classes of scale-free graphs [21].…”

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Characteristic times of biased random walks on complex networks

2014

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We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree k is proportional to k α , where α is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely: i) the time the walker needs to come back to the starting node, ii) the time it takes to visit a given node for the first time, and iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of α which minimizes the three characteristic times is different from the value αmin = −1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of αmin in the range [−1, −0.5], while disassortative networks have αmin in the range [−0.5, 0]. We derive an analytical relation between the degree correlation exponent ν and the optimal bias value αmin, which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks, by means of an appropriate tuning of the motion bias.PACS numbers: 89.75. Hc, 05.40.Fb, 89.75.Kd In the last decade or so the quantitative analysis of networks having different origin and function, including social networks, the human brain, the Internet, the World Wide Web, has revealed that all these systems exhibit comparable structural properties at different scales, and are more similar to each other than expected [1,2]. It has been found that the structural complexity of networks from the real world usually has a significant impact on the dynamical processes occurring over them, including opinion dynamics [3], epidemics [4] and synchronization [5].Random walks are the simplest way to explore a network, and are one of the most widely studied class of processes on complex networks [6,7]. Different kinds of random walks have been used to implement efficient local search strategies [8,9], and also to reveal the presence of hierarchies and network communities [10,11]. Particular attention has been devoted to the study of the characteristic times associated to random walks, such as the mean return times, or the mean first passage times, respectively the average time the walker takes to come back to the starting node or to hit a given node [12]. Such characteristic times can be determined analytically for random walks on regular lattices [13], but their calculation for graphs with heterogeneous structures is still the object of active research [14,15]. Recent results include the derivation of analytic expressions for the characteristic times of unbiased random walks on Erdös-Rényi random graphs [16], on fractal networks [17][18][19][20] and on particular classes of scale-free graphs [21]. To date, only approximate solutions are available for random walks on real networks [22][23][24][25].A class of random walks which is particularly int...

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confidence: 99%

Characteristic times of biased random walks on complex networks

2014

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“…At this point, we call the chain reversible. Now, we shift attention to random walks on weighted networks [35,36]. We consider a finite nonbipartite network (or graph) = ( , ) with nodes (or vertices, sites) and edges connecting them.…”

Section: Preliminaries and Terminologiesmentioning

confidence: 99%

“…The sum , called the strength of node , runs over the set ( ) of all the connected neighbors of . Such a chain is reversible with the stationary distribution [35,36] …”

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Mean First Passage Time of Preferential Random Walks on Complex Networks with Applications

Zheng

Xiao

Wang

et al. 2017

Mathematical Problems in Engineering

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This paper investigates, both theoretically and numerically, preferential random walks (PRW) on weighted complex networks. By using two different analytical methods, two exact expressions are derived for the mean first passage time (MFPT) between two nodes. On one hand, the MFPT is got explicitly in terms of the eigenvalues and eigenvectors of a matrix associated with the transition matrix of PRW. On the other hand, the center-product-degree (CPD) is introduced as one measure of node strength and it plays a main role in determining the scaling of the MFPT for the PRW. Comparative studies are also performed on PRW and simple random walks (SRW). Numerical simulations of random walks on paradigmatic network models confirm analytical predictions and deepen discussions in different aspects. The work may provide a comprehensive approach for exploring random walks on complex networks, especially biased random walks, which may also help to better understand and tackle some practical problems such as search and routing on networks.

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“…Note that most previous works on trapping dendrimers based on discrete time random walks focused on the unbiased random walk. However, sometimes it is more suitable to describe some particular problems by a biased random walk than the unbiased one 46 , since the transition probability depends on not only network topologies but also other properties relevant to the diffusion dynamics 47 . Among numerous biased random walks, maximal entropy random walk (MERW) 48 , maximizes the entropy of paths, has been studied recently 49 .…”

Section: Introductionmentioning

confidence: 99%

Maximal entropy random walk improves efficiency of trapping in dendrimers

Peng

Zhang

2014

The Journal of Chemical Physics

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We use maximal entropy random walk (MERW) to study the trapping problem in dendrimers modeled by Cayley trees with a deep trap fixed at the central node. We derive an explicit expression for the mean first passage time from any node to the trap, as well as an exact formula for the average trapping time (ATT), which is the average of the source-to-trap mean first passage time over all non-trap starting nodes. Based on the obtained closed-form solution for ATT, we further deduce an upper bound for the leading behavior of ATT, which is the fourth power of ln N , where N is the system size. This upper bound is much smaller than the ATT of trapping depicted by unbiased random walk in Cayley trees, the leading scaling of which is a linear function of N . These results show that MERW can substantially enhance the efficiency of trapping performed in dendrimers.

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Random walks on weighted networks

Cited by 116 publications

References 55 publications

Characteristic times of biased random walks on complex networks

Characteristic times of biased random walks on complex networks

Mean First Passage Time of Preferential Random Walks on Complex Networks with Applications

Maximal entropy random walk improves efficiency of trapping in dendrimers

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