Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights

Riascos, Alejandro P.; Mateos, José L.

doi:10.1103/physreve.90.032809

Cited by 87 publications

(177 citation statements)

References 50 publications

(68 reference statements)

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“…It can be seen that the smaller αβ the more narrowly the waiting time PDF is concentrated at small t-values close to t = 0 . This behavior can also be identified in view of Laplace transform (34) 35) and (36). We depict with dashed lines the case when αβ = 1.…”

Section: Generalization Of the Fractional Poisson Processmentioning

confidence: 68%

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Generalized Space–Time Fractional Dynamics in Networks and Lattices

Michelitsch¹,

Riascos²,

Collet³

et al. 2020

Advanced Structured Materials

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We analyze generalized space-time fractional motions on undirected networks and lattices. The continuoustime random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time-fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density. We analyze 'generalized space-time fractional diffusion' in the infinite d -dimensional integer lattice Z d . We obtain in the diffusion limit a 'macroscopic' space-time fractional diffusion equation. Classical CTRW models such as with Laskin's fractional Poisson process and standard Poisson process which occur as special cases are also analyzed. The developed generalized space-time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems. of the mean-square displacement in anomalous diffusion [5]. It has been demonstrated that such anomalous diffusive behavior is well described by a random walk subordinated to the fractional generalization of the Poisson process. This process which was to our knowledge first introduced by Repin and Saichev [13], was developed and analyzed by Laskin who called this process the 'fractional Poisson process' [14,15]. The Laskin's fractional Poisson process was further generalized in order to obtain greater flexibility to adopt real-world situations [16,17,18]. We refer this renewal process to as 'generalized fractional Poisson process' (GFPP). Recently we developed a CTRW model of a normal random walk subordinated to a GFPP [17,18].The purpose of the present paper is to explore space fractional random walks that are subordinated to a GFPP. We analyze such motions in undirected networks and as a special application in the multidimensional infinite integer lattice Z d .

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Section: Generalization Of the Fractional Poisson Processmentioning

confidence: 68%

“…, N . The topology of the network is described by the positive-semidefinite N × N Laplacian matrix which is defined by [8,12,36,37,38,39,40] L…”

Section: Continuous-time Random Walk On Networkmentioning

confidence: 99%

Generalized Space–Time Fractional Dynamics in Networks and Lattices

Michelitsch¹,

Riascos²,

Collet³

et al. 2020

Advanced Structured Materials

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“…This issue has been already addressed in the previous literatures1314 and is not the main scope of this paper. Nonetheless, we think one can use several excellent algorithms such as the preprocessing algorithm31, which is one possible solution for this problem.…”

Section: Discussionmentioning

confidence: 88%

“…This limitation is circumvented by the natural of Lévy walks model1112. Recently, intensive attention has been devoted to anomalous random walks on networks, such as Lévy walks131415, traditional web surfing16, and even electric signals transmitted in brain networks10. One striking feature of anomalous random walks is having the long-range hopping (i.e., the walker can hop to far away nodes not directly connected to its current position).…”

mentioning

confidence: 99%

Navigation by anomalous random walks on complex networks

Weng

Zhang

Khajehnejad

et al. 2016

Sci Rep

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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.

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“…In this work, we address this matter, following the framework introduced in Refs. [16][17][18]. Our study considers the fractional diffusion of node-centric continuous-time RWs on undirected multiplex with two layers.…”

Section: Introductionmentioning

confidence: 99%

Fractional dynamics on circulant multiplex networks: optimal coupling and long-range navigation for continuous-time random walks

Allen‐Perkins¹,

Andrade²

2019

J. Stat. Mech.

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This work analyzes fractional continuous-time random walks on two-layer multiplexes. A nodecentric dynamics is used, in which it is assumed a Poisson distribution of a walker to become active, while a jump to one of its neighbors depends on the connection weight. Synthetic multiplexes with well known topology are used to illustrate dynamical features obtained by numerical simulations, while exact analytical expressions are presented for multiplexes assembled by circulant layers with finite number of nodes. Special attention is given to the effect of inter-D x and intra-layer D i coefficients on the system's behavior. In opposition to usual discrete time dynamics, the relaxation time has a well defined minimum at an optimal value of D x /D i . It is found that, even for the enhanced diffusion condition, the walkers mean square displacement increases linearly with time.

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Fractional dynamics on networks: Emergence of anomalous diffusion and Lévy flights

Cited by 87 publications

References 50 publications

Generalized Space–Time Fractional Dynamics in Networks and Lattices

Generalized Space–Time Fractional Dynamics in Networks and Lattices

Navigation by anomalous random walks on complex networks

Fractional dynamics on circulant multiplex networks: optimal coupling and long-range navigation for continuous-time random walks

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