Random walks on networks with stochastic resetting

Riascos, Alejandro P.; Boyer, Denis; Herringer, Paul; Mateos, José L.

doi:10.1103/physreve.101.062147

Cited by 74 publications

(87 citation statements)

References 68 publications

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“…Bonomo and Pal [54] derived a criterion that dictates when restart remains beneficial in discrete space and time restarted processes, and then applied the result to a symmetric and a biased random walker in one-dimensional lattice confined within two absorbing boundaries. Riascos et al [62] studied random walks on arbitrary networks subject to resetting with a constant probability. They derived the exact expressions of the stationary probability distribution and the MFPT by the spectral representation of the transition matrix without resetting.…”

Section: Introductionmentioning

confidence: 99%

First passage in discrete-time absorbing Markov chains under stochastic resetting

Chen,

Li,

Huang

2021

Preprint

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First passage of stochastic processes under resetting has recently been an active research topic in the field of statistical physics. However, most of previous studies mainly focused on the systems with continuous time and space. In this paper, we study the effect of stochastic resetting on first passage properties of discrete-time absorbing Markov chains, described by a transition matrix Q between transient states and a transition matrix R from transient states to absorbing states. Using a renewal approach, we exactly derive the unconditional mean first passage time (MFPT) to either of absorbing states, the splitting probability the and conditional MFPT to each absorbing state. All the quantities can be expressed in terms of a deformed fundamental matrix Zγ = [I − (1 − γ)Q] −1 and R, where I is the identity matrix, and γ is the resetting probability at each time step. We further show a sufficient condition under which the unconditional MPFT can be optimized by stochastic resetting. Finally, we apply our results to two concrete examples: symmetric random walks on one-dimensional lattices with absorbing boundaries and voter model on complete graphs.

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Section: Introductionmentioning

confidence: 99%

First passage in discrete-time absorbing Markov chains under stochastic resetting

Chen,

Li,

Huang

2021

Preprint

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“…The study of random walks on networks under the influence of resetting is relatively new and has focused mainly on one-dimensional lattices [23,47,48,49,50,51,52]. In a recent work, we have considered random walks under resetting in discrete time on arbitrary connected networks and established general relationships between several basic observables and the spectral representation of the transition matrix that defines the random walk without resetting [50]. These results were further extended to the case of resetting to multiple nodes [53].…”

Section: Introductionmentioning

confidence: 99%

“…This section summarizes the methods introduced in Refs. [49,50] and the analytical expressions for the stationary distribution and the mean-first passage times. In Section 3 we derive a condition for optimal resetting from which we obtain numerical values for the optimal resetting probability (at which the mean-first passage time to a given node is minimum) in the case of local random walks on rings and Cayley trees.…”

Section: Introductionmentioning

confidence: 99%

Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

Riascos,

Boyer,

Mateos

2021

Preprint

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The spectral theory of random walks on networks of arbitrary topology can be readily extended to study random walks and Lévy flights subject to resetting on these structures. When a discrete-time process is stochastically brought back from time to time to its starting node, the mean search time needed to reach another node of the network may be significantly decreased. In other cases, however, resetting is detrimental to search. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, we derive a general criteria for finite networks that establishes when there exists a non-zero resetting probability that minimises the mean first passage time at a target node. Right at optimality, the coefficient of variation of the first passage time is not unity, unlike in continuous time processes with instantaneous resetting, but above 1 and depends on the minimal mean first passage time. The approach is general and applicable to the study of different discretetime ergodic Markov processes such as Lévy flights, where the long-range dynamics is introduced in terms of the fractional Laplacian of the graph. We apply these results to the study of optimal transport on rings and Cayley trees.

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“…There is already a vast amount of specialized literature on various aspects of the subject, see, e.g., [28,[30][31][32][33]. For instance, models have been developed to describe Lévy flight dynamics and random walks with long-range jumps on undirected graphs [34,35], random walks with stochastic resetting on graphs [36] and models for long-range mobility in cities [37], among many others.…”

Section: Introductionmentioning

confidence: 99%

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

Michelitsch

Polito²,

Riascos

2020

Fractal Fract

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We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process `space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the `state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

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Random walks on networks with stochastic resetting

Cited by 74 publications

References 68 publications

First passage in discrete-time absorbing Markov chains under stochastic resetting

First passage in discrete-time absorbing Markov chains under stochastic resetting

Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

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