Mortal random walks on a family of treelike regular fractals with a deep trap

2020 ◽  
Vol 34 (06) ◽  
pp. 2050031
Author(s):  
Zikai Wu ◽  
Guangyao Xu
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Network Models ◽  
Deep Trap ◽  
Initial Position ◽  
Single Step ◽  
The Mean ◽  
Walk Dimension ◽  
Mean Time ◽  
Temporal Factor

Due to the ubiquitous occurrence of evanescence in many physical, chemical and biological scenarios, mortal random walks that incorporate evanescence explicitly have drawn more and more attention. It has been a hot topic to study mortal random walks on distinct network models. In this paper, we study mortal random walks on T fractal and a family of treelike regular fractals with a trap located at central node (i.e., innermost node). First, with self-similar setting composed of T fractal, initial position of the walker and location of trap, the total trapping probability of the mortal walker reduces to a function of walker’s single-step survival parameter [Formula: see text]. In more detail, the total trapping probability is expressed by the [Formula: see text]th iteration of map (scaling function) of [Formula: see text]. Based on the map, the analytical expression of total trapping probability’s dominant behavior, the mean time to trapping (MFPT) and temporal factor are obtained, which are related to random walk dimension. Last, we extend the analysis to a family of treelike regular fractals. On them, the total trapping probability is still expressed as the [Formula: see text]th iteration of the map scaling [Formula: see text]. Accordingly, dominant behavior of total trapping probability, MFPT and temporal factor are determined analytically. Both analytical results obtained on T fractal and more general treelike regular fractals show that the mean time to trapping and desired random walk dimension can be obtained by tuning the survival probability parameter [Formula: see text]. In summary, the work advances the understanding of mortal random walks on more general deterministic networks.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

Series expansions for the all-time maximum of α-stable random walks

2016 ◽  
Vol 48 (3) ◽  
pp. 744-767
Author(s):  
Clifford Hurvich ◽  
Josh Reed
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Queueing Theory ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Series Expansions ◽  
Stable Random Variables ◽  
The Mean ◽  
The Stability

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

On the current enhancement at the edge of a crack in a lattice of resistors

1998 ◽  
Vol 30 (2) ◽  
pp. 342-364 ◽  
Author(s):  
Howard M. Taylor ◽  
Dennis E. Sweitzer
Keyword(s):  
Random Walk ◽  
Vertical Direction ◽  
Potential Gradient ◽  
Domain Of Attraction ◽  
Lattice Points ◽  
Fourier Methods ◽  
The North ◽  
The Mean ◽  
Mean Time ◽  
Cauchy Process

Consider a network whose nodes are the integer lattice points and whose arcs are fuses of 1Ω resistance. Remove a horizontal segment ofNadjacent vertical arcs, forming a ‘crack’ of lengthN. Subject the network to a uniform potential gradient ofvvolts per arc in the north-south (or vertical) direction and measure the current in one of the two vertical arcs at the ends of the crack. We write this current in the forme(N)v, and calle(N) thecurrent enhancement.We show that the enhancement grows at a rate that is the order of the square root of the crack length. Our method is to identify the enhancement with the mean time to exit an interval for a certain integer valued random walk, and then to use some of the well-known Fourier methods for studying random walk. Our random walk has no mean or higher moments and is in the domain of attraction of the Cauchy law. We provide a good approximation to the enhancement using the explicitly known mean time to exit an interval for a Cauchy process. Weak convergence arguments together with an estimate of a recurrence probability enable us to show that the current in an intact fuse, that is in the interior of a crack of lengthN, grows p roportionally withN/logN.

Uncovering the impact of delay phenomenon on random walks in a family of weighted m-triangulation networks

2021 ◽  
pp. 2150234
Author(s):  
Junhao Guo ◽  
Zikai wu
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Analytical Solutions ◽  
Network Models ◽  
Trapping Efficiency ◽  
Trapping Time ◽  
Delay Phenomenon ◽  
The Impact ◽  
Insight Into ◽  
Trapping Process

Uncovering the impact of special phenomena on dynamical processes in more distinct weighted network models is still needed. In this paper, we investigate the impact of delay phenomenon on random walk by introducing delayed random walk into a family of weighted m-triangulation networks. Specifically, we introduce delayed random walk into the networks. Then one and three traps are deployed, respectively, on the networks in two rounds of investigation. In both rounds of investigation, average trapping time (ATT) is applied to measure trapping efficiency and derived analytically by harnessing iteration rule of the networks. The analytical solutions of ATT obtained in both investigations show that ATT increases sub-linearity with the size of the network no matter what value the parameter [Formula: see text] manipulating delayed random walk takes. But [Formula: see text] can quantitatively change both its leading scaling and prefactor. So, introduction of delay phenomenon can control trapping efficiency quantitatively. Besides, parameters [Formula: see text] and [Formula: see text] governing networks’ evolution quantitatively impact both the prefactor and leading scaling of ATT simultaneously. In summary, this work may provide incremental insight into understanding the impact of observed phenomena on special trapping process and general random walks in complex systems.

scholarly journals Classes of random walks on temporal networks with competing timescales

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Julien Petit ◽  
Renaud Lambiotte ◽  
Timoteo Carletti
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Stationary State ◽  
Mathematical Analysis ◽  
Random Walk Model ◽  
Temporal Networks ◽  
Underlying Network ◽  
The Mean ◽  
Rich Spectrum ◽  
General Stochastic

Abstract Random walks find applications in many areas of science and are the heart of essential network analytic tools. When defined on temporal networks, even basic random walk models may exhibit a rich spectrum of behaviours, due to the co-existence of different timescales in the system. Here, we introduce random walks on general stochastic temporal networks allowing for lasting interactions, with up to three competing timescales. We then compare the mean resting time and stationary state of different models. We also discuss the accuracy of the mathematical analysis depending on the random walk model and the structure of the underlying network, and pay particular attention to the emergence of non-Markovian behaviour, even when all dynamical entities are governed by memoryless distributions.

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

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Zhongtuan Zheng ◽  
Gaoxi Xiao ◽  
Guoqiang Wang ◽  
Guanglin Zhang ◽  
Kaizhong Jiang
Keyword(s):  
Complex Networks ◽  
Random Walks ◽  
First Passage Time ◽  
Network Models ◽  
Passage Time ◽  
Main Role ◽  
Mean First Passage Time ◽  
Eigenvalues And Eigenvectors ◽  
First Passage ◽  
The Mean

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.

OPTIMAL CONTROL OF THE BLOWUP TIME OF A DIFFUSION

1996 ◽  
Vol 06 (05) ◽  
pp. 665-687 ◽  
Author(s):  
E.N. BARRON ◽  
R. JENSEN ◽  
W. LIU
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Bellman Equation ◽  
Initial Position ◽  
Diffusion Matrix ◽  
Blowup Time ◽  
Markov Diffusion ◽  
Matrix Formula ◽  
The Mean ◽  
Mean Time

If we have a controlled Markov diffusion which may explode in finite time, the problem arises regarding using the control in order to maximize the mean time to explosion, i.e. the blowup time. The maximal mean blowup time, u(x), as a function of the initial position x∈ℝn is characterized as the unique continuous viscosity solution of a Bellman equation, satisfying the boundary condition that u vanishes at infinity. Then we consider the problem of convergence of the maximal mean blowup time uε(x) corresponding to a diffusion matrix [Formula: see text], as ε → 0. We establish that, in general, the stochastic mean blowup time does not converge to the deterministic blowup time. However, the certainty equivalent blowup time does converge to the deterministic blowup time.

On the current enhancement at the edge of a crack in a lattice of resistors

1998 ◽  
Vol 30 (02) ◽  
pp. 342-364 ◽  
Author(s):  
Howard M. Taylor ◽  
Dennis E. Sweitzer
Keyword(s):  
Random Walk ◽  
Vertical Direction ◽  
Potential Gradient ◽  
Domain Of Attraction ◽  
Lattice Points ◽  
Fourier Methods ◽  
The North ◽  
The Mean ◽  
Mean Time ◽  
Cauchy Process

Consider a network whose nodes are the integer lattice points and whose arcs are fuses of 1Ω resistance. Remove a horizontal segment ofNadjacent vertical arcs, forming a ‘crack’ of lengthN. Subject the network to a uniform potential gradient ofvvolts per arc in the north-south (or vertical) direction and measure the current in one of the two vertical arcs at the ends of the crack. We write this current in the forme(N)v, and calle(N) thecurrent enhancement.We show that the enhancement grows at a rate that is the order of the square root of the crack length. Our method is to identify the enhancement with the mean time to exit an interval for a certain integer valued random walk, and then to use some of the well-known Fourier methods for studying random walk. Our random walk has no mean or higher moments and is in the domain of attraction of the Cauchy law. We provide a good approximation to the enhancement using the explicitly known mean time to exit an interval for a Cauchy process. Weak convergence arguments together with an estimate of a recurrence probability enable us to show that the current in an intact fuse, that is in the interior of a crack of lengthN, grows p roportionally withN/logN.

A description of random walks with collision anisotropy and with a nonconstant mean free path

1990 ◽  
Vol 68 (9) ◽  
pp. 912-917 ◽  
Author(s):  
Thomas Goulet ◽  
Isabelle Mattél ◽  
Jean-Paul Jay-Gerin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Path ◽  
Radial Distribution ◽  
Mean Square Displacement ◽  
Mean Free Path ◽  
Mean Square ◽  
Satisfactory Description ◽  
Average Value ◽  
The Mean

We study various characteristics of a particle's random walk both analytically and with the help of Monte-Carlo simulation techniques. With the analytical approach, we derive the expression[Formula: see text]which relates the mean-square displacement [Formula: see text] to (i) the number of steps N in the walk, (ii) the mean-square displacement [Formula: see text] on each of the steps, and (iii) a coefficient of collision anisotropy A defined as the average value of the cosine of the scattering angle θ. This expression is general in the sense that it holds for any value of N and A. It is, however, restricted to cases where the mean free path is constant throughout the random walk. The results of the simulations allow a further generalization to random walks with a nonconstant mean free path. They also allow the study of the radial distribution f(r) of particles after the walk. We find that a set of six functions fi(r) is necessary to give a satisfactory description of the particles' radial distribution for arbitrary values of N and A.

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