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.

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.

Deterministic Random Walks on the Two-Dimensional Grid

2009 ◽  
Vol 18 (1-2) ◽  
pp. 123-144 ◽  
Author(s):  
BENJAMIN DOERR ◽  
TOBIAS FRIEDRICH
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Walk Model ◽  
Two Dimensional ◽  
Expected Number ◽  
Fixed Order ◽  
The Difference ◽  
Propp Machine

Jim Propp's rotor–router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbours in a fixed order. We analyse the difference between the Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8 if all vertices serve their neighbours in clockwise or order, and 7.3 otherwise. This result in particular shows that the order in which the neighbours are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

scholarly journals Large Deviations for Continuous Time Random Walks

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 697 ◽  
Author(s):  
Wanli Wang ◽  
Eli Barkai ◽  
Stanislav Burov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Exponential Decay ◽  
Continuous Time ◽  
Large Deviation ◽  
Waiting Times ◽  
Continuous Time Random Walk ◽  
Random Walk Model ◽  
Logarithmic Correction ◽  
Continuous Time Random Walks

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

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.

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.

A Multi-Label Classification Algorithm Based on Random Walk Model

2010 ◽  
Vol 33 (8) ◽  
pp. 1418-1426 ◽  
Author(s):  
Wei ZHENG ◽  
Chao-Kun WANG ◽  
Zhang LIU ◽  
Jian-Min WANG
Keyword(s):  
Random Walk ◽  
Classification Algorithm ◽  
Random Walk Model

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

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