Random walks on fractals: higher moments

Argyrakis, P.; Anacker, L. W.; Kopelman, Raoul

doi:10.1088/0305-4470/21/2/036

Cited by 5 publications

(7 citation statements)

References 10 publications

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“…Since the variance of the first passage time scales as (26), we relate the variance to the mean by calculating the reduced moment R(N T ) = Var(T * ) 1/2 / T * . This type of approach has been used in the study of the variance of the range of an n-step random walk [8,21]. As in [21] we expect that lim NT →∞ R(N T ) exists and remains finite, for every fractal structure with a spectral dimension of d < 2.…”

Section: / Dmentioning

confidence: 99%

“…. This is the natural extension of equation ( 5), with a n and b n defined in equation (8). The solution of the equations are…”

Section: Appendix A: Corner Concentrationsmentioning

confidence: 99%

“…The method is exact, and can be applied to any finitely ramified deterministic fractal, making the exploration of higher moments of the GFPT possible. There is interest in higher moments of random walk quantities [8,9] as such results give further insight into the nature of transport on fractals. Knowledge of these moments is also useful to investigate the variance of the bi-molecular reaction A + B → A [10].…”

Section: Introductionmentioning

confidence: 99%

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Global first-passage times of fractal lattices

Haynes

Roberts

2008

Phys. Rev. E

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The global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time T . We find simple expressions for the mean global first passage time T for five fractals: the d-dimensional Sierpinski gasket, T-fractal, hierarchical percolation model, Mandelbrot-Given curve and a deterministic tree. We also find an exact expression for the second moment T 2 and show that the variance of the first passage time, Var(T ), scales with the number of nodes within the fractal N such that Var(T ) ∼ N 4/d , whered is the spectral dimension.

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Section: / Dmentioning

confidence: 99%

“…. This is the natural extension of equation ( 5), with a n and b n defined in equation (8). The solution of the equations are…”

Section: Appendix A: Corner Concentrationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global first-passage times of fractal lattices

Haynes

Roberts

2008

Phys. Rev. E

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“…The quantities we are interested in are the first passage time (FPT), which is the time it takes a random walker to reach a given site for the first time, and first return time (FRT), which is the time it takes a random walker to return to the starting site for the first time [25][26][27][28][29][30][31][32] . In the past several years, a lot of work was devoted to analyze the mean 12,[33][34][35][36][37][38][39][40][41][42] and the variance 25,[43][44][45][46] of the two random variables on different networks. The mean provides valuable estimate of the random variable and the variance is good measure on whether the estimate provided by the mean is reliable.…”

Section: Introductionmentioning

confidence: 99%

Scaling properties of first-passage quantities on the fractal and transfractal scale free networks

Peng

2016

Preprint

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Fractal (or transfractal) and scale free characters are common properties of real life network systems. It is of great significance to uncover the effect of these characters on the dynamic processes taking place on complex medias. In this paper, we consider the random walk process on a kind of fractal (or transfractal) scale free networks, which also called as (u, v) flowers, and we focus on the global first passage time (GFPT) and first return time (FRT). Here, we present method to derive exactly the probability generation function, mean and variance of the GFPT and FRT for a given hub (i.e., node with the highest degree) and then the scaling properties of the mean and the variance of the GFPT and FRT are disclosed. Our results show that, for the case of u > 1, while the networks are fractals, the mean of the GFPT scales with the volume of the network as, where * denotes the mean of random variable * , N t is the volume of the network with generation t and d s is the spectral dimension of the network; but, for the case of u = 1, while the networks are nonfractals, the mean of the GFPT scales as GM F P T t ∼ N 2/ ds t , where ds is the transspectral dimension of the network, which is introduced in this paper. Results also show that, the variance of the GFPT scales as V ar(GF P T t ) ∼ GF P T t 2 , where V ar( * ) denotes variance of of random variable * ; whereas the variance of the FRT scales as V ar(F RT t ) ∼ F RT t GF P T t . Our results imply that for the case that the networks are nonfractals, the mean and the variance of the GFPT are not controlled by the spectral dimension(i.e., d s = 2), but they are controlled by the transspectral dimension. In order to evaluate the fluctuation of the GFPT and FRT, we also calculate the reduced moments of the the GFPT and FRT and find that, in the limit of large size, the reduced moment of the FRT tends to be infinite, whereas the reduced moment of the GFPT is almost a const.Therefore, on the (u, v) flowers of large size, the fluctuation of the FRT is huge, whereas the fluctuation of the GFPT is much smaller.

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“…On the contrary, a high variance or reduced moment indicates that the random variable are spread out over a wider range of values and shows the estimate provided by its mean is unreliable. In the past several years, valuable results for the variance of the FPT are also obtained on different networks, such as some tree and comb structure [5,44,45], T-fractal, Sierpinski gasket, hierarchical percolation model and etc [43,46]. But for the variance and the reduced moment of FRT and FPT on the networks with scale-free and small-world effect, to the best of our knowledge, explicit results are seldom reported.…”

Section: Introductionmentioning

confidence: 99%

Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

Peng

2015

Preprint

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In this paper, we study random walks on a small-world scale-free network, also called as pseudofractal scale-free web (PSFW), and analyze the volatilities of first passage time (FPT) and first return time (FRT) by using the variance and the reduced moment as the measures. Note that the FRT and FPT are deeply affected by the starting or target site. We don't intend to enumerate all the possible cases and analyze them. We only study the volatilities of FRT for a given hub (i.e., node with highest degree) and the volatilities of the global FPT (GFPT) to a given hub, which is the average of the FPTs for arriving at a given hub from any possible starting site selected randomly according to the equilibrium distribution of the Markov chain. Firstly, we calculate exactly the probability generating function of the GFPT and FRT based on the self-similar structure of the PSFW. Then, we calculate the probability distribution, the mean, the variance and reduced moment of the GFPT and FRT by using the generating functions as a tool. Results show that: the reduced moment of FRT grows with the increasing of the network order N and tends to infinity while N → ∞; but for the reduced moments of GFPT, it is almost a constant(≈ 1.1605) for large N . Therefore, on the PSFW of large size, the FRT has huge fluctuations and the estimate provided by MFRT is unreliable, whereas the fluctuations of the GFPT is much smaller and the estimate provided by its mean is more reliable. The method we propose can also be used to analyze the volatilities of FPT and FRT on other networks with selfsimilar structure, such as (u, v) flowers and recursive scale-free trees.

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Random walks on fractals: higher moments

Cited by 5 publications

References 10 publications

Global first-passage times of fractal lattices

Global first-passage times of fractal lattices

Scaling properties of first-passage quantities on the fractal and transfractal scale free networks

Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

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