Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket. II. The eigenvalue spectrum

Bentz, Jonathan L.; Turner, J. W.; Kozak, John J.

doi:10.1103/physreve.82.011137

Cited by 55 publications

(53 citation statements)

References 17 publications

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“…(7) and (8) and Eqs. (9) and (10) for RWs to the boundary] can be compared with analytic solutions obtained previously for random walks on two deterministic fractals, the Sierpinski "gasket" [11,14] with fractal dimension d f = ln3/ln2 ≈ 1.584, and the Sierpinski "tower" with d f = 2 [12]. The analytic solution for the mean walk length of a particle performing a RW on a finite, planar Sierpinski gasket with a single trap at one vertex is given by…”

Section: Discussionmentioning

confidence: 99%

“…There exists an extensive literature both on disordered (random) fractals [8][9][10] and deterministic fractals such as the Sierpinski gasket [11][12][13][14][15][16][17]. In this article, we introduce a fractal-like geometry which yields analytic results in two complementary cases: diffusion from satellite sites to a centrosymmetric trap, and diffusion from a centrosymmetric source to boundary sites.…”

Section: Introductionmentioning

confidence: 99%

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Lattice statistical theory of random walks on a fractal-like geometry

2014

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We have designed a two-dimensional, fractal-like lattice and explored, both numerically and analytically, the differences between random walks on this lattice and a regular, square-planar Euclidean lattice. We study the efficiency of diffusion-controlled processes for flows from external sites to a centrosymmetric reaction center and, conversely, for flows from a centrosymmetric source to boundary sites. In both cases, we find that analytic expressions derived for the mean walk length on the fractal-like lattice have an algebraic dependence on system size, whereas for regular Euclidean lattices the dependence can be transcendental. These expressions are compared with those derived in the continuum limit using classical diffusion theory. Our analysis and the numerical results quantify the extent to which one paradigmatic class of spatial inhomogeneities can compromise the efficiency of adatom diffusion on solid supports and of surface-assisted self-assembly in metal-organic materials.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lattice statistical theory of random walks on a fractal-like geometry

2014

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“…It is good model to mimic reality systems [5][6][7]. Random walks on fractals , which can be applied as model for transport in disordered media [8,9], has attracted lots of interests [10][11][12][13]. The range of applicability and of physical interest is enormous [14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

“…In the past several years, MFPT for random walks on fractals have been extensively studied [13,[19][20][21][22][23][24]. For example, the MTT for some special nodes were obtained for different fractals(or networks) such as Sierpinski gaskets [19], Apollonian network [25], pseudofractal scale-free web [26], deterministic scale-free graph [27] and some special trees [28][29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Effects of node position on diffusion and trapping efficiency for random walks on fractal scale-free trees

Peng¹,

Xu²

2014

J. Stat. Mech.

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Abstract. We study unbiased discrete random walks on the FSFT based on the its self-similar structure and the relations between random walks and electrical networks. First, we provide new methods to derive analytic solutions of the MFPT for any pair of nodes, the MTT for any target node and MDT for any starting node. And then, using the MTT and the MDT as the measures of trapping efficiency and diffusion efficiency respectively, we analyze the effect of trap's position on trapping efficiency and the effect of starting position on diffusion efficiency. Comparing the trapping efficiency and diffusion efficiency among all nodes of FSFT, we find the best (or worst) trapping sites and the best (or worst) diffusing sites. Our results show that: the node which is at the center of FSFT is the best trapping site, but it is also the worst diffusing site. The nodes which are the farthest nodes from the two hubs are the worst trapping sites, but they are also the best diffusion sites. Comparing the maximum and minimum of MTT and MDT, we found that the maximum of MTT is almost4m 2 +4m+1 times of the minimum of MTT, but the the maximum of MDT is almost equal to the minimum of MDT. These results shows that the position of target node has big effect on trapping efficiency, but the position of starting node almost has no effect on diffusion efficiency. We also conducted numerical simulation to test the results we have derived, the results we derived are consistent with those obtained by numerical simulation.

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“…That is to say, the MFRT of node v is 2m/d v on any finite connected network, where m is the total numbers of edges of the network and d v is the degree of node v [21,22]. For the MFPT, general formula to calculate the MPFT between any two nodes in any finite networks was presented [22]; exact results of the MFPT to some special nodes and the mean trap time (i.e., the MFPTs to a given target averaged over all the starting nodes) have obtained on different networks, such as Sierpinski gaskets [23,24], Apollonian network [25], scale-free Koch networks [26,27], deterministic recursive trees [28][29][30][31][32][33][34][35][36][37][38] and some other deterministic networks [39][40][41][42].However, the MFPT and the MFRT aren't always the good estimates of the FPT and the FRT. Whether the estimates provided by the MFPT and the MFRT are reliable is subject to the volatilities of the two random variables.…”

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confidence: 99%

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

Peng¹

2016

J. Stat. Mech.

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Abstract.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.Volatilities of FPT and FRT on a small-world scale-free network 2 IntroductionFirst 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, are two important quantities in the random walk literature [1][2][3][4]. The importance lies in the fact that many physical processes are controlled by first passage events [5][6][7][8][9][10][11][12], and that FRT can model the time intervals between two successive extreme events [13][14][15][16][17], such as traffic jams in roads, the floods, the droughts, and power blackouts in electrical power grid, etc [18][19][20]. Both FPT and FRT are random variables which can not be determined exactly and researchers can only try to find suitable quantities to estimate them. A first step consists in the analysis of the mean of the two random variables, the mean first-passage time (MFPT) and the mean first return time (MFRT). For the MFRT, it can be calculated from the stationary distribution directly. That is to say, the MFRT of node v is 2m/d v on any finite connected network, where m is the total numbers of edges of the network and d v is the degree of node v [21,22]. For the MFPT, general formula to calculate the MPFT between any two nodes in any finite networks was presented [...

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Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket. II. The eigenvalue spectrum

Cited by 55 publications

References 17 publications

Lattice statistical theory of random walks on a fractal-like geometry

Lattice statistical theory of random walks on a fractal-like geometry

Effects of node position on diffusion and trapping efficiency for random walks on fractal scale-free trees

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

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