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Register a new userExact calculations of first-passage properties on the pseudofractal scale-free web
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In this paper, we consider discrete time random walks on the pseudofractal scale-free web (PSFW) and we study analytically the related first passage properties. First, we classify the nodes of the PSFW into different levels and propose a method to derive the generation function of the first passage probability from an arbitrary starting node to the absorbing domain, which is located at one or more nodes of low-level (i.e., nodes with large degree). Then, we calculate exactly the first passage probability, the survival probability, the mean and the variance of first passage time by using the generating functions as a tool. Finally, for some illustrative examples corresponding to given choices of starting node and absorbing domain, we derive exact and explicit results for such first passage properties. The method we propose can as well address the cases where the absorbing domain is located at one or more nodes of high-level on the PSFW, and it can also be used to calculate the first passage properties on other networks with self-similar structure, such as $(u, v)$ flowers and recursive scale-free trees.
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The explicit determinations of the mean first-passage time (MFPT) for trapping problem are limited to some simple structure, e.g., regular lattices and regular geometrical fractals, and determining MFPT for random walks on other media, especially complex real networks, is a theoretical challenge. In this paper, we investigate a simple random walk on the the pseudofractal scale-free web (PSFW) with a perfect trap located at a node with the highest degree, which simultaneously exhibits the remarkable scale-free and small-world properties observed in real networks. We obtain the exact solution for the MFPT that is calculated through the recurrence relations derived from the structure of PSFW. The rigorous solution exhibits that the MFPT approximately increases as a power-law function of the number of nodes, with the exponent less than 1. We confirm the closed-form solution by direct numerical calculations. We show that the structure of PSFW can improve the efficiency of transport by diffusion, compared with some other structure, such as regular lattices, Sierpinski fractals, and T-graph. The analytical method can be applied to other deterministic networks, making the accurate computation of MFPT possible.
Levy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the jump lengths---are drawn from an $alpha$-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Levy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Levy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Levy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index $alpha$ and the skewness (asymmetry) parameter $beta$. The other approach is based on the stochastic Langevin equation with $alpha$-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.
With nontrivial entropy production, first passage process is one of the most common nonequilibrium process in stochastic thermodynamics. Using one dimensional birth and death precess as a model framework, approximated expressions of mean first passage time (FPT), mean total number of jumps (TNJ), and their coefficients of variation (CV), are obtained for the case far from equilibrium. Consequently, uncertainty relations for FPT and TNJ are presented. Generally, mean FPT decreases exponentially with entropy production, while mean TNJ decreases exponentially first and then tends to a starting site dependent limit. For forward biased process, the CV of TNJ decreases exponentially with entropy production, while that of FPT decreases exponentially first and then tends to a starting site dependent limit. For backward biased process, both CVs of FPT and TNJ tend to one for large absolute values of entropy production. Related properties about the case of equilibrium are also addressed briefly for comparison.
We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $dot{x}_t=sqrt{2 D_0 V(B_t)},xi_t$, where $xi_t$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V(B_t)$ is a stochastic diffusivity (noise strength), which itself is a functional of independent Brownian motion $B_t$. We derive exact, compact expressions for the probability density functions (PDFs) of the first passage time (FPT) $t$ from a fixed location $x_0$ to the origin for three different realisations of the stochastic diffusivity: a cut-off case $V(B_t) =Theta(B_t)$ (Model I), where $Theta(x)$ is the Heaviside theta function; a Geometric Brownian Motion $V(B_t)=exp(B_t)$ (Model II); and a case with $V(B_t)=B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF has exactly the Levy-Smirnov form (specific for standard Brownian motion) for Model II, which concurrently exhibits a strongly anomalous diffusion. For Models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the Levy-Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target.
An efficient searcher needs to balance properly the tradeoff between the exploration of new spatial areas and the exploitation of nearby resources, an idea which is at the core of scale-free Levy search strategies. Here we study multi-scale random walks as an approximation to the scale- free case and derive the exact expressions for their mean-first passage times in a one-dimensional finite domain. This allows us to provide a complete analytical description of the dynamics driving the asymmetric regime, in which both nearby and faraway targets are available to the searcher. For this regime, we prove that the combination of only two movement scales can be enough to outperform both balistic and Levy strategies. This two-scale strategy involves an optimal discrimination between the nearby and faraway targets, which is only possible by adjusting the range of values of the two movement scales to the typical distances between encounters. So, this optimization necessarily requires some prior information (albeit crude) about targets distances or distributions. Furthermore, we found that the incorporation of additional (three, four, ...) movement scales and its adjustment to target distances does not improve further the search efficiency. This allows us to claim that optimal random search strategies in the asymmetric regime actually arise through the informed combination of only two walk scales (related to the exploitative and the explorative scale, respectively), expanding on the well-known result that optimal strategies in strictly uninformed scenarios are achieved through Levy paths (or, equivalently, through a hierarchical combination of multiple scales).
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