Determining mean first-passage time on a class of treelike regular fractals

Relatively general techniques for computing mean first-passage time (MFPT) of random walks on networks with a specific property are very useful, since a universal method for calculating MFPT on general graphs is not available because of their complexity and diversity. In this paper, we present techniques for explicitly determining the partial mean first-passage time (PMFPT), i.e., the average of MFPTs to a given target averaged over all possible starting positions, and the entire mean first-passage time (EMFPT), which is the average of MFPTs over all pairs of nodes on regular treelike fractals. We describe the processes with a family of regular fractals with treelike structure. The proposed fractals include the $T$ fractal and the Peano basin fractal as their special cases. We provide a formula for MFPT between two directly connected nodes in general trees on the basis of which we derive an exact expression for PMFPT to the central node in the fractals. Moreover, we give a technique for calculating EMFPT, which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials. Making use of the proposed methods, we obtain analytically the closed-form solutions to PMFPT and EMFPT on the fractals and show how they scale with the number of nodes. In addition, to exhibit the generality of our methods, we also apply them to the Vicsek fractals and the iterative scale-free fractal tree and recover the results previously obtained.Comment: Definitive version published in Physical Review

The Computational Complexity of Generating Random Fractals

In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation that can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics.Comment: 28 pages, LATEX, 8 Postscript figures available from [email protected]

Zipf's law, 1/f noise, and fractal hierarchy

Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within the specific scientific domains. All these observations are associated with scaling laws and have caused a broad research interest in the scientific circle. However, the inherent relationships between these scaling phenomena are still pending questions remaining to be researched. In this paper, theoretical derivation and mathematical experiments are employed to reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution. First, the multifractal process follows the generalized Zipf's law empirically. Second, a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the occurrence of large catastrophic events can be described with similar exponential laws and power laws. The self-similar hierarchy is a more general framework or structure which can be used to encompass or unify different scaling phenomena and rules in both physical and social systems such as cities, rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The mathematical laws on the hierarchical structure can provide us with a holistic perspective of looking at complexity such as self-organized criticality (SOC).Comment: 20 pages, 9 figures, 3 table

Collective behavior and evolutionary games - An introduction

This is an introduction to the special issue titled "Collective behavior and evolutionary games" that is in the making at Chaos, Solitons & Fractals. The term collective behavior covers many different phenomena in nature and society. From bird flocks and fish swarms to social movements and herding effects, it is the lack of a central planner that makes the spontaneous emergence of sometimes beautifully ordered and seemingly meticulously designed behavior all the more sensational and intriguing. The goal of the special issue is to attract submissions that identify unifying principles that describe the essential aspects of collective behavior, and which thus allow for a better interpretation and foster the understanding of the complexity arising in such systems. As the title of the special issue suggests, the later may come from the realm of evolutionary games, but this is certainly not a necessity, neither for this special issue, and certainly not in general. Interdisciplinary work on all aspects of collective behavior, regardless of background and motivation, and including synchronization and human cognition, is very welcome.Comment: 6 two-column pages, 1 figure; accepted for publication in Chaos, Solitons & Fractals [the special issue is available at http://www.sciencedirect.com/science/journal/09600779/56

From Turing instability to fractals

Complexity focuses on commonality across subject areas and forms a natural platform for multidisciplinary activities. Typical generic signatures of complexity include: (i) spontaneous occurrence of simple patterns (e.g. stripes, squares, hexagons) emerging as dominant nonlinear modes [1], and (ii) the formation of a highly complex pattern in the form of a fractal (with comparable levels of detail spanning decades of scale). Recently, a firm connection was established between these two signatures, and a generic mechanism was proposed for predicting the fractal generating capacity of any nonlinear system [2]. The mechanism for fractal formation is of a very general nature: any system whose Turing threshold curves exhibit a large number of comparable spatial-frequency instability minima are potentially capable of generating fractal patterns. Spontaneous spatial fractals were first reported for a very simple nonlinear system: the diffusive Kerr slice with a single feedback mirror [3]. These Kerr-slice fractals are distinct from both the transverse fractal eigenmodes of unstable-cavity lasers [4], and also from the fractals found in optical soliton-supporting systems [5,6]. On the one hand, unstable-cavity fractals may be regarded as a linear superposition of diffraction patterns with different scale lengths, each of which arises from successive round-trip magnifications of an initial diffractive seed. On the other hand, fractals formed in the Kerr slice result entirely from intrinsic nonlinear dynamics (i.e. light-matter coupling leading to harmonic generation and/or four-wave mixing cascades). These processes conspire to generate new spatial frequencies that, in turn, can produce optical structure on smaller and smaller scales, down to the order of the optical wavelength. Here we report the first predictions of spontaneous fractal patterns inside driven damped ring cavities containing a thin slice of nonlinear material. Both dispersive (i.e. diffusive-relaxing Kerr [3]) and absorptive (i.e. Maxwell- Bloch saturable absorber [7]) are considered. New linear analyses have shown that the transverse instability spectra of these two cavity systems possess the requisite comparable minima that predict the capacity for the spontaneous generation of fractal patterns. Extensive numerical simulations, in both one and two transverse dimensions, have verified that both the dispersive and absorptive cavities do indeed give rise to nonlinear optical fractals in the transverse plane. Our results confirm that the mechanism for fractal formation has independence with respect to the details of the nonlinearity. An essential ingredient for the generation of fractals is the presence of a feedback mechanism [2]. Feedback drives the cascade process that is responsible for the creation of higher spatial wavenumbers, and which ultimately leads to the “structure across decades of scale” character of the fractal pattern. Cavity geometries are therefore ideal candidates as potential optical fractal generators. The simplest dispersive nonlinearity is provided by the relaxing-diffusing Kerr effect. The threshold curves possess the qualitative features necessary for the generation of spontaneous fractal patterns: successive and comparable spatial frequency minima. Rigorous simulations have shown that the Kerr cavity is indeed capable of generating fractal patterns. In a single-K configuration, where the filter attenuates all those spatial wavenumbers outside the first instability band, it is found that simple stripe patterns emerge. Once this stationary pattern has been reached, the spatial filter is removed to allow all waves to propagate. Energy is transferred to higher spatial frequencies, and the simple strip pattern acquires successive level of fine detail at a rate that depends upon the system parameters. By analysing the power spectrum P(K) it can be seen that a fractal pattern emerges relatively rapidly. Eventually, the system enters a dynamic equilibrium (within typically less than a hundred transits) where the average power spectrum remains unchanged even though the pattern continues to evolve in real space. When this statistically invariant state is attained, the pattern is referred to as a scale-dependent fractal. An appreciable portion of the dynamic state is well described by a linear relationship ln P(K) = a + bK, where a and b are constants, and this type of behaviour is one of the characteristics of a fractal pattern [2]. We have recently found that a thin-slice Maxwell-Bloch saturable absorber [7] can also generate fractal patterns. This system can be either purely absorptive or purely dispersive. Linear analysis, together with a generalized boundary condition (which allows for attenuation), yields the threshold condition for Turing instability. One finds that the threshold spectrum comprises a series of adjacent instability islands. Simulations have revealed that the Maxwell-Bloch system can also support fractals. The single-K patterns turn out to be hexagonal arrays, familiar from conventional pattern formation [1,3]. Once this state has been reached, the spatial filter is removed and one can observe a rapid transition toward a fractal pattern. The qualitative behaviour of fractals patterns in both dispersive and absorptive systems are found to be the same, confirming the assertion of independence with respect to nonlinearity. References: [1] J. B. Geddes et al., “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 5, 3471-3485 (1994). [2] J. G. Huang and G. S. McDonald, “Spontaneous optical fractal pattern formation,” Phys. Rev. Lett. 94, 174101 (2005). [3] G. D’Alessandro and W. J. Firth, “Hexagonal spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537-548 (1992). [4] J. G. Huang et al., “Fresnel diffraction and fractal patterns from polygonal apertures,” J. Opt. Soc. Am. A 23, 2768-2774 (2006). [5] M. Soljacic and M. Segev, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048-R1051 (2000). [6] S. Sears et al., “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902-1905 (2000). [7] A. S. Patrascu et al., “Multi-conical instability in the passive ring cavity: linear analysis,” Opt. Commun. 91, 433-443 (1992)