Deterministic Walks in Random Media

Lima, Gilson F.; Martinêz, Alexandre Souto; Kinouchi, Osame

doi:10.1103/physrevlett.87.010603

Cited by 87 publications

(111 citation statements)

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“…Several characteristics of random walks on complex networks have been analyzed in connection with diffusion and exploration processes [23][24][25]. In this context, it is known that some processes, such as navigation and exploratory behavior are neither purely random nor totally deterministic [26], and can be described by walks on graphs [27,28].…”

Section: Introductionmentioning

confidence: 99%

Self-avoiding walks on scale-free networks

Herrero

2005

Phys. Rev. E

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Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are expected to be more suitable than unrestricted random walks to explore various kinds of real-life networks. Here we study long-range properties of random SAWs on scale-free networks, characterized by a degree distribution P (k) ∼ k −γ . In the limit of large networks (system size N → ∞), the average number sn of SAWs starting from a generic site increases as µ n , with µ = k 2 / k − 1. For finite N , sn is reduced due to the presence of loops in the network, which causes the emergence of attrition of the paths. For kinetic growth walks, the average maximum length, L , increases as a power of the system size: L ∼ N α , with an exponent α increasing as the parameter γ is raised. We discuss the dependence of α on the minimum allowed degree in the network. A similar power-law dependence is found for the mean self-intersection length of non-reversal random walks. Simulation results support our approximate analytical calculations.

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Section: Introductionmentioning

confidence: 99%

Self-avoiding walks on scale-free networks

Herrero

2005

Phys. Rev. E

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“…When greater values of μ are considered, the cycle distribution is no longer peaked at p min = μ + 1, but presents a whole spectrum of cycles with period p ≥ p min [15,16,20].…”

Section: Introductionmentioning

confidence: 99%

“…The agent performs a partially self-avoiding walk, where the self-avoidance is limited to the memory window τ = μ − 1. The walker's behavior depends strictly on the data set configuration and on the starting site [15,16]. The walker's movements are entirely performed based on a neighborhood table, so that the distances among the sites are simply a way of ranking their neighbors.…”

Section: Introductionmentioning

confidence: 99%

Color Texture Analysis and Classification: An Agent Approach Based on Partially Self-avoiding Deterministic Walks

Backes

Martinêz

Bruno

2010

Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications

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Abstract. Recently, we have proposed a novel approach of texture analysis that has overcome most of the state-of-art methods. This method considers independent walkers, with a given memory, leaving from each pixel of an image. Each walker moves to one of its neighboring pixels according to the difference of intensity between these pixels, avoiding returning to recent visited pixels. Each generated trajectory, after a transient time, ends in a cycle of pixels (attractor) from where the walker cannot escape. The transient time (t) and cycle period (p) form a joint probability distribution, which contains image pixel organization characteristics. Here, we have generalized the texture based on the deterministic partially self avoiding walk to analyze and classify colored textures. The proposed method is confronted with other methods, and we show that it overcomes them in color texture classification.

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“…Some processes, such as navigation and exploratory behavior are neither purely random nor totally deterministic, and can be also described by walks on graphs [4]. In this context, the generic properties of deterministic navigation [5] and directed self-avoiding walks [6] in random networks have been analyzed recently.…”

Section: Introductionmentioning

confidence: 99%

Self-avoiding walks and connective constants in small-world networks

Herrero

Saboyá

2003

Phys. Rev. E

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Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one-and two-dimensional regular lattices. The number of SAW's un was obtained from numerical simulations as a function of the number of steps n on the considered networks. The so-called connective constant, µ = limn→∞ un/un−1, which characterizes the long-distance behavior of the walks, increases continuously with disorder strength (or rewiring probability, p). For small p, one has a linear relation µ = µ0 +ap, µ0 and a being constants dependent on the underlying lattice. Close to p = 1 one finds the behavior expected for random graphs. An analytical approach is given to account for the results derived from numerical simulations. Both methods yield results agreeing with each other for small p, and differ for p close to 1, because of the different connectivity distributions resulting in both cases.

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Deterministic Walks in Random Media

Cited by 87 publications

References 10 publications

Self-avoiding walks on scale-free networks

Self-avoiding walks on scale-free networks

Color Texture Analysis and Classification: An Agent Approach Based on Partially Self-avoiding Deterministic Walks

Self-avoiding walks and connective constants in small-world networks

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