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

Herrero, Carlos P.; Saboyá, Martha

doi:10.1103/physreve.68.026106

Cited by 27 publications

(31 citation statements)

References 42 publications

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“…the mean value obtained (for each n) by averaging over the network sites and over different network realizations (for given γ, k 0 , and N ). For Erdös-Rényi random networks with poissonian distribution of degrees, one has s rd n = k n [34], and therefore the connective constant is µ = k . In connection with this, we note that for a Bethe lattice (or Cayley tree) with connectivity k 0 , the number of SAWs is given by s BL n = k 0 (k 0 − 1) n−1 , and one has µ BL = k 0 − 1.…”

Section: Basic Definitions and Methodsmentioning

confidence: 99%

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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: Basic Definitions and Methodsmentioning

confidence: 99%

“…Universal constants for SAWs on regular lattices have been discussed by Privman et al [33]. In our context of complex networks, the asymptotic properties of SAWs have been studied recently in small-world networks [34].…”

Section: Introductionmentioning

confidence: 99%

Self-avoiding walks on scale-free networks

Herrero

2005

Phys. Rev. E

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“…These studies provided much insight on the structure and thermodynamics of polymers [6,7]. However, SAWs on networks have not attracted much attention [32,33,34,23,35].…”

Section: Introductionmentioning

confidence: 99%

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

Tishby¹,

Biham²,

Katzav³

2016

J. Phys. A: Math. Theor.

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Abstract. We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a deadend node from which they cannot proceed. Focusing on Erdős-Rényi networks we show that the pruned networks maintain a Poisson degree distribution, p t (k), with an average degree, k t , that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, n T (ℓ), increases dramatically as a function of ℓ. We also obtain analytical results for the path-length distribution, P (ℓ), of the SAW paths which are actually pursued, starting from a random initial node. It turns out that P (ℓ) follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.

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“…The connective constant depends upon the particular topology of each lattice, and is known with high accuracy for two-and three-dimensional lattices [45,48]. Self-avoiding walks on small-world networks have been studied in recent years [38]. In particular, for networks generated from the square lattice, the connective constant µ was found to rise from µ = 2.64 for rewiring probability p = 0 (regular lattice) to µ = 3.70…”

Section: B Self-avoiding Walksmentioning

confidence: 99%

“…They are also useful to characterize complex crystal structures [36] and networks in general [37]. In particular, the asymptotic properties of SAWs on small-world networks were studied in connection with the so-called connective constant or long-distance effective connectivity [38]. Recently, kinetic-growth self-avoiding walks on uncorrelated complex networks were considered, with particular emphasis upon the influence of attrition on the maximum length of the paths [39,40].…”

Section: Introductionmentioning

confidence: 99%

Kinetic-growth self-avoiding walks on small-world networks

Herrero

2007

Eur. Phys. J. B

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Kinetically-grown self-avoiding walks have been studied on Watts-Strogatz small-world networks, rewired from a two-dimensional square lattice. The maximum length L of this kind of walks is limited in regular lattices by an attrition effect, which gives finite values for its mean value L . For random networks, this mean attrition length L scales as a power of the network size, and diverges in the thermodynamic limit (system size N → ∞). For small-world networks, we find a behavior that interpolates between those corresponding to regular lattices and randon networks, for rewiring probability p ranging from 0 to 1. For p < 1, the mean self-intersection and attrition length of kinetically-grown walks are finite. For p = 1, L grows with system size as N 1/2 , diverging in the thermodynamic limit. In this limit and close to p = 1, the mean attrition length diverges as (1 − p) −4 . Results of approximate probabilistic calculations agree well with those derived from numerical simulations.

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

Cited by 27 publications

References 42 publications

Self-avoiding walks on scale-free networks

Self-avoiding walks on scale-free networks

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

Kinetic-growth self-avoiding walks on small-world networks

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