Efficient Monte Carlo Algorithm and High-Precision Results for Percolation

Newman, M. E. J.; Ziff, Robert M.

doi:10.1103/physrevlett.85.4104

Cited by 468 publications

(497 citation statements)

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“…R L is the probability that for a site occupation p there exists a contiguous cluster of occupied sites which crosses completely the square lattice of size L. p c is the probability of occupation at the percolation threshold. There are several ways [8] to define R L . We use two of them: R e L is the probability that there exits a cluster crossing either the horizontal or the vertical direction, and R b L is the probability that there exits a cluster crossing both directions.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Anisotropy and percolation threshold in a multifractal support

Lucena¹,

Freitas²,

Corso³

et al. 2003

Braz. J. Phys.

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Recently a multifractal object, Q mf , was proposed to allow the study of percolation properties in a multifractal support. The area and the number of neighbors of the blocks of Q mf show a non-trivial behavior. The value of the probability of occupation at the percolation threshold, pc, is a function of ρ, a parameter of Q mf which is related to its anisotropy. We investigate the relation between pc and the average number of neighbors of the blocks as well as the anisotropy of Q mf . I IntroductionDue to the work of many physicists, and particularly to the contributions of Dietrich Stauffer, percolation theory has became a powerful tool in Science describing phenomena in many areas as geology, biology, magnetism, or social phenomena [1,2]. Despite the enormous success of percolation it has been a theory studied in a support (lattice) that has a single dimension. The only references relating percolation and multifractality concern to the multifractal properties of some quantities of the spanning cluster at the percolation threshold [3,4].Recently, a model to study percolation in a multifractal was proposed [5] in the literature. In fact, the authors have created an original multifractal object, Q mf , and an efficient way to estimate its percolation properties. In this work we study in detail the method to estimate p c for this multifractal and discuss the relation between p c , some topologic characteristics, and the anisotropy of Q mf .The multifractal object we develop, Q mf , is an intuitive generalization of the square lattice [5]. Suppose that in the construction of the square lattice we use the following algorithm: take a square of size 1 and cut it symmetrically with vertical and horizontal lines. Repeat this process n-times; at the n th step we have a regular square lattice with 2 n × 2 n cells. The setup algorithm of Q mf is quite similar, the main difference is that we do not cut the square in a symmetric way. In section 2 we explain in detail this algorithm.The development of Q mf has a twofold motivation. Firstly, there are systems like oil reservoirs that show multifractal properties [6] and are good candidates to be modeled by such object. Secondly, there is indeed a much more general scope: we want to study percolation phenomena in lattices that are not regular, but that are multifractal in the geometrical sense. It is important to know how site percolation transition happens in lattices in which the cells vary in size and also in the number of neighbors.In this work we analyse some geometric and topologic properties of the percolation cluster generated on Q mf at the percolation threshold. The paper is organized as follows: in Section II we present the process of construction of Q mf and the algorithm for the estimation of p c , in Section III we show the numerical simulations concerning the percolation threshold p c and the topologic properties of Q mf ; and finally in Section IV we present our final remarks and comments. II The modelIn this section we show the process of building the multifrac...

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Section: Numerical Resultsmentioning

confidence: 99%

“…Each time a block of Q mf is chosen the algorithm check if the occupied cells at the underlying square lattice are connected in such a way to form a spanning percolation cluster. The algorithm to check percolation is similar to the one used in [7,8,9,10].…”

Section: The Modelmentioning

confidence: 99%

Anisotropy and percolation threshold in a multifractal support

Lucena¹,

Freitas²,

Corso³

et al. 2003

Braz. J. Phys.

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“…Note that we use an adaptation of the Newman-Ziff algorithm (Newman and Ziff, 2000) in all the computations we perform which is significantly faster than the usual breadth-first search, and that the GCC can be computed using standard network algorithms applied to the adjacency matrix of the hypernetwork. To determine the percolation threshold we employ the network susceptibility function as defined in (Radicchi, 2015), which is given by…”

Section: Percolation In Hypernetworkmentioning

confidence: 99%

Complexity and robustness in hypernetwork models of metabolism

Pearcy

Chuzhanova

Crofts

2016

Journal of Theoretical Biology

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Metabolic reaction data is commonly modelled using a complex network approach, whereby nodes represent the chemical species present within the organism of interest, and connections are formed between those nodes participating in the same chemical reaction. Unfortunately, such an approach provides an inadequate description of the metabolic process in general, as a typical chemical reaction will involve more than two nodes, thus risking oversimplification of the the system of interest in a potentially significant way. In this paper, we employ a complex hypernetwork formalism to investigate the robustness of bacterial metabolic hypernetworks by extending the concept of a percolation process to hypernetworks. Importantly, this provides a novel method for determining the robustness of these systems and thus for quantifying their resilience to random attacks/errors. Moreover, we performed a site percolation analysis on a large cohort of bacterial metabolic networks and found that hypernetworks that evolved in more variable environments displayed increased levels of robustness and topological complexity.

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“…Last, it is very ambiguous 29 in most experimental studies whether the extracted 0  from eqn (1) coincides with the critical value c  at the transition of structural connectedness which can be determined independently. [30][31][32] Through the same Monte Carlo simulations as in our early work, 33 we have studied systems comprising width-less conductive sticks (a simple model for carbon nanotubes or metal nanowires) and found that when the stick number density N ranges from 7 to 60 (N is much higher than the percolation threshold 31 Nc ≈ 5.64), eqn (1) gives perfect fitting to all the simulation data, but N0, as well as s, significantly varies with the resistance ratio Rj/Rs and in general evidently deviates from the critical values, as shown in Fig. 1.…”

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confidence: 99%

Conductivity scaling in supercritical percolation of nanoparticles – not a power law

Östling

2015

Nanoscale

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PostprintThis is the accepted version of a paper published in Nanoscale. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.Citation for the original published paper (version of record):Li, J., Östling, M. (2015) Conductivity scaling in supercritical percolation of nanoparticles: not a power law. The power-law behavior widely observed in supercritical percolation systems of conductive nanoparticles may merely be a phenomenological approximation to the true scaling law not yet discovered. In this work, we derive a comprehensive yet simple scaling law and verify its extensive applicability to various experimental and numerical systems. In contrast to the power law which lacks theoretical backing, the new scaling law is explanatory and predictive, and thereby helpful to gain more new insights into percolation systems of conductive nanoparticles.

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Efficient Monte Carlo Algorithm and High-Precision Results for Percolation

Cited by 468 publications

References 18 publications

Anisotropy and percolation threshold in a multifractal support

Anisotropy and percolation threshold in a multifractal support

Complexity and robustness in hypernetwork models of metabolism

Conductivity scaling in supercritical percolation of nanoparticles – not a power law

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