The structure of site percolation models on three-dimensional square lattices

Abstract

Получено 23 мая 2013 г., после доработки 4 июля 2013 г.В работе рассматривается структура моделей перколяции узлов на трехмерных квадратных решетках при различных формах (1, π)-окрестности. Для этих моделей предложены изо-и анизотропные модификации алгоритма инвазивной перколяции с (1, 0)и (1, π)-окрестностями. Все рассмотренные алгоритмы являются частными случаями анизотропного алгоритма инвазивной перколяции на n-мерной решетке с (1, π)-окрестностью. Данный алгоритм положен в основу библиотеки SPSL, выпущенной под лицензией GNU GPL-3 с использованием свободного языка программирования R.Ключевые слова: перколяция узлов, n-мерная квадратная решетка, неметрическое расстояние Минковского, язык программирования R, библиотека SPSL Abstract. -In this paper we consider the structure of site percolation models on three-dimensional square lattices with various shapes of (1, π)-neighborhood. For these models, are proposed iso-and anisotropic modifications of the invasion percolation algorithm with (1, 0)and (1, π)-neighborhoods. All the above algorithms are special cases of the anisotropic invasion percolation algorithm on the n-dimensional lattice with a (1, π)-neighborhood. This algorithm is the basis for the package SPSL, released under GNU GPL-3 using the free programming language R.

Information Technology of Problems Solutions Support in a Complex System Management

“…and "What do we want?". The use of such a model for solving elementary problems is confirmed by the modern capabilities of the method, because percolation problems are now known, posed and solved on two-dimensional [4], three-dimensional [5] and even seven-world lattices [6].…”

“…These models, as a rule, are quite complex, but their main sign is that these models are most often based on only one variable: temperature, flow rate, electric current, etc. [5,6]. This greatly simplifies the calculation and execution of control actions.…”

On the convergence of percolation probability functions to cumulative distribution functions on square lattices with (1,0)-neighborhood

“…This package contains several functions for generating percolation clusters and their samples on 2D and 3D square lattices with different sizes, neighborhood types, occupation probabilities and starting site subsets, but with a standard uniform weighting distribution S ∼ Unif(0, 1). The basic characteristics of percolation models realized by functions from the SPSL package have been described by the author in previously published works [16,17,18] in Russian. The scope of these percolation models is limited by such stochastic phenomena that are satisfactorily described by uniformly distributed random variables with linear cumulative distribution functions.…”

“…In this case, the sample of clusters on the percolation lattice can be interpreted as a cross-section of the random process that generates these clusters. The first numerical results with statistical analysis of site percolation models on 2D and 3D square lattices with (1, d)-neighborhoods were previously published by the author in Russian [16,17,18].…”

Statistical estimation of the average number of cluster sites on uniformly weighted square lattices

“…where k is a parameter that determines the angular coefficient of the lower slant asymptote; p c , s are the shift and scale parameters of the model (3) that determine the position of the inflection point and the radius of the neighborhood within which the second derivative of the approximating function (3) differs significantly from zero. The slant dashed lines in Figure 1 All results were obtained using the "ssi20()" function from the "SPSL" package released by the author for the R system under the GNU GPL-3 license [3]. To estimate the parameters of the regression model, the "gsl_nls()" function from the "gslnls" package, released under the GNU GPL-3 license for the R system, was used [4].…”