Percolation on Fractal Networks: A Survey

Martínez-Cruz, Miguel A.; Ortíz, Julián Patiño; Ortíz, Miguel Patiño; Balankin, Alexander S.

doi:10.3390/fractalfract7030231

Cited by 10 publications

(4 citation statements)

References 107 publications

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“…When the concentration of open sites exceeds the percolation threshold, there is a spanning cluster of pores (pore network) in which each pore is connected to at least one neighbor pore. The fractal properties of the critical percolation cluster (CPC) are known to be universal and determined by the lattice dimension [101][102][103]. Nonetheless, most of the percolation models of fracture networks are based on the continuum percolation with regard to the fracture distributions within the rocks (as can be seen in Refs.…”

Section: Attribute Definitionmentioning

confidence: 99%

“…where the critical exponents α, σ, and τ control the scaling behaviors of the total number of finite clusters, the size of the largest cluster, and the finite cluster-size distribution, respectively, [101][102][103]. However, the fractal properties of percolation clusters in the scalefree networks were studied only above the critical embedding dimension d c ≥ 4 [108], whereas the fracture networks are embedded into three or two dimensions, and so the scale-free networks embedded in d < d c represent a special interest.…”

Section: Attribute Definitionmentioning

confidence: 99%

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Fractal Features of Fracture Networks and Key Attributes of Their Models

et al. 2023

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This work is devoted to the modeling of fracture networks. The main attention is focused on the fractal features of the fracture systems in geological formations and reservoirs. Two new kinds of fracture network models are introduced. The first is based on the Bernoulli percolation of straight slots in regular lattices. The second explores the site percolation in scale-free networks embedded in the two- and three-dimensional lattices. The key attributes of the model fracture networks are sketched. Surprisingly, we found that the number of effective spatial degrees of freedom of the scale-free fracture network models is determined by the network embedding dimension and does not depend on the degree distribution. The effects of degree distribution on the other fractal features of the model fracture networks are scrutinized.

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Section: Attribute Definitionmentioning

confidence: 99%

Section: Attribute Definitionmentioning

confidence: 99%

Fractal Features of Fracture Networks and Key Attributes of Their Models

et al. 2023

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“…Therefore, in order to unambiguously characterize the inherent features of fractals, one needs to employ more dimension numbers, though yet there is no consensus about the number and identities of basic dimension numbers (see, for review, Refs. [46][47][48][49][50] and references therein).…”

Section: Introductionmentioning

confidence: 99%

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Ortíz

Martínez-Cruz

et al. 2023

Fractal Fract

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The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

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“…The revelation emerged that the effective spatial degrees of freedom in scale-free networks are dictated by the embedding dimension, in contrast to the degree distribution [30]. The fractal characteristics impact percolation within self-similar networks [31]. The effects of geometric confinement on point statistics in a quasi-low-dimensional system were studied.…”

Section: Introductionmentioning

confidence: 99%

Non-standard analysis for fractal calculus

Golmankhaneh

Welch

Serpa

et al. 2023

J Anal

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In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for solving α-order differential equations. Notably, we extend our analysis to solve Fractal Bernoulli differential equations. The applications of our findings are then showcased through the solutions of problems such as fractal compound interest, the escape velocity of the earth in fractal space and time, and the estimation of time of death incorporating fractal time. Visual representations of our results are also provided to enhance understanding.

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Percolation on Fractal Networks: A Survey

Cited by 10 publications

References 107 publications

Fractal Features of Fracture Networks and Key Attributes of Their Models

Fractal Features of Fracture Networks and Key Attributes of Their Models

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Non-standard analysis for fractal calculus

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