Hodge's idea in percolation: percolation threshold estimation by the unit cell

Gallyamov, S.R.; Mel'chukov, S.A.

doi:10.20537/vm110405

Cited by 2 publications

(4 citation statements)

References 5 publications

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“…In works [35][36][37] based on Shklovsky and de Gennes' representations of the topological structure of the connecting cluster ("skeleton and dead" ends), the following function Y(ξ, L), describing the conditional probability of flow into the network, was obtained:…”

Section: Theoretical Methods Within Percolation Theorymentioning

confidence: 99%

“…When describing percolation using Equation (1), the main task is to define the polynomial degree i and its coefficients. Combined use of function (1) and methods of algebraic Hodge geometry [42] and Kadanoff-Wilson similarity theory [43,44] using renormalization groups (for example see [19] enables us (in some cases) to calculate theoretical values of the percolation threshold for some regular structures [35][36][37]. According to Hodge theory, algebraic varieties (sets composed of subsets, each of which is a set of solutions to any polynomial equations) are considered.…”

Section: Theoretical Methods Within Percolation Theorymentioning

confidence: 99%

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The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold

2019

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Analyses of the processes of information transfer within network structures shows that the conductivity and percolation threshold of the network depend not only on its density (average number of links per node), but also on its spatial symmetry groups and topological dimension. The results presented in this paper regarding conductivity simulation in network structures show that, for regular and random 2D and 3D networks, an increase in the number of links (density) per node reduces their percolation threshold value. At the same network density, the percolation threshold value is less for 3D than for 2D networks, whatever their structure and symmetry may be. Regardless of the type of networks and their symmetry, transition from 2D to 3D structures engenders a change of percolation threshold by a value exp{−(d − 1)} that is invariant for transition between structures, for any kind of network (d being topological dimension). It is observed that in 2D or 3D networks, which can be mutually transformed by deformation without breaking and forming new links, symmetry of similarity is observed, and the networks have the same percolation threshold. The presence of symmetry axes and corresponding number of symmetry planes in which they lie affects the percolation threshold value. For transition between orders of symmetry axes, in the presence of the corresponding planes of symmetry, an invariant exists which contributes to the percolation threshold value. Inversion centers also influence the value of the percolation threshold. Moreover, the greater the number of pairs of elements of the structure which have inversion, the more they contribute to the fraction of the percolation threshold in the presence of such a center of symmetry. However, if the center of symmetry lies in the plane of mirror symmetry separating the layers of the 3D structure, the mutual presence of this group of symmetry elements do not affect the percolation threshold value. The scientific novelty of the obtained results is that for different network structures, it was shown that the percolation threshold for the blocking of nodes problem could be represented as an additive set of invariant values, that is, as an algebraic sum, the value of the members of which is stored in the transition from one structure to another. The invariant values are network density, topological dimension, and some of the elements of symmetry (axes of symmetry and the corresponding number of symmetry planes in which they lie, centers of inversion).

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Section: Theoretical Methods Within Percolation Theorymentioning

confidence: 99%

Section: Theoretical Methods Within Percolation Theorymentioning

confidence: 99%

The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold

2019

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“…In [32][33][34] based on the topological structure of binding clusters proposed by Schklovskiy and de Zhen ("skeleton and dead ends"), the function of conditional flow probability (percolation) in grid Y(ξ, L) was obtained as follows:…”

Section: Calculating the Dependency Of The Percolation Threshold Dependency On The Network Density (Average Number Of Links Per Crossroadmentioning

confidence: 99%

“…The main problem when describing percolation using Equation ( 1) is indicating the polynomial degree i and its coefficients. The shred use of Equation ( 1) and Hodge algebraic geometry methods [35], as well as Kadanoff-Wilson renormalization theory [36,37] with groups (see, e.g., [18]), enables us (in all cases) to calculate theoretical values of the percolation threshold for any regular structures [32][33][34]. In Hodge theory, algebraic varieties are studied (varieties, consisting of subsets, any of which comprise a set of solutions to any polynomial equations).…”

Section: Calculating the Dependency Of The Percolation Threshold Dependency On The Network Density (Average Number Of Links Per Crossroadmentioning

confidence: 99%

The Influence of Transport Link Density on Conductivity If Junctions and/or Links Are Blocked

Aleshkin¹

2021

Mathematics

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This paper examines some approaches to modeling and managing traffic flows in modern megapolises and proposes using the methods and approaches of the percolation theory. The author sets the task of determining the properties of the transport network (percolation threshold) when designing such networks, based on the calculation of network parameters (average number of connections per crossroads, road network density). Particular attention is paid to the planarity and nonplanarity of the road transport network. Algorithms for building a planar random network (for modeling purposes) and calculating the percolation thresholds in the resulting network model are proposed. The article analyzes the resulting percolation thresholds for road networks with different relationship densities per crossroad and analyzes the effect of network density on the percolation threshold for these structures. This dependence is specified mathematically, which allows predicting the qualitative characteristics of road network structures (percolation thresholds) in their design. The conclusion shows how the change in the planar characteristics of the road network (with adding interchanges to it) can improve its quality characteristics, i.e., its overall capacity.

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Hodge's idea in percolation: percolation threshold estimation by the unit cell

Cited by 2 publications

References 5 publications

The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold

The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold

The Influence of Transport Link Density on Conductivity If Junctions and/or Links Are Blocked

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