Scale-free coordination number disorder and multifractal size disorder in weighted planar stochastic lattice

Abstract

The square lattice is perhaps the simplest cellular structure. In this work, however, we investigate the various structural and topological properties of the kinetic and stochastic counterpart of the square lattice and termed them as kinetic square lattice (KSL) and weighted planar stochastic lattice (WPSL) respectively. We find that WPSL evolves following several non-trivial conservation laws,where xi and yi are the length and width of the ith block. The KSL, on the other hand, evolves following only one conservation law, namely the total area, although one find three apparently different conserved integrals which effectively the total area. We show that one of the conserved quantity of the WPSL obtained either by setting n = 1 or n = 4 can be used to perform multifractal analysis. For instance, we show that if the ith block is populated with either pi ∼ x 3 i or pi ∼ y 3 i then the resulting distribution in the WPSL exhibits multifractality. Furthermore, we show that the dual of the WPSL, obtained by replacing each block with a node at its center and common border between blocks with an edge joining the two vertices, emerges as a scale-free network since its degree distribution exhibits power-law P (k) ∼ k −γ with exponent γ = 5.66. It implies that the coordination number distribution of the WPSL is scale-free in character as we find that P (k) also describes the fraction of blocks having k neighbours.

A planar network between short and long range behaviour

“…In contrast, Refs. [21,22] explore a full random version of this algorithm for an aleatory parameter ρ. The section ratio ρ regulate the asymmetry of the multifractal.…”

Multi-multifractality and dynamic scaling in stochastic porous lattice

“…It be seen from the plots that the results matches perfectly with the analytical prediction given by Eqs. ( 13) and (14). Moreover, fitted straight Fig.…”

“…The resulting lattice is called weighted planar stochastic lattice or in short WPSL2 where the factor 2 refers to the fact that two mutually perpendicular partitioning lines are used. It has much richer properties than the square lattice and kinetic square lattice [13,14]. One of the emergent behavior of the lattice is that the coordination number distribution function of WPSL2 follows inverse power law which immediately implies that the degree distribution of the network corresponding to its dual follow the same inverse power-law P (k) ∼ k −γ with the same exponent γ = 5.66 [13].…”

“…Multifractal formalism is an answer to this question. In 2010, Hassan et al proposed a weighted planar stochastic lattice(WPSL) which has been shown to have many non-trivial properties [13,14]. One of the properties is that it is not only a multifractal but infinitely many multifractal spectrum.…”

Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice

“…Recently, one of us Hassan et al proposed a space-filling weighted planar stochastic lattice (WPSL) where the coordination number distribution function follows powerlaw [20,21]. This is in sharp contrast to many of the cellular structures that we are familiar with.…”