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

Abstract

In this article, we show that the block size distribution function in the weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network, exhibits dynamic scaling. We verify it numerically using the idea of data-collapse. As the WPSL is a space-filling cellular structure, we thought it was worth checking if the Lewis and the Aboav-Weaire laws are obeyed in the WPSL. To this end, we find that the mean area < A > k of blocks with k neighbours grow linearly up to k = 8, and hence the Lewis law is obeyed. However, beyond k > 8 we find that < A > k grows exponentially to a constant value violating the Lewis law. On the other hand, we show that the Aboav-Weaire law is violated for the entire range of k. Instead, we find that the mean number of neighbours of a block adjacent to a block with k neighbours is approximately equal to six, independent of k.

Explosive percolation on a scale-free multifractal weighted planar stochastic lattice

“…This is equivalent to the coordination number distribution in the WPSL. We find that P (k) decays obeying a powerlaw [30,32]. Thus we see that the WPSL in one hand has the properties of networks since unlike a typical lattice its coordination number distribution follow power-law.…”

“…Moreover, the dynamics of the growth of this lattice is governed by infinitely many conservation laws, one of which being the trivial conservation of total area. One more interesting property of the WPSL is that each of the non-trivial conservation law can be used as a multifractal measure and hence it is also a multimultifractal [30]. Krapivsky and Ben-Naim also showed that it exhibits multiscaling [31].…”

“…Besides, if we regard m k as the mean or typical number of neighbors of only those blocks which has exactly k neighbours then we find that km k is a constant (statistical sense). It implies that the Aboav-Weaire law, km k ∝ k, is violated in the WPSL for the entire range of k [30].…”

“…It describes the concentration of blocks of the area within the size range a and a+da at time t. We have recently shown that it exhibits dynamic scaling [30]. Note that the WPSL is a disordered lattice that emerges through evolution and hence it can only be useful if the snapshots taken at different late stages are similar.…”

“…On the other, unlike lattices, its dual display the property of networks as its coordination number distribution follows a power-law. Besides, unlike regular lattice, the size of its cells are not equal rather the distribution of the area size of its blocks obeys dynamic scaling [30]. Moreover, the dynamics of the growth of this lattice is governed by infinitely many conservation laws, one of which being the trivial conservation of total area.…”