Self-similarity, Aboav-Weaire's and Lewis' laws in weighted planar stochastic lattice

TL;DR

This paper investigates the structural laws and scaling behavior of the weighted planar stochastic lattice (WPSL), revealing that while some classical laws are obeyed in certain regimes, others are violated, and the block size distribution exhibits dynamic scaling.

Contribution

The study demonstrates that the WPSL exhibits dynamic scaling in its block size distribution and analyzes the applicability of Lewis and Aboav-Weaire laws, revealing their limitations in this structure.

Findings

Block size distribution in WPSL shows dynamic scaling.

Lewis law holds for blocks with up to 8 neighbors.

Aboav-Weaire law is violated across all neighbor ranges.

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 $k$.