Self-similarity, Aboav-Weaire's and Lewis' laws 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 $k$.