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 $k$.

6 pages, 6 figures

References in corpus (4)

• Anderson Transitions
• Contact process on a Voronoi triangulation
• Emergence of fractal in aggregation with stochastic self-replication
• Dyadic Cantor set and its kinetic and stochastic counterpart

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• A weighted planar stochastic lattice with scale-free, small-world and multifractal properties
• Multi-multifractality and dynamic scaling in stochastic porous lattice
• Contact Process on Weighted Planar Stochastic Lattice
• Nonuniversal critical dynamics on planar random lattices with heterogeneous degree distributions