A weighted planar stochastic lattice with scale-free, small-world and multifractal properties

TL;DR

This paper introduces a weighted planar stochastic lattice with multifractal, scale-free, and small-world properties, revealing complex conservation laws and hierarchical network structures.

Contribution

It presents a novel stochastic lattice model with unique multifractal and conservation properties, and demonstrates its dual network's scale-free and small-world characteristics.

Findings

The lattice exhibits infinitely many conservation laws with multifractal measures.

The dual network has a power-law degree distribution with exponent 4.13.

The network is small-world with high clustering and logarithmic path length growth.

Abstract

We investigate a class of weighted planar stochastic lattice (WPSL1) created by random sequential nucleation of seed from which a crack is grown parallel to one of the sides of the chosen block and ceases to grow upon hitting another crack. It results in the partitioning of the square into contiguous and non-overlapping blocks. Interestingly, we find that the dynamics of WPSL1 is governed by infinitely many conservation laws and each of the conserved quantities, except the trivial conservation of total mass or area, is a multifractal measure. On the other hand, the dual of the lattice is a scale-free network as its degree distribution exhibits a power-law $P(k)\sim k^{-\gamma}$ with $\gamma=4.13$. The network is also a small-world network as we find that (i) the total clustering coefficient $C$ is high and independent of the network size and (ii) the mean geodesic path length grows logarithmically with $N$. Besides, the clustering coefficient $C_k$ of the nodes which have degree $k$ decreases exactly as $2/(k-1)$ revealing that it is also a nested hierarchical network.