Scale-free coordination number disorder and multifractal size disorder in weighted planar stochastic lattice

The square lattice is perhaps the simplest cellular structure. In this work, however, we investigate the various structural and topological properties of the kinetic and stochastic counterpart of the square lattice and termed them as kinetic square lattice (KSL) and weighted planar stochastic lattice (WPSL) respectively. We find that WPSL evolves following several non-trivial conservation laws, $\sum_i^N x_i^{n-1} y_i^{{{4}\over{n}}-1}={\rm const.}\ \forall \ n$, where $x_i$ and $y_i$ are the length and width of the $i$th block. The KSL, on the other hand, evolves following only one conservation law, namely the total area, although one find three apparently different conserved integrals which effectively the total area. We show that one of the conserved quantity of the WPSL obtained either by setting $n=1$ or $n=4$ can be used to perform multifractal analysis. For instance, we show that if the $i$th block is populated with either $p_i\sim x_i^3$ or $p_i\sim y_i^3$ then the resulting distribution in the WPSL exhibits multifractality. Furthermore, we show that the dual of the WPSL, obtained by replacing each block with a node at its center and common border between blocks with an edge joining the two vertices, emerges as a scale-free network since its degree distribution exhibits power-law $P(k)\sim k^{-\gamma}$ with exponent $\gamma=5.66$. It implies that the coordination number distribution of the WPSL is scale-free in character as we find that $P(k)$ also describes the fraction of blocks having $k$ neighbours.