Scale-free network topology and multifractality in a weighted planar stochastic lattice

We propose a weighted planar stochastic lattice (WPSL) formed by the random sequential partition of a plane into contiguous and non-overlapping blocks and we find that it evolves following several non-trivial conservation laws, namely \sum_i^N x_i^{n-1} y_i^{4/n-1} is independent of time ∀n, where x_i and y_i are the length and width of the ith block. Its dual on the other hand, obtained by replacing each block with a node at its centre and the common border between blocks with an edge joining the two vertices, emerges as a network with a power-law degree distribution P(k)~k^{- γ}, where γ=5.66 reveals scale-free coordination number disorder since P(k) also describes the fraction of blocks having k neighbours. To quantify the size disorder, we show that if the ith block is populated with p_i~x_i³, then its distribution in the WPSL exhibits multifractality.