Weighted planar stochastic lattice

Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell has exactly the same number of nearest, next nearest, nearest of next nearest etc. neighbors and hence they are called regular lattice. Often physicists and mathematicians study phenomena which require disordered lattice where each cell do not have exactly the same number of neighbors rather the number of neighbors can vary wildly. For instance, if one wants to study the spread of disease, viruses, rumors etc. then the last thing one would look for is the square lattice. In such cases a disordered lattice is necessary. One way of constructing a disordered lattice is by doing the following.

Starting with a square, say of unit area, and dividing randomly at each step only one block, after picking it preferentially with respect to ares, into four smaller blocks creates weighted planar stochastic lattice (WPSL). Essentially it is a disordered planar lattice as its block size and their coordination number are random.

Description

In applied mathematics, a weighted planar stochastic lattice (WPSL) is a structure that has properties in common with those of lattices and those of graphs. Recently, Hassan et al proposed a lattice, namely the weighted planar stochastic lattice. For instance, unlike a network or a graph, it has properties of lattices as its sites are spatially embedded. On the other hand, unlike lattices, its dual (obtained by considering the center of each block of the lattice as a node and the common border between blocks as links) display the property of networks as its degree distribution follows a power law. Besides, unlike regular lattices, the sizes of its cells are not equal; rather, the distribution of the area size of its blocks obeys dynamic scaling, whose coordination number distribution follows a power-law.

A snapshot of the weighted stochastic lattice.

Construction of WPSLs

The construction process of the WPSL can be described as follows. It starts with a square of unit area which we regard as an initiator. The generator then divides the initiator, in the first step, randomly with uniform probability into four smaller blocks. In the second step and thereafter, the generator is applied to only one of the blocks. The question is: How do we pick that block when there is more than one block? The most generic choice would be to pick preferentially according to their areas so that the higher the area the higher the probability to be picked. For instance, in step one, the generator divides the initiator randomly into four smaller blocks. Let us label their areas starting from the top left corner and moving clockwise as $a_{1},a_{2},a_{3}$ and $a_{4}$. But of course the way we label is totally arbitrary and will bear no consequence to the final results of any observable quantities. Note that $a_{i}$ is the area of the $i$th block which can be well regarded as the probability of picking the $i$th block. These probabilities are naturally normalized $\sum _{i}a_{i}=1$ since we choose the area of the initiator equal to one. In step two, we pick one of the four blocks preferentially with respect to their areas. Consider that we pick the block $3$ and apply the generator onto it to divide it randomly into four smaller blocks. Thus the label $3$ is now redundant and hence we recycle it to label the top left corner while the rest of three new blocks are labelled $a_{5},a_{6}$ and $a_{7}$ in a clockwise fashion. In general, in the $j$th step, we pick one out of $3j-2$ blocks preferentially with respect to area and divide randomly into four blocks. The detailed algorithm can be found in Dayeen and Hassan and Hassan, Hassan, and Pavel.

Area size distribution function C(a,t) for three different sizes of the WPSL. In the inset the same data is plotted on the self-similar coordinates and the data collapse of all the data taken at different times collapse into one universal curve. It reveals that the area size distribution function obeys dynamic scaling.

Degree distribution of the dual of the WPSL and coordination number distribution of the WPSL itself.