Weighted planar stochastic lattice

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In applied mathematics, a weighted planar stochastic lattice (WPSL) is type of graph that generalizes the concept of a lattice.^[1][2][3] The construction of a WPSL involves progressively subdividing a unit square into smaller and smaller regions. The graph defined by assigning a vertex to each region and drawing an edge between the vertices for adjacent regions has a power-law degree distribution.^[4]

The construction process of the WPSL can be described as follows.^[5][6] 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 ${\displaystyle a_{1},a_{2},a_{3}}$ and ${\displaystyle a_{4}}$. 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 ${\displaystyle a_{i}}$ is the area of the ${\displaystyle i}$th block which can be well regarded as the probability of picking the ${\displaystyle i}$th block. These probabilities are naturally normalized ${\displaystyle \sum _{i}a_{i}=1}$ 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 ${\displaystyle 3}$ and apply the generator onto it to divide it randomly into four smaller blocks. Thus the label ${\displaystyle 3}$ is now redundant and hence we recycle it to label the top left corner while the rest of three new blocks are labelled ${\displaystyle a_{5},a_{6}}$ and ${\displaystyle a_{7}}$ in a clockwise fashion. In general, in the ${\displaystyle j}$th step, we pick one out of ${\displaystyle 3j-2}$ blocks preferentially with respect to area and divide randomly into four blocks.