In applied mathematics, a weighted planar stochastic lattice (WPSL) is type of

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

that generalizes the concept of a lattice. The construction of a WPSL involves progressively subdividing a

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...

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.

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. Log aC vs a 1 copy

References

{{Reflist Applied mathematics Stochastic models Network theory Graph theory