Weighted planar stochastic lattice
For instance, if one wants to study the spread of disease, viruses, rumors etc.
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.
In general, space-filling planar cellular structures can be useful in a wide variety of seemingly disparate physical and biological systems.
Examples include grain in polycrystalline structures, cell texture and tissues in biology, acicular texture in martensite growth, tessellated pavement on ocean shores, soap froths and agricultural land division according to ownership etc.
[1][2][3] The question of how these structures appear and the understanding of their topological and geometrical properties have always been an interesting proposition among scientists in general and physicists in particular.
In general, cellular structures appear through random tessellation, tiling, or subdivision of a plane into contiguous and non-overlapping cells.
For instance, Voronoi diagram and Apollonian packing are formed by partitioning or tiling of a plane into contiguous and non-overlapping convex polygons and disks respectively.
[5] That is, the distribution is peaked about the mean where it is almost impossible to find cells which have significantly higher or fewer coordination number than the mean.
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,[6] whose coordination number distribution follows a power-law.
The generator then divides the initiator, in the first step, randomly with uniform probability into four smaller blocks.
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
But of course the way we label is totally arbitrary and will bear no consequence to the final results of any observable quantities.
Consider that the substrate is a square of unit area and at each time step a seed is nucleated from which two orthogonal partitioning lines parallel to the sides of the substrate are grown until intercepted by existing lines.
It results in partitioning the square into ever smaller mutually exclusive rectangular blocks.
Note that the higher the area of a block, the higher is the probability that the seed will be nucleated in it to divide that into four smaller blocks since seeds are sown at random on the substrate.
The emergence of a network-based framework has brought a fundamental change, offering a much much better pragmatic skeleton than any time before.
Today epidemic models is one of the most active applications of network science, being used to foresee the spread of influenza or to contain Ebola.
