Weighted planar stochastic lattice

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. 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 thirst for understanding their topological and geometrical properties have always been an interesting proposition among scientists in general and physicists in particular. There exists a number of models that prescribe how to generate cellular structures. Often these structures can mimic directly the structures found in nature and they are able to capture the essential properties that we find in natural structures. 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.[4][5]

Regular planar lattices like square lattices, triangular lattices, honeycomb lattices, etc., are the simplest example of the cellular structure in which every cell has exactly the same size and the same coordination number. The planar Voronoi diagram, on the other hand, has neither a fixed cell size nor a fixed coordination number. Its coordination number distribution is rather Poissonian in nature.[6] 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. 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 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,[7] whose coordination number distribution follows a power-law.[8][9]

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 and . 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 is the area of the th block which can be well regarded as the probability of picking the th block. Interestingly, these probabilities are naturally normalized 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 and apply the generator onto it to divide it randomly into four smaller blocks. Thus the label is now redundant and hence we recycle it to label the top left corner while the rest of three new blocks are labelled and in a clockwise fashion. In general, in the th step, we pick one out of blocks preferentially with respect to area and divide randomly into four blocks. The detailed algorithm can be found in Dayeen and Hassan[10] and Hassan, Hassan, and Pavel.[11]

This process of lattice generation can also be described as follows. 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. It can also describes kinetics of fragmentation of two-dimensional objects.[12][13]

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.

Properties of WPSLs

Before 2000 epidemic models, for instance, were studying by applying them on regular lattices like square lattice assuming that everyone can infect everyone else in the same way. 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. The WPSL can be a good candidate for applying epidemic like models since it has the properties of graph or network and the properties of traditional lattice as well.


References

  1. ↑ M. Rao, S. Sengupta and H. K. Sahu Phys. Rev. Lett. {\bf 75}, 2164 (1995).
  2. ↑ A. Okabe, B. Boots, K. Sugihara and S. N. Chiu, {\it Spatial Tessellations - Concepts and Applications of Voronoi Diagrams} (Chicester: Wiley, 2000).
  3. ↑ E. Ben-Naim and P. L. Krapivsky, Phys. Rev. Lett. 76, 3234 (1996).
  4. ↑ A. Okabe, B. Boots, K. Sugihara and S. N. Chiu, {\it Spatial Tessellations - Concepts and Applications of Voronoi Diagrams} (Chicester: Wiley, 2000).
  5. ↑ G. W. Delaney, S. Hutzler and T. Aste, Phys. Rev. Lett. {\bf 101} 120602 (2008).
  6. ↑ M. M. de Oliveira, S. G. Alves, S. C. Ferreira, and R. Dickman, "Contact process on a Voronoi triangulation", Phys. Rev. E 78 031133 (2008)
  7. ↑ F. R. Dayeen and M. K. Hassan “Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice” Chaos, Solitons & Fractals 91 228 (2016)
  8. ↑ M. K. Hassan, M. Z. Hassan and N. I. Pavel, “Scale-free network topology and multifractality in a weighted planar stochastic lattice” New Journal of Physics 12 093045 ( 2010) http://doi:10.1088/1367-2630/12/9/093045
  9. ↑ M. K. Hassan, M. Z. Hassan and N. I. Pavel, Scale-free coordination number disorder and multifractal size disorder in weighted planar stochastic lattice, J. Phys: Conf. Ser, 297 012010 (2011)
  10. ↑ F. R. Dayeen and M. K. Hassan “Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice” Chaos, Solitons & Fractals 91 228 (2016)
  11. ↑ M. K. Hassan, M. Z. Hassan and N. I. Pavel, Scale-free coordination number disorder and multifractal size disorder in weighted planar stochastic lattice, J. Phys: Conf. Ser, 297 012010 (2011)
  12. ↑ P. L. Krapivsky, E. Ben-Naim, "Scaling and multiscaling in models of fragmentation", Phys. Rev. E 50 3502 (1994)
  13. ↑ P. L. Krapivsky, S. Redner and E. Ben_naim, A kinetic View of Statistical Physics (Cambridge University Press, New York, 2010)
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