Hierarchical aggregation in percolation model

There is a growing belief that the complex dynamics of seismicity can be better understood by studying the collective behavior of numerous lithosphere instability sources rather than focusing on the details of each of them. Classical site-percolation is a simple and tractable model which exhibits such important general features of complex systems as criticality and phase transitions of second kind. It also illustrates the mechanism of hierarchical aggregation, which is very important for explaining collective phenomena in material fracture and earthquake nucleation processes. We study the dynamics of a 2D site percolation model on a square lattice using the hierarchical approach introduced by Gabrielov et al., Phys. Rev. E., 5293–5300, 1999. The key elements of the approach are the tree representation of clusters and the Horton–Strahler scheme for cluster ranking. Accordingly, the evolution of percolation model is considered as a hierarchical inverse cascade of cluster aggregation. We analyzed the growth of the percolation cluster and established the time-dependent rank distribution of its subclusters, as well as corresponding laws for its mass, rank, and their relationship. We report several phenomena premonitory to the onset of percolation that complement the traditional power-law increase of the model's observables. In addition, we have shown that the Tokunaga side-branching constraint uniquely determines the mass–rank relationship for a general aggregation process (not necessarily originated from the percolation model). The results can be used for development and improvement of earthquake prediction techniques.