Self-Organized Criticality on Self-Similar Lattice: Exponential Time Distribution between Extremes
In 1987, Bak, Tang, and Wiesenfeld introduced a mechanism (hereafter, the BTW mechanism) that underlies self-organized critical systems. Extreme events generated by the BTW mechanism are be- lieved to exhibit an unpredictable occurrence. In spite of this general opinion, the largest events in the original BTW model are efficiently predictable by algorithms that exploit information that is hidden in ap- plications. Intending to relate the predictability of self-organized critical systems with the level of its asymmetry, we examine the inter-event dis- tribution of extreme avalanches generated by the BTW mechanism on symmetrical and asymmetrical self-similar lattices. Initially, we claim that the main part of the size-frequency relationship is power-law in- dependent of the asymmetry, but the asymmetry reduces the range of scale-free avalanches in the domain of small avalanches. Further, we turn to extremes and claim that they are located on the downward bend of the distribution of the avalanches over their sizes. Finally, we compare the probability distribution of waiting time between two successive extremes with the exponential distribution. The latter gives the reference point of the complete unpredictability naturally measured in terms of the sum of two rates related to type I and II statistical errors: the rate of the unpredicted avalanches and the alarm time rate. We posit that the devi- ations of the observed probability distribution from the exponential one do not affect the unpredictability of extremes drawn from the waiting time between them.