InterJournal Complex Systems, 602 Status: Accepted |
Manuscript Number: [602] Submission Date: 20628 Revised On: 21023 |
Self Organized Criticality in State Transition Systems |
Subject(s): CX.07, CX.13
Category: Brief Article
Abstract:
The presence of self-organized criticality in a completely representative class of discrete monotonic state transition systems is demonstrated by the presence of a power law distribution in computational experiments involving convergence to fixed points. In order to produce the power law distribution as seen in the sandpile effect, we explored a resemblance between (1) incrementing programs by ascending a natural and completely representative lattice of monotonic state-transition systems with associated ascents in the lattice of global fixed point states of the system and (2) dropping sand grains on a sand pile with avalanches observed at the surface of the sandpile. The monotonic state-transition systems under consideration are positive logic programs over a finite signature $Sigma$. That is, functions $p: Sigma o 2^{2^Sigma}$ which describe the evolution of interpretations of $Sigma$ by the cumulative one step iteration operator $hat{T}_p(I) = I cup { x in Sigma mid exists y in p(x) [ y subseteq I ] } $. For any $p$, $hat{T}_p$ has a fixed point. A wider goal of these investigations is to determine just how these fixed points depend on $p$. We know, for example, that positive logic programs exist as points in a lattice and that incremental ascents through the program lattice also raises the fixed point of $hat{T}_p$ in the lattice of subsets of $Sigma$. Raising $p$ may, however, leave the fixed point of $hat{T}_p$ fixed, just as adding a grain to a pile of sand may not cause an avalanche. This resemblance of our transition system to Bak-Sneppen models of evolution was reinforced by considering the difference in the size of fixed points as the size of an avalanche. The log of the frequency of avalanches was plotted against the log of the sizes of the avalanches and it is clear that the plot is evidence of a power law distribution. In order to get enough points for a plot, it is essential that the program reach a fixed point ( under $hat{T}$ ) only after a rather long sequence of raisings. This was achieved by adjusting the parameters of the program, e.g. signature size, in a suitable manner. One obtains approximately $2^{10}$ avalanches with a signature size of $2^{13}$.
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