[General Statement of Bak's SOC].
The theory of self-organized criticality posits that complex behavior in nature
emerges from the dynamics of extended, dissipative systems that evolve through a sequence of meta-stable states into a critical state, with long range
spatial and temporal correlations. Minor disturbances lead to intermittent events of all sizes. These events organize the system into a complex state that
cannot be reduced to a few degrees of freedom. This type of ``punctuated equilibrium'' dynamics has been observed in astrophysical, geophysical, and
biological processes, as well as in human social activity.
[Renormalization is called Blockmodeling in Social Network Research - here, forest fires are found not to be SOC but ordinary critical, dependent on parameter fine tuning.] We introduce a Renormalization scheme for the one and two dimensional Forest-Fire models in order to characterize the nature of the critical state and its scale invariant dynamics. We show the existence of a relevant scaling field associated with a repulsive fixed point. This model is therefore critical in the usual sense because the control parameter has to be tuned to its critical value in order to get criticality. It turns out that this is not just the condition for a time scale separation. The critical exponents are computed analytically and we obtain $\nu=1.0$, $\tau=1.0$ and $\nu=0.65$, $\tau=1.16$ respectively for the one and two dimensional case, in very good agreement with numerical simulations.
[FF model arguing SOC] The forest fire model is a reaction-diffusion model where energy, in the form of trees, is injected uniformly, and burned (dissipated) locally. We show that the spatial distribution of fires forms a novel geometric structure where the fractal dimension varies continuously with the length scale. In the three dimensional model, the dimensions varies from zero to three, proportional with $log(l)$, as the length scale increases from $l \sim 1$ to a correlation length $l=\xi$. Beyond the correlation length, which diverges with the growth rate $p$ as ${\xi} \propto p^{-2/3}$, the distribution becomes homogeneous. We suggest that this picture applies to the ``intermediate range'' of turbulence where it provides a natural interpretation of the extended scaling that has been observed at small length scales. Unexpectedly, it might also be applicable to the spatial distribution of luminous matter in the universe. In the two-dimensional version, the dimension increases to D=1 at a length scale $l \sim 1/p$, where there is a cross-over to homogeneity, i. e. a jump from D=1 to D=2.
[Renormalization again with FF model]
[Patch size affects second order phase transition from ordinary critical to percolation-like.... SOC-like results due to superposition of pre- and post-phase transition...] We study finite-size effects in the self-organized critical forest-fire model by numerically evaluating the tree density and the fire size distribution. The results show that this model does not display the finite-size scaling seen in conventional critical systems, but rather a rearrangement of the tree structure from patches of different density to a more homogeneous system with large density fluctuations in time.
[HOT models of FFs and other phenomena as another general theory for power-law distributions in evolving or designed systems] We introduce a family of robust design problems for complex systems in uncertain environments which are based on tradeoffs between resource allocations and losses. Optimized solutions yield the ``robust, yet fragile'' features of Highly Optimized Tolerance (HOT) and exhibit power law tails in the distributions of events for all but the special case of Shannon coding for data compression. In addition to data compression (DC), we construct specific solutions for world wide web traffic (WWW) and forest fires(FF), and obtain excellent agreement with measured data.
[Bak-type SOC model of FF with slopes that differ from actual data]
[Briefly put: HOT can evolve thru evolutinary algorithms] We introduce a mechanism for generating power law distributions, referred to as {\it highly optimized tolerance} (HOT), which is motivated by biological organisms and advanced engineering technologies. Our focus is on systems which are optimized, either through natural selection or engineering design, to provide robust performance despite uncertain environments. We suggest that power laws in these systems are due to tradeoffs between yield, cost of resources, and tolerance to risks. These tradeoffs lead to highly optimized designs that allow for occasional large events. We investigate the mechanism in the context of percolation and sand pile models in order emphasize the sharp contrasts between HOT and self organized criticality (SOC), which has been widely suggested as the origin for power laws in complex systems. Like SOC, HOT produces power laws. However, compared to SOC, HOT states exist for densities which are higher than the critical density, and the power laws are not restricted to special values of the density. The characteristic features of HOT systems include: (1) high efficiency, performance, and robustness to designed-for uncertainties, (2) hypersensitivity to design flaws and unanticipated perturbations, (3) nongeneric, specialized, structured configurations, and (4) power laws. The first three of these are in contrast to the traditional hallmarks of criticality, and are obtained by simply adding the element of design to percolation and sand pile models, which completely changes their characteristics.
[HOT links complexity to robustness in designed systems, and can arise naturally through biological mechanisms] The relative importance of exogenous and endogenous effects in extinction has been widely debated. We present a unifying picture based on a biotic community of evolving lattice organisms, where mutation and selection of the fittest leads to the evolution of specialized internal structure reflecting common environmental disturbances. This is an example of Highly Optimized Tolerance (HOT), a theoretical framework supporting observations of repeatable patterns in the fossil record, in which large extinction events are triggered by rare environmental disturbances, most strongly effecting the most highly evolved, specialized organisms, followed by periods of rapid growth and diversification.
[shows various equivalences among different models] We turn the stochastic critical forest-fire model introduced by Drossel and Schwabl (PRL 69, 1629, 1992) into a deterministic threshold model. This new model has many features in common with sandpile and earthquake models of Self-Organized Criticality. Nevertheless our deterministic forest-fire model exhibits in detail the same macroscopic statistical properties as the original Drossel and Schwabl model. We use the deterministic model and a related semi-deterministic version of the model to elaborate on the relation between forest-fire, sandpile and earthquake models.