Powerlaws and distributions with heavy tails are common features of many experimentally studied complex systems, like the distribution of the sizes of earthquakes and solar flares, or the duration of neuronal avalanches in the brain. It had been tempting to surmise that a single general concept may act as a unifying underlying generative mechanism, with the theory of self organized criticality being a weighty contender.
On the theory side there has been, lively activity in developing new and extended models. Three classes of models have emerged. The first line of models is based on a separation between the time scales of drive and dissipation, and includes the original sandpile model and its extensions, like the dissipative earthquake model. Within this approach the steady state is close to criticality in terms of an absorbing phase transition. The second line of approach is based on external drives and internal dynamics competing on similar time scales and includes the coherent noise model, which has a non-critical steady state characterized by heavy-tailed distributions. The third line of modeling proposes a non-critical state which is self-organizing, being guided by an optimization principle, such as the concept of highly optimized tolerance.
We present a comparative overview regarding distinct modeling approaches together with a discussion of their potential relevance as underlying generative models for real-world phenomena. The complexity of physical and biological scaling phenomena has been found to transcend the explanatory power of individual paradigmal concepts, like the theory of self-organized criticality, the interaction between theoretical development and experimental observations has been very fruitful, leading to a series of novel concepts and insights.
Powerlaws and Self-Organized Criticality in Theory and Nature
Dimitrije Markovic, Claudius Gros