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Saturday, 1 June 2013

82. Self-Organized Criticality


How do complex adaptive systems move towards the edge-of-chaos regime (cf. Part 81), and then manage to stay there? Per Bak (1996) supplied a clear answer in terms of his important notion of self-organized criticality (SOC).


This is a very powerful idea in complexity science. In fact, according to Bak (1996), SOC is so far the only known general mechanism to generate 'robust' complexity.

Bak defined a complex system as that which consists of a large number of components, has large variability, and the variability exists on a wide range of length / time scales. According to him, large 'avalanches', not gradual change, can lead to qualitative changes of behaviour, and may form the basis for emergent phenomena and complexity. A surprisingly large number of phenomena, processes and behaviours can be understood in terms of this very basic feature of complex systems.

We can understand the essence of the SOC idea by considering the very simple sandpile experiment. Imagine a tabletop on which grains of sand are drizzling down steadily. To start with, the flat sandpile just grows thicker with time, and the sand grains remain close to where they land. A stage comes when the sand starts cascading down the sides of the table. The pile gets steeper and steeper with time, and there are more and more sandslides. With time the sandslides (avalanches or catastrophes) become bigger and bigger, and eventually some of the sandslides may span all or most of the pile. The average slope now becomes constant with time, and we speak of a stationary state.


This is a system far removed from equilibrium. Its behaviour has become collective. Falling of just one more grain on the pile may cause a huge avalanche (or it may not). The sandpile is then said to have reached a self-organized critical state, which can be reached from the other side also by starting with a very big pile: the sides would just collapse until all the excess sand has fallen off.

The state is self-organized because it is a property of the system and no outside influence has been brought in. It is critical because the grains are critically poised: The edges and surfaces of the grains are interlocked in a very intricate pattern, and are just on the verge of giving way. Even the smallest perturbation can lead to a chain reaction (avalanche), which has no relationship to the smallness of the perturbation; the response is unpredictable, except in a statistical-average sense. The period between two avalanches is a period of tranquillity (stasis) or equilibrium; punctuated equilibrium is a more appropriate description of the entire sequence of events.

Just like the constant input drizzle of sand in that system, a steady input of energy, or water, or electrons, can drive systems towards criticality, and then they self-organize into criticality by repeated spontaneous pullbacks from super-criticality, so that they are always poised at or near the edge between chaos and order. The openness of the system (to the inputs) is a crucial factor. A closed system (and there are not many around) need not have this tendency to move towards an SOC state.

Nature operates at the SOC state. As in the sandpile experiment, large systems may start from either end, but they tend to approach the SOC state.

Biological and other types of evolution are nothing but examples of self-organization. The constant influx of information-carrying energy (or rather Gibbs free energy) leads to the evolution of higher and higher levels of complexity and order.

Bak (1996) theorized that even life is a self-organized critical phenomenon. Similarly, in biological evolution, mass extinctions, as also punctuated equilibria, can be understood as SOC phenomena.

In general, complex behaviour is exhibited mainly by open systems far from equilibrium. For closed equilibrium systems, complex universal behaviour can arise only under some very specific conditions. 'Critical phenomena' (cf. Wadhawan 2000) at continuous phase transitions in crystals are an example of this. At the critical point, the system passes from a disordered state to an ordered state. The important point is that the system has to be brought very close to the critical point to observe complex behaviour, namely scale-free fluctuations of the order parameter, giving transient ordered domains of all sizes. Such complex criticality is not robust; it occurs only at or very near the critical point, and not at other temperatures (Bak 1996).

According to Bak (1996), chaos is another such example of nonrobust criticality, and therefore chaos theory cannot explain complexity completely. Chaos theory explains how simple, deterministic systems can sometimes exhibit unpredictable behaviour, but complex-looking behaviour occurs in such systems only for some specific range of values of the control parameter; the complexity is not robust.

A third example of criticality which is not self-organized, and therefore not robust, is that of a nuclear-fission reactor. The reactor is kept critical (neither subcritical nor supercritical) by the human operator, and cannot, on its own, tend towards (and be stable around) a state of self-organized criticality.

Robustness is an essential feature of SOC. Fragile criticality is not conducive to the adaptive evolution of a complex system.

Many examples of SOC have been identified.