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Per Bak.
How Nature Works: the science of self-organized criticality.
OUP. 1997

rating : 2.5 : great stuff
review : 6 July 1999

An unlikely event is likely to happen because there are so many unlikely events that could happen.

Per Bak is one of the founding fathers of the science of self-organised critical systems. These are systems that are neither too "cold" (frozen into motionlessness), nor too "hot" (swirling in memoryless chaotic motion), but rather poised on the edge; and that, importantly, self-organise into this poised state, with no external tuning or organisation required. One feature that characterises such systems is a power law distribution of the characteristic events (avalanches, quakes, crashes, jams, ...). Although large events are comparatively rare, events can and do happen on all scales, with no different mechanism needed to explain the rare large events than that which explains the smaller, more common ones.

Systems in Balance are not Complex:
     In the past, it has often been tacitly assumed that large systems, such as those we find in biology and economics, are in a stable balance ... Contingency is irrelevant. Small freak events can never have dramatic consequences. ...
     A general equilibrium theory has not been explicitly formulated for biology, but a picture of nature as being in "balance" often prevails. Nature is supposed to be something that, can, in principle, be conserved; this idea motivates environmentalists and conservationists. No wonder: in a human lifetime the natural world changes very little, so equilibrium concepts may seem natural or intuitive. However, if nature is in balance, how did we get here in the first place? ... Does nature as we see it now ... have any preferential status from an evolutionary point of view? Implicitly, the idea of nature being in balance is intimately related to the view that humans are at the centre: our natural world is the "right one".

Chaos is not Complexity:
     ... Chaos theory shows how simple systems can have unpredictable behaviour.
     Chaos signals have a white noise spectrum, not 1/f. ... Chaotic systems have no memory of the past and cannot evolve. However, precisely at the "critical" point where the transition to chaos occurs, there is complex behaviour, with a 1/f-like signal. The complex state is at the border between predictable periodic behaviour and unpredictable chaos. Complexity occurs only at one very special point, and not ... where there is real chaos. ...
     In short, chaos theory cannot explain complexity.

Power laws, characteristic of self-organising critical systems, have an interesting property. Events are not periodic (although they appear to be, because large events happen rarely, and we are keen to force a periodic structure onto them.) Neither is the occurrence of an event statistically independent of the occurrence of earlier events (unlike tossing a coin). And neither is an event "more likely" to occur if it has not happened for a while ("we haven't has an earthquake for ages -- one must be due soon"). Rather counter-intuitively, the opposite holds:

The power law indicates that the longer you have waited since a large earthquake at a given location, the longer you can expect still to have to wait... Earthquakes are clustered in time, not periodic.

Even when we do recognise the clustering effect, as with buses -- "nothing for ages, then three come along at once!" -- it is taken as a sign of the perversity of the universe behaving in an unnatural way merely to inconvenience us, not as a recognition of a universal law.

Bak explains the theory of self-organised criticality partly by describing the historical process that led him and his coworkers to develop it. Crucial to their understanding was the building of extremely simple models that nevertheless exhibited the interesting properties. Initially they started with a 2D network of connected twisted pendulums. Although that model did exhibit self-organising criticality, it was still too far from an intuitive set-up to give them the needed insight. But then they hit upon what has become the trademark example: the sandpile, with its avalanches self-organising it to the critical slope.

Perhaps our ultimate understanding of scientific topics is measured in terms of our ability to generate metaphoric pictures of what is going on. Maybe understanding is coming up with metaphoric pictures. ... the [sandpile] picture led to a vastly improved intuitive understanding of the phenomenon.

A sandpile is a simple, everyday system that is easy to visualise and understand. [Personally, I feel that scientific understanding requires both the metaphoric picture, and the mathematical manipulation. I well remember my General Relativity courses where I had good pictures and metaphors -- bending light beams and tipped light cones falling into black holes -- but I never managed to set up or solve the relevant equations -- so I felt I never really understood it.]

Bak also has some acid comments to make about "big" science. After all, he and his colleagues opened up a new branch of scientific study by looking at sand. He thinks too much money is spent on expensive equipment that does not necessarily lead to good, insightful science. [Maybe the answer lies at neither extreme, but rather that a self-organised scientific establishment with a power law distribution of experiments "on all scales" is best?]

Now that the multibillion dollar funding for the Texas superconducting super collider has vanished, experiments based more on thinking and imagination and less on the blind and mindless use of costly hardware, as had prevailed for thirty years, wouldn't be such a bad outcome. I suspect that more insight will come out of sandpile-type experiments than would ever have come out of the super collider, at a cost reduced by a factor of 10,000. But we will never know. ... None of the funding was transferred to other projects ... But since thoughts and sand are free, our research is more resilient.

In fact, one of Bak's coworkers came up with a model that is even simpler than sandpile avalanches, and yet still exhibits self-organised criticality:

... the model was probably simpler than any model that anybody had ever written for anything: Random numbers are arranged in a circle. At each time step, the lowest number, and the numbers at its two neighbours, are each replaced by new random numbers. ... This simple scheme leads to rich behaviour beyond what we could image.

This model was simple enough that they could run computer simulations, and discover the events that led to big avalanches. But at the time they happened, those events appeared no different from any other event -- there was no indication of the enormous effect they were to have. And that is because the avalanche is caused not only by that event, but also by the entire history of the system as it self-organises into a critical state.

Because we can explain with utmost precision what has happened does not mean that we are able to predict what will happen.

After discussing sandpiles and other simple models, to build up a picture of just what a self-organised critical system is, and how it behaves (events on all scales), Bak goes on to discuss a variety of phenomena that appear to be self-organised (because they exhibit power law behaviours). These include stock market crashes, traffic jams, solar flares:

The theory explains why huge solar flares that disrupt telecommunications occur, on average, every 10 to 20 years. The large events are not periodic ... If we have patience enough, we are bound to experience even larger flares with more devastating effects, with a frequency given by extrapolating the critical behaviour further.

evolution in general:

... the idea of a poised state operating between a frozen and a disordered, chaotic state makes an appealing picture for evolution. A frozen state cannot evolve. A chaotic state cannot remember the past. That leaves the critical state as the only alternative.

and the extinction of the dinosaurs in particular:

It seems to be a widespread assumption that some cataclysmic impact must be responsible for mass extinction, so the debate has been about which external force is responsible.

The point is that no external force, or anomalous internal force, is necessary to explain the events. It is just that a self-organised critical system naturally has infrequent large events, be they mass extinctions, enormous avalanches, stock market crashes, massive earthquakes, or whatever. [Although I must admit, having grown up during a time when everyone was puzzled by what caused the inexplicable extinction of the dinosaurs, I now view the plethora of competing explanations with some amusement -- what with meteor impacts, climate change, vegetation change, shrews eating their eggs, local supernovae, solar flares, volcanic eruptions, collapse of Deccan Traps, ... how did they manage to live so long?]

This is a great book. It isn't perfect: the later chapters do rush some of the explanations (in particular, a bunch of rather similar graphs have too brief descriptions of what they portray); some of the personal anecdotes are a little heavy-handed; and the typeface is revolting. Ignoring these minor defects, however, there is a clear explanation of the process of self-organised criticality, with a good range of examples, some excellent intuition priming, and some great stuff about the nature of economics and the ability to control it that I wish (vainly, I know) that politicians could absorb, understand, and act on.