Saturday, September 20, 2008

Power law article finished

Catastrophic Power Law and Everyday Life

A recent massive Toronto propane explosion, and the city's pitiful response has inspired me to resurrect my Claptrap article and beef it up for a Knol. I went back to it, after the big Canadian Listeria disaster.

Power Law is the fundamental force behind most large-scale disasters, such as earthquakes, and industrial accidents. It is a consequence of the fractal, self-similar ordering of most natural systems, and human societies. Disasters are a consequence of Power Law coming against the natural linear thinking of the human brain.

Fractal Ordering and Nature

Take a look at most natural scenes, and you will find them pretty. A rugged coastline, a cliff face, a beautiful tree, and a sunflower. All of them appeal to our sense of proportion.

These are all fractal patterns, and are created from the application of simple growth rules, applied over and over again. When this concept first came out, I studied it intensely, and could never get over the complicated math that accompanied it. I dismissed it as merely a way to make pretty pictures. Over the years, I have become to realize that it is one of the most fundamental forces of nature, and that the concept is very simple.

Take an everyday tree. You can derive some very simple rules for making that tree, simply by specifying the degree of branching (two, three, etc), the length of the base twigs, and then state that a new sprout is simply another tree growing off the shaft. The tree can become immensely complex! The 'roughness' of the tree is a consequence of the branching rules.

Geology is my love, and here's where fractals totally rule! Take this pretty picture, for example.

You can't tell the scale! Is it a polished mineral on a ring? Is it a rock face? It is actually a satellite image from the ASTER collection. That is why all photos of geology have something applied for scale, such as a notebook or a person.

Fractals have some important attributes that affect us in everyday life. Look at the pretty clouds, nice and fluffy. You can zoom in on the clouds, and still see the same patterns as in the larger clouds. Most clouds have the same fractal roughness, but limiting dimensions are important here! There is no practical limit on the small side, since you could zoom in forever, and it would be fractal, until you hit the scale of condensation droplets. But on the large side, there are varying limits.

On a nice day, the fluffy clouds are limited to a certain size, before the fractal pattern breaks down. In Ontario this summer, we have had rough fractal clouds with no limit on the large side. That means you constantly experience all types of weather on the same day! The cloud patterns are perfectly fractal on the satellite and radar. The sunny breaks and the rain follow Power Law. Lots of little ones, and then some super big thunderstorms or sunny periods. Very pretty cloud formations, very depressing weather!

Human societies love fractals. We organize ourselves in families, villages, regions, countries, etc. Most companies also organize in self-similar hierarchies. Every smaller group resembles the larger group as a whole.

Power Law

Power Law can be a truly horrible thing. It proceeds exactly like earthquakes. One day you are chugging along with just some minor rumblers, and POW! the Big One hits. That is because each smaller Magnitude is ten times more frequent, and each Magnitude upgrade is 30 times more damaging. This results in rare, damaging events that are totally not foreseen by most human minds. Think of it, just 1% of a given number of earthquakes has a 1000 times the hitting power!

This happens with all power-law disasters, even those that are totally man-made. It is normal human comprehension to think of things in linear terms - the pot is nearly boiling, a little more and it starts to boil. We always think we are going to get some warning of trouble. We always rely on a linear extrapolation of our past experience: "Oh, this hurricane could be a bit worse than last year's."

Inevitable Power-Law Disasters

Even earthquake and hurricane disasters, are in a sense man-made. Earthquakes make buildings fall on people, hurricanes breach levies. I find that power-law disasters are inevitable when you have an unstable fractal organization (self-similar).

Take a stable fractal organization: a tree. A twig breaks off, and the whole tree does not become weaker, in fact it is more stable because of the lightened load. But introduce a linear weakening mechanism, such as rot or disease, and then things become interesting. The branches falling down become power law. You get a distribution of many small branches falling, with one in ten being significantly larger. As the owner of the house right under it, you better chop it down soon, because there is a chance of major branches, or the whole tree falling.

A totally self-similar organization can act as a tree. Under certain circumstances, each small failure can set things up for a larger failure, especially if everybody acts in 'crisis mode', and just puts in short-term solutions. As an example, consider NASA before the first Shuttle failure. Everything was self-similar right from the top, everybody engaged in 'group think', and the failures were building up in a power law distribution. There were lots of minor failures of the o-rings.

How can this be prevented? First of all, as with earthquakes, by careful monitoring. Are the failures following power law? This is very difficult to do from the outside, since nearly all such self-similar organizations hide these things, from both themselves, and the press. Take for example, the French or Japan nuclear organizations, they would never reveal the countless minor incidents.

The public could demand more transparency from such organizations. The only cure is to cut down the maximum fractal dimension, by adding some 'hard blocks', or transparency. I believe that NASA is now doing this.

No comments: