A Lévy flight is a random walk where the size of the steps has an infinite variation – they can be small, and tend to be, but there is no limit to how large they can become, and while large steps get smaller and smaller in frequency, they never die out - and where the amount of time between steps is not related to distance. This is the property known as having a heavy tail, and many mathematicians distinguish flights from walks by saying that walks have steps whose time taken is in proportion to the distance travelled.

Looking for Lévy flights is one of the cardinal skills of the modern world. Most people don't even know what they are, assuming that events follow an ordinary distribution – described by what is called the "Central Limit Theory". Lévy himself is similarly obscure – most people have not heard of Paul Pierre Lévy even though he is one of the mathematicians who most describes the world we live in. Wikipedia, which can go on to voluminous detail about every city mentioned in a vaguely obscure fantasy series, can't even manage a coherent biography of a man whose mathematics moves markets and describes the shape of life and death.

I'm going to be explicit about the point of this post, to get people to look for Lévy flight behavior. Whether in music or in the movement of counterfeit money - the Lévy flight pattern is pervasive in biological systems. And everything that people do, is biological.

My own interest in Levy flight behavior was sparked at the age of 9, when I noticed that in the north field, plants of a given kind were found in clusters. The pattern implied a mathematics, but mathematics implied something underneath. It would be a long time later that I would understand that there is a locative relationship involved in selection –closer clusters select for individuals with better intra-species competition, that is selfish and sexual signally behavior – while long jump individuals have better inter-species competition. When you are with neighbors of the same kind, holding off the others is easier, but your neighbors are also your competitors for everything including the attention of bees. Alone, you have the grasses and other competitors of other species. This implies that the balance of inter/intra species competition will be influenced by the long jump strategy a species has.

Even when I was young I concluded that traits could be "piggy backed" onto a jumper – a teacher laughed at me until I brought in a parcel of four leafed clovers – products of one seed landing and spreading. Four leaf clover have no particular advantage, but without competition – whatever trait that made the long jumper live, is now attached to the traits that it just happened to have.

The features of Lévy flight distribution show up with a vengeance with the advent social networking software, because the internet and personal telecom facilitate the "long jumps" needed to create a Levy flight distribution. thus allowing insurgencies to metastasize whether cultural or military – the long jumps radically reduce the distance between two nodes, and the cost of a long jump determines the cohesiveness of the clusters. The cheaper the jump, the greater the cohesion. A random "walk" means that the amount of time for a given move is proportional to distance, a "flight" on the other hand means that all steps take the same amount of time. A "random flight" down Wall Street comes to different conclusions than a random walk – something which Bouchard and Potters detail in their useful text on financial risk and drivative pricing.

Consider if you will an article from today's Washington Post by author Larry Kahaner on the AK-47. The original "submachine gun" was the Thompson, which was a marvel of engineering inefficiency, but based around the concept of the "trench broom", lots of killing power in a small volume. The power of the automatic hand carried weapon is based on the Lévy flight behavior of infantry – long periods of walking around, combined with very close in combat in a very concentrated area. The most likely place to find something, is next to lots of other things just like it.

The AK-47 does not have the accurate range or sophistication of the M-16, it is designed around the long jump/close cluster concept of Lévy flight – even though the mathematics were not known to its creators. The armoured column or mechanized infantry crossed long distances, and then broomed at close quarters. The AK-47 and its descendants are still in use today, and they are effective as guerilla weapons, precisely because the problem of a guerilla is not shooting at individuals at long range, but doing as much damage as quickly as possible from close range.

The US military, by contrast, uses range of weapons to cover long jumps, as a result, in the rolling battle environment, there are no competitors to the US military – before they can kill us, we can kill them. This same strategy, however, does poorly in the clustered combat environment, where the distinctions between "us" and "them" are not clear or have broken down.

But Lévy flight behavior is all around us. Think for a moment on the question of drawing legislative districts. The right wing loves "compact" districts that follow "traditional lines" – lines that were often gerrymandered when created. A "compact" district favors normal distribution over Lévy distributions. However, consider the case of urban dwellers or groups that specialize in forms of labor that cluster, they will be systematically discriminated against, since their Ld will focus on small towns far apart. Many of the "gerrymanders" aren't, in fact, abnormal, but follow the long jump taken by a particular group of people – often a road, valley or river.

Random flights, versus random walks, then, show up in our finances, in our politics, and in our military situation. We know emprically that core science takes place in a few journals - which is the incentive for a Lévy flight search - once you have gotten to the point of diminishing returns, the best move is to move on.

Recognizing this – that we are moving from a "random walk" world to a "random flight" world – leads us to look back at other times when there were similar transitions. The printing press in Europe is a well worn example, as is the age of discovery. The late Roman Republic failed to deal with the shift from walk driven Italy politics, to flight driven military and political dynamics that came from being masters of the inland sea.

The expectation of random walk versus the reality of random flight which often causes mismatches and anomalies. Random flights often can send cascades of letters to a congressmen where only a few were expected. Mistaking random flight behavior for random walk behavior leads people to look for conspiracies, because a random flight move is so far out of their normal expectations, that it seems to be produced by design. Conversely, governments will take brute force approaches to shutting down long jumps – for example the United States Government is about to require pre-approval of all international travel – no one will get out, without the being cleared by the growing secret police system of the United States.

The paranoia over long jumps, in fact, leads to the urging of invisible restraints – because the lone seed, unbound by the intra-species competition that has the chance to wildly propogate. Indeed most living creatures have long jump/close cluster strategies – dominate a local area, but send some shots off into the void, hoping that they will find another favorable place.

The interstate system is one such flight generator - travel by US routes some time - the time to travel does not vary as much between city and country driving as driving on interstates - where it can take as long to go from New Haven to Boston as to cross the George Washington Bridge. Controlling access to flight generators is one key means by which gatekeepers maintain power - whether it is phone calls, college admissions or media coverage - if it creates a jump, it creates a discontinuity. In fact, what many people think of as a "tipping point" from numbers, is caused, in reality, by the establishment of a series of long jump mechanisms at lower energy than before. Not smart mobs, but smart flies. One person locates something - a web video, a cheap airfare, a flame war - and tells lots of others. Suddenly dead quiet is lethally noisy.

Flights are very efficient at rapidly locating large numbers of different kinds of resources - because finding something is likely next to other things, it means that a close examination of an area will rapidly yield a catalog of its most prevalent resources - a close search is statistically close to the total contents of that one area. By gathering clustered data from separated clusters both the total population, and the "clumpiness" of the population can be found. Paradoxically, this makes it a bad pattern for finding a single lost item - we don't lose our keys near other sets of keys.

Internet politics is in the business of joining tight clusters of activity with long leaps that join them - getting people upset with Lieberman and people upset with Conrad Burns to see themselves as part of the same moment - and moving resources rapidly to the point where they will do the most good. "Top Down" and "Bottom Up" structures both want to restrict most people to normal and slow distributions, where, like sheep, they stay in the same places. Lévy flight distribution overturns the entire random walk of equal steps paradigm of social normalcy and organization, whether the space traversed is physical or political.

So, in summary, Paul Pierre Lévy gave us many tools to think about the world, he, and those tools, are only now becoming known, but once an individual has a feel for their signature, they look at the world differently. Suddenly patterns which seem random or coincidental become joined by how the long jump and short cluster relate to each other, finding a game theory matrix where the two forms are joined by some common trade off – the choice that the jumpers make is the opposite of the choice the clusterers make. Since Lévy flights are "natural" in the sense that natural processes result in Lévy distrubtions, and human beings participate in this because we too, are natural, there is no reason to be biased against them in our social organization or legal system, except, of course, that some interests benefit from having people bound all up in close clusters, or by walks rather than taking flight on distributive wings.