Power Law Distributions

Imagine that your city has lost its electricity, but the telephone network is intact. You are asked to create a telephone tree such that everyone can be kept up to date on the situation as efficiently as possible. What would the telephone tree look like? First you would need to decide how many people each person would contact. If each person only contacts two other people the number of layers needed before everyone is contacted becomes too large to be efficient. If each person is asked to contact 25 other people there will be much less layers, but everyone will complain about how long it is taking to contact their people. The ideal is probably about 10 other people to phone. The number of layers is manageable and the requirements of each individual are not too onerous. So, the first person phones 10 people and each of them phone 10 people and so on until everyone in the city has been contacted. It would not make sense to have people in the second layer contact 2 people each, 17 in the third layer, 46 in the fourth layer and so on. This would cause bottlenecks that would slow the message reaching people. The most efficient system would be for each layer to have the most efficient number of people to contact. That means the proportions between layers is constant. In this case each layer has 10 times the number of people as the layer before. You can see that this structure is therefore self-similar. This relationship is called a power law relationship and as we have seen in the fractal examples of trees, lungs and coast lines, is often found in nature because it is effective.

In any area where earthquakes are felt, there will be a very large number of small earthquakes, most of which actually go unnoticed. Next, there is a smaller number of medium sized earthquakes and very few very large earthquakes. This relationship between the different sizes of earthquakes is not just a rough tendency, but also conforms to a power law distribution.

If we make a graph of the pattern of earthquake sizes with the number of earthquakes of a given size on the horizontal axis and the size of earthquakes on the vertical axis, we find the graph forms a nearly straight line. A point on the line near the top left hand part of the line describes a small number of large earthquakes, while at the bottom right end of the line describes a large number of very small earthquakes. Of course, all the intermediate values of earthquake sizes also fit on the graph to make the straight line.

The graphs are usually drawn as a log-log graph, where the logarithm of the numbers are used rather than the numbers themselves, because of the huge range of numbers used. It just makes the graphs much easier to see, without actually changing the mathematical patterns themselves.

While we cannot predict how big the next earthquake will be, we do know that, overall, it will fit within the mathematical pattern of the power law distribution.

The interesting thing is that this mathematical relationship is found in many other seemingly unrelated parts of our world. For example, internet use has been found to fit power law distributions. There is a small number of websites that attract an extremely large number of hits (e.g. Microsoft, Google, eBay). Next there is a medium number of websites with a medium level of hits and finally, literally many millions of sites, like my own, that only attract a few hits.

The number of species and how abundant each one is in a given area of land also fits a power law. There will be a few species that are very abundant, a medium sized number of medium abundance and many species that are not very abundant.

If we put a sand pile on a tray and drop more sand slowly to the top, bit by bit and measure the amount of sand that slides off the edges each time there is a 'sand slide', we find the same power law graph is created (Bak, 1996). There is a small number of slides where large quantities of sand are tipped over the edge of the tray, a medium sized number of slides with medium amounts of sand being tipped over and a large number of slides with a small amount of sand falls. As with the graph of the earthquakes, if we measured any other point between the top and the bottom of the line, the amounts would fit on their place on the straight line. Interestingly, the sand on the sand pile tends to rearrange it self so that every time there is a sand slide, the angle of the slope readjusts itself back to the same slope.

While nobody is organising it, it turns out that the ranking of a person according to their income within their community fits a power law distribution known as Zipf’s law. If we find the income of everyone in the community and rank them so the richest is ranked as 1, the next richest is ranked 2, and the person with the lowest income has the last ranking.

If Zipf’s operates, the ranking, denoted by N, will link to the amount of money possessed by the person of ranking N as 1/N times the amount of the top amount. That means we can write the income and rank of people like this:

Rank                Rank fraction               Money  possessed by person

1                      1/1                               $1,000,000.00
2                      1/2                               $   500,000.00
3                      1/3                               $   333,333.00
4                      1/4                               $   250,000.00
etc.

Obviously it is very unlikely that the people will have the exact amounts as predicted, but the real life examples are actually close enough that the predicted values to be valid. The very same Zipf law fits the populations of cities.

Below is the chart for the ten top cities in the USA in 2005. New York is ranked 1st down to San Jose ranked as 10^th. The darker line shows the actual populations while the lighter line describes the population as calculated by the Zipf formula. As you see the lines are not exactly the same, but are eerily close. What unknown force is operating so that millions of people making their own decisions about where they live, each with their own complex reasons, somehow conform to the mathematical pattern of Zipf’s Law.
George Zipf was a linguist at Harvard University, who first found the mathematical connection in relation to words found in text. From a large piece of text the number of times each word is found is recorded. Generally the most common word found is “the”, followed by “of”, then “and”. The number of times any particular word appears follows the same pattern as for city populations and charts a similar graph.

Vilfredo Pareto developed another power law distribution, now known as the Pareto distribution. A simplified rule of thumb based on the distribution is known as the Pareto principle states that 80% of customer I a business generate 20% of the business and 20% of the customers generate 80% of the business. File size distributions on the internet, clusters of Bose-Einstein condensates near absolute zero, the size of oil fields, meteorites and areas burnt in forest fires are all approximately at least exhibiting Pareto’s Law. The 80/20 rule is also found in areas such as criminal offending. 80% of offending is committed by 20% of criminals and 80% of criminal are only responsible for 20% of offending.

When looking generally at power law distributions, we also find that the time spent in traffic jams, the bank accounts of people, the number of people killed in wars and the number of sexual partners a person has in their lifetime and a host of other seemingly unrelated patterns in nature follow the mathematics of power law distributions.

There are also power law distributions which occur over time, such as in a dripping tap or even our heartbeat. Careful measurement shows our heartbeats are not regular as we might assume, but the slight variations from a strictly regular beat follow power law distributions, but those formed over time are called 1/f noise.

There is a link between power law distributions and fractals. If we look at a fractal tree structure, we find a very small number of very thick branches coming out from the one trunk. Those branches divide to a larger number of smaller branches right down to a very large number of twigs. An ideal mathematical fractal reduces by the same proportion at each new level of magnification. Ideal mathematical power law distributions also reduce by constant proportions through each level of magnification in the same way. 

The river system discussed above demonstrates power law distributions in the number of rivers of each size. The river course is determined by the geography of the landscape, which also proves to be fractal. Mountains are self similar, so it makes sense that fractal mountains generate fractal rivers.

In power law distributions in the real world the straight-line graph often curves down towards the horizontal axis at the bottom of the graph. This may be related to the fact that natural fractals such as a fern hit physical limits to keeping the fractal pattern.

Power law distributions operate within complex systems, which display diversity and freedom. The individual agents that make up the system have diverse natures and the freedom to operate autonomously. Clay Shirky interestingly notes that diversity plus freedom creates inequality and the greater the diversity, the greater the inequality. In other words, the more different people are, the more variety there will be in how they choose to live their lives. The more variety there is the wider the difference and the greater the inequality as noted by the income gap whatever other measure of wellbeing we choose.

This has interesting implications when we look at societal structures. We often bewail the problems caused by inequalities between people in terms of political power, income, access to health care, education, etc. The mathematics of power laws would suggest, however that this inequality is inherent in human complex systems. Not only that, but the more we become diverse and able to act as free agents, the greater the degree of inequality will grow.

The inequality is produced because of positive feedback loops forming in the population. If everyone in the society starts out more or less even, those with even a very modestly larger amount of money, or natural skill, or makes a slightly greater effort will be more likely to succeed, further increasing their advantage. Over time this extremely small advantage can grow into a huge increase in inequality. An equal system with diversity will naturally tend towards greater diversity and inequality.

This supports the old adage that money makes money and the rich grow richer, and the poor grow poorer. The longer the interactions continue and the more people who join in, the more striking will be the difference between rich and poor. This also links to the principle of Chaos Theory, that such systems are very dependent on initial conditions. A small advantage at the beginning is far more likely to result in a high ranking than a small advantage later on, when other agents have already gained significant advantages.

You may remember Beta video recorders. Early on, there was a battle between Beta and VHS. Once VHS appeared to be gaining ascendancy, even if only by a small amount, more and more people bought VHS and fewer and fewer people bought Beta. Even though Beta is generally accepted as the best technologically, VHS gained a monopoly on the market. The rise of Microsoft followed the same dynamics. The more it became accepted as the leader, the more it consolidated its position. Competitors such as Linux, though often accepted as more stable and available free still struggles to gain acceptability.

Communism had the truly noble intention of creating an equal society. Unfortunately it never worked. By forcing equality, when human diversity naturally creates inequality, problems emerged. Almost always, the dreamed of equality was taken over by a tyrannical leader taking autocratic control. Do power law distributions have an influence pulling the dynamics back to inequality? Do power laws have the ability to change events in order to maintain their structure like a complex adaptive system?

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