Sunday, January 15, 2012

The Power Law and Its Importance

Since I am starting the first topic in this blog, I thought it would be apt to begin by saying something about the title of the blog - "Power Law".

As was introduced in class, the power law looks quite simple - P(k) = k-a
What is so special about this equation?

One look at it and people would just ignore it as some normal equation of no much significance. But if you start researching and thinking about it, you would find that human behaviour is mostly governed by this equation. If you start looking around, you would find the presence of this equation in almost every daily activity of people.

According to this law, 20% of the population is 80% popular. Why does this happen? Let us take a simple scenario – there are 10 courses and you are asked to select 5 out of them. If you make the assumption that the students are making the decision of their own, independent of others, then every course has equal probability of getting selected. But in reality, this is not the case.

People’s choices tend to be affected by the choices of others. Suppose X is the first one to choose his courses – he selects 5 courses unaffected by others choices. Now when Y comes to choose his courses, there is high probability that he would select some of the courses that X had selected. Now when Z comes to choose his courses, there is higher probability that he would select some of the courses which were chosen by both X and Y. In this way the process will continue. Thus, a particular course might have higher number of students while another course might have very few takers. In this way one course becomes popular while another falls below average popularity.

This happens mostly due to the tendency of human nature to live in clusters. People mostly follow the crowd. The power law is also evident in the fact that only 20% of the population holds 80% of the wealth. This is the reason that lots of people live below the poverty line.

Even Zipf’s Law states that the word frequency also follows the power law pattern, with a few words occurring with a high frequency (I, of, the), a moderate number of common words (book, cup, cat) and a huge number of low frequency words (esculent, acquiesce).

So it is quite interesting to note that such a small equation, that looks so trivial, can be of so much importance. 

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