Lecture 10: Power laws

The last Tuesday, we learned Power laws distribution. To understand the character of Power laws distribution, we compare it with normal distribution. Normal distribution has a peak and do not has heavy tail. For example, heights of human males follow the normal distribution with a peak around 180 cm. The power laws distribution is a straight line with negative slop in the log-log scale and it has a heavy tail. For example, the city population size is a power laws distribution, and the number of the city which has a huge population is very small, but many small towns. And the equation of power laws distribution is:

ln⁡(p(x))=c-aln(x)

Where p(x) is the probability of observing an item of size x; c and are two constants.
Power laws distribution is everywhere. Work frequency, web hits, book sold, telephone calls received and earthquake magnitude all follow the power distribution. During the analysis the power laws distribution, the most common and not very accurate method is do the linear fit line of the whole range in the log-log scale to get the value of a and c. But sometimes, because of the barrier condition or the noise in the heavy tail, this method will cause a big error. There are many way to solve this problem.

First solution is logarithmic binning. Bin data into exponentially wider bins and normalized by the width of the bin. This method can get a good fit line with information lost. Second solution is cumulative binning. The cumulative probability of a power laws probability distribution is also power laws but with an exponent a-1. No loss of information of this method. Third way to do this is chose a good start point to do the fitness, like the distribution of citations to papers, power laws is evident only in the tail. The fourth way is maximum likehood fitting, it is the best way. But to use this method you have to make sure that you have a power laws distribution.

A power laws looks the same no matter what scale we look at it. We call this is scale free.

At the end of our class, to understand the power laws distribution further, we learn a lot of example with power laws distribution and other distribution, like cumulative distribution, power-law with an exponential cutoff, zipf’s law and pareto principle.