# Is there any relation between Power Law and Negative Binomial distribution?

In a social experiment that I was conducting, I was trying to count the number of people each user contacted in a period of 10 days. The population size was 100 for the experiment. Based on the values that I calculated, I fit a negative binomial distribution to the data (the Q-Q plot is given below).

Conventional wisdom says that most networks amongst humans follow a power law distribution. I am guessing that my population size is too small to make a full conclusion about anything but is there any kind of relation between a negative binomial distribution and a power law distribution? I am asking this because I read a few days back that Normal Distribution and Gamma distribution (whose discrete analogue is the negative binomial) have a special role in that many other distributions can be derived from the Gamma distribution. I am wondering if this is true even with the power law distribution. I am a beginner in statistics so kindly point me in the right direction if I am out of track.

• Might you be able to remember where you 'read a few days back'? – onestop Dec 1 '10 at 20:48
• @onestop: Sorry. I read that in some research paper but I found a quick link on wiki http://en.wikipedia.org/wiki/Generalized_gamma_distribution for a generalized gamma distribution. Please feel free to correct me though. – Legend Dec 1 '10 at 21:04
• This probability plot strongly suggests a mixture of two distributions, one located between 5 and 60 and the other between 50 and 150+. This casts doubt on any simpler characterization, like negative binomial. – whuber Dec 1 '10 at 21:27
• @whuber: That is interesting. Would you have any suggestions on how to proceed? Do I start looking for ways to separate the distributions? – Legend Dec 2 '10 at 1:42
• Not necessarily. It depends on why you're doing this fitting. At this stage it seems you are exploring the data. Let the data behavior guide you. Can you find any possible cause or explanation for a mixture? Would there be a meaningful difference between values in the 0-60 range versus values in the 40-150+ range? In short, what can you learn about your social experiment from this? – whuber Dec 2 '10 at 13:50