http://www-personal.umich.edu/~mejn/courses/2006/cmplxsys899/powerlaws.pdf

M.E.J. Newman from the University of Michigan at Ann Arbor discusses power laws in nature and the reasons why some distributions follow power laws. He gives examples of some power laws such as the frequency of words in English text, number of telephone calls in a day, diameter of moon craters, frequency of family names, and populations of cities. Other distributions are exponential laws, such as lengths of relationships between couples and number of email contacts.
He also uses a quantitative analysis in looking at these power laws. One reason why distributions follow power laws with exponents greater than 1 is that they can be normalized. For a distribution p(k), integrating p(k) from kmin to infinity equals 1 for some kmin, below which the power law does not hold. However, this integral would diverge if p(k) fell off as 1/k, and would converge for exponents greater than 1. Newman also discusses how to find what is likely to be the largest value of k in a set of measurements.      Another reason why p(k) follows a power law is the “Yule process,” or the rich-get-richer mechanism that professor Kleinberg discussed in class.
The fact that so many large samples of measurements follow power laws makes me wonder about what all of these samples have in common. Why is web page popularity a power law and why is length of relationships between couples an exponential law? The “Yule process” provides some justification, but it is not intuitive. Intuitively, it seems like a rich-get-richer mechanism could lead to exponential laws as well. I could not find any better explanation or proof of the origins of power laws, but several articles explained that this subject is currently debated in the scientific community. I would also be interested to see the differential equations look at large samples of measurements to see what support it provides for power laws. From Newman’s paper, it seems like there is a lot of interest in these systems and a lot to be discovered.