We recently learned that the distribution of links on the Web is best described by a power law as opposed to a Gaussian distribution as we might have guessed at first. We then explained this observation by constructing a model which has the feature that the rich get richer.
A paper by Luis Bettenourt et all titled “Growth, innovation, scaling, and the pace of life in cities” shows empirical evidence that many properties of cities such as new patents, number of inventors, total wages, GDP, crimes, new AIDS cases among others are power law functions of population size. Based of these findings the authors of the paper proceed to make a model that described the population of a city as a function of size. The model predicts  a number of things such as that major innovations or adaptations must occur at an accelerated rate to avoid the collapse of the city in question.
It is interesting to me that the authors of this paper constructed the model for the population of the city, and used it to make predictions about the city, based on the assumption that these power laws always occur but did not give an explanation or even proposed a model that would show why these power laws arise. They do say at the end of the paper that there is much to be explored in generalizing the empirical observations but they do not make any suggestions as to how to begin thinking about the possible generalizations.
The intuition behind the model we used in class was that new websites tend to have links to websites that are already popular. In this way it is not that as difficult as we might think for websites to become extremely popular since according to the model they are becoming more popular because they are already popular. We could clearly apply the same model for the growth of cities. Bigger cities grow faster! Also, the Rich-Get_Richer model could also be applied to many of the properties of cities mentioned above. For example, the more new cases of AIDS we have today, the more new cases of AIDS we will have tomorrow.
The fact that the Rich-Get-Richer model could be applied to both the population size and the properties of cities does not explain the fact that these properties are power laws of population size but it seems like good place to start in trying to come up with a model that does explain it. It would be a good research problem to come up with it.
The paper mentioned can be found at http://www.pnas.org/cgi/reprint/0610172104v1.