4. Heterogeneity

Polyna Urn

The Polyna Urn scheme is a thought experiment. Think of a cannister of balls of various colors. At each timestep we will pull out a ball and replace this ball, as well as an additional ball of the same color. For example, if we pulled out a red ball, we would replace this red ball and then add another red ball. Notice how this affects the probability that we will choose the red ball the next time. The red ball now has slightly better odds. If we iterate this process many times, a heterogeneity between the balls that are initially picked begins to increase. The balls of the most frequent color will amplify and continue to result in more balls of this color.

Economics of Celebrities

This is a measurement of the economics of celebrity. Socail media analysis gives us access to an interesting dataset, which would have thrilled early social network scientists. We have information on what everybody likes. From this we can create distributions, clustering analysis, topological dynamic analysis to understand the nontrivial properties of the resultant network.

Matthew Effect

The Matthew Effect, or Cumulative Advantage are two names of a similar pheonomeon as the Polyna urn scheme. In each case the rich get richer, and this loop continues to permeate.

Power Law, Zipf’s Law, Distribution

If we are to measure the distribution of this frequency, of the balls, of the distribution of city sizes, of the result of these kinds of cumulative advantage. We see that the cumulative advantage is related to the future frequencies of selections. On a log-log plot the power law obeys a particular straight line. This means that the logarithmic properties hold across the scale of the network.