Tuesday, March 06, 2012

The Mystery of Power-Law Distributions

One criticism of sociology, and the macro social sciences more generally (such as political science, anthropology, and economics), is that there are very few "laws" of social reality. There are, however, some sociological regularities that are as yet not fully explained, and which seem bizarre. The most enduring and puzzling of these are power-law distributions (a well-known special case of this is "Zipf's Law"), which is the fact that  "large" instances of things are extremely rare, while "small" occurrences of things are extremely common (where size can refer to frequency in a population, population size, geographic space, and so on). In practice this means that a handful of words are much more frequent than other words (and most words are rarely used), wealth is concentrated in a small number of people (and most people are poor), there are a handful of really popular songs (and a vast number of unpopular tunes), and so on. Even the sizes of sand particles on a beach follow a power-law distribution: how often have you seen a boulder on a beach?

What might explain the ubiquity of power-law distributions? As far as I can tell, nobody is entirely sure, although we have some good guesses. For example, the sociologist Herbert Simon outlined a theory of preferential growth attachment (also known as the "rich get richer" effect), in which songs that are already fairly popular will become more popular, cities that are already large will become even larger, and words already used widely will become even more widely used. Note that this explanation hinges on a positive feedback effect: the probability that any thing gets "larger" is directly proportional to the current "largeness" of the thing; or, to put it another way, large values get amplified rather than cancelled out (as in a normal distribution).

Power-law distributions have important cultural, statistical, and political implications.

Culturally, there are several implications. First, most cultural constructs  are rarely used and only a handful are common among any group of people. To put it another way, the shared part of culture is likely to be relatively small, while the particular part of culture is vast. Second, frequently used cultural constructs are particularly stable over time; that is, 500 years from the word "the" will still be used, while "sesquipedalian" has a more uncertain future. Third, the stability of a cultural system is derived from the more frequently used cultural constructs, while the dyanmism is among the less frequently used constructs. Fourth, initial conditions are extremely important for the frequency and hence durability of cultural constructs: for instance, small, random fluctuations led to the popularity of "the" in the English language. Finally, following from the previous point, the consequences of initial conditions are highly unpredictable; given small initial changes English speakers today might instead be using the word "tha" or "se" instead of "the." 

Statistically, the presence of power-law distributions is a reminder that classical linear regression (based on the normal distribution) is not always the appropriate fit to a scatter plot of two variables, and that summarizing a distribution as a mean or median can be highly misleading.

Politically, power-law distributions have a unique implication for efforts to deal with wealth inequality: one effective way to alter the distribution of wealth is to remove the positive feedback effects from wealth. The desired distribution of wealth would thus be described by a normal rather than power law function. Importantly, removing the positive feedback effects of wealth would not lead to the removal of inequality, but rather a change in the distribution so that the mean, median, and mode are the same. From this perspective, policies should be in place so that (in principle) a person's change in wealth is independent of their current level of wealth. Such policies might include very high taxes on capital gains, restrictions on the influence of wealth in political decision-making, rules specifying equal monetary amounts from promotions for all occupational levels in a firm, and so on.