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Universality And Scaling
The work of Plerou et al. [1999], [2002] established the existence of universality and scaling in financial data. Their work has been extended by several authors, including Zumbach [2004], Li et al., [2011] and Kertesz and Eisler [2012]. It is the existence of universality and, particularly, scaling that we show refutes market-cap breakpoints being—in general—empirically based.
To begin our discussion of scaling, we note that many empirical quantities cluster around a typical value: the speeds of cars on a highway, the weights of apples in a store, air pressure, sea level, the temperature in New York at noon on July 4. All of these things vary somewhat, but their distributions place a negligible amount of probability far from the typical value, making the typical value representative of most observations. For instance, it is a useful statement to say that an adult male American is about 71 inches tall, because few deviate very far from this size. Even the largest deviations, which are exceptionally rare, are still only about a factor of 2 from the mean in either direction; hence, the distribution can be well characterized by stating just its mean and standard deviation.
Not all distributions fit this pattern, however; while those that do not are often considered problematic or defective, they can be some of the most interesting observations. The fact that they cannot be characterized as simply as other measurements is often a sign of a complex underlying process that merits further study.
Among such distributions, the power law has attracted particular attention over the years for its mathematical properties, which sometimes lead to surprising physical consequences. The power law appears in a diverse range of natural and man-made phenomena. For example, the populations of cities, the intensities of earthquakes and the sizes of power outages are all thought to have power-law distributions. Quantities such as these are not well characterized by their typical or average values. For instance, according to the 2000 U.S. Census, the average population of a city, town or village in the United States is 8,226. This average is not a useful one for most purposes because a significant fraction of the total population lives in cities (New York, Los Angeles, etc.) whose population is larger by several orders of magnitude.
The main property of scaling (or power) laws is their scale invariance. Given a relation f(x) = axk, scaling the argument x by a constant factor c causes only a proportionate scaling of the function itself. That is,
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