Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker These exponent 1 and exponent 3 descriptions of the sizes of rocks become rather similar if we graph them in a certain way. The idea is to use log-log graph paper. A different type of power law arises with natural sounds such as the babbling of a brook, the whispering of the wind in the trees, the crashing of the ocean, or the splashing of rain. We can view these sounds as being made up of different-sized pieces in the sense that they’re combinations of waves of different frequencies or cycles per second. The deep sounds have low frequencies and the shrill sounds have high wavelengths. Here we view the “size” of a wave as being its frequency, and we view its intensity as the amplitude of the sound’s contribution. Now it so happens that most natural sounds obey the power law. Sound component amplitude = 1 / frequency Fractals can be characterized as systems that obey power laws with fractional exponents. The Koch curve we can measure the size of a piece by its linear length, and the intensity of that size by the number of components of that size. It turns out that we get levels like this, but with the levels overlaid upon each other. 1 piece of length 1, 4 pieces of volume 1/3, 16 pieces of volume 1/9 ... and 4 L pieces of volume 1 / 3 L Now it turns out that if D = log(4)/log(3), then 3 D = 4. The nineteenth century Italian economist Vilfredo Pareto observed a kind of power law in income distribution, with people’s income being proportional to the inverse of some power of their rank. He came up with a simple kind of model to suggest how this could come about. In the more general case, suppose that each employee has K subordinates, and each subordinate earns 1/L as much as his or her immediate boss. If we generate the organizational chart down to N levels, viewing level 0 as being the single supreme boss at the top, then the people at the Nth level will be getting a salary S of (1/L) N = (1/L N ) times the supreme boss’s salary. The number of people at the Nth level will be K N . Their rank R will also be on the order of K N , because if we sum up the rows of employees ranking higher than the Nth level people, we are looking at the sum with i ranging from 0 to N-1 of K i , which is (K N - 1) / K-1, a quantity which is roughly K N-1 at the beginning of the row, but K N at the end of the row. Now define D to be (log L) / (log K), then K D is L and if we say R is K N , then R D = (K N ) D = L N . Then S = 1/L N = 1/(K N ) D ~ 1 / R D or, quite simply, S ~ 1 / R D . Inverse Power Law Examples Inverse power laws are all over the place; here’s a few more examples. p. 88
Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker Category Job Company Author Movie Artist Web page City Person Success Salary Net Income Average book advance Gross receipts on opening weekend Average price paid for a work Number of links to this page Population Number of close acquaintances In each case we can draw a graph of how the success value drops off as we move from the higher ranking towards the lower ranking individuals in a class. As it turns out, these graphs tend to drop off more steeply than one might expect. They obey distribution rules known as power laws. Zipf’s law is an especially simple power law. In the more general case, the rank may be raised to some power D. So we expect for each individual in the sample, Success measure of an individual I = C / (Success rank of the individual I) D Now in many situations, it turns out not to be practical to rank the entire sample in order of success. It can be more useful to express a power law in terms of the probability of finding an individual of a certain level. In terms of salaries, for instance, we might take a large sample of people, determine their salaries, and determine the probability of finding a person at each salary level. In this case the power law could be written in this form. Success measure S = A / (Probability of finding an object with success measure S) D The two kinds of power laws are roughly equivalent. In order to graph these kinds of laws, scientists often use what’s called logarithmic graph paper. That is, the scales along the two axes can be set to, say, powers of ten instead of measuring uniformly along the axis. If we do this we are in effect plotting log(y) against log(x), which has the effect of turning power-law graphs into downward-sloping straight lines which are more or less steep depending on the size of D. 7 7 Suppose I take my logarithms to base ten, and u and v are, respectively the log base ten of y and x. This means that x = 10 u and y = 10 v . In this case, a power law of the form y = C/x D has the form y = x -D because a negative exponent means division. We can convert this step by step to a linear equation giving v as a function of u. 10 v = (C * 10 u ) -D log(10 v ) = log((C * 10 u ) -D ) because the logs of equals are still equal. log(10 v ) = log(C) + -D * log((10 u )) because logarithms turn products into sums and exponents into products. v = log(C) - D*u because the log base ten of 10 a is always just a. v = B - D*u, writing B for the constant log(C). p. 89
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