Complexity_ A Guided Tour - Melanie Mitchell [129]
The Kleiber scaling law is valid but the metabolic scaling theory is wrong. Others have argued that metabolic scaling theory is oversimplified, that life is too complex and varied to be covered by one overreaching theory, and that positing fractal structure is by no means the only way to explain the observed power-law distributions. One ecologist put it this way: “The more detail that one knows about the particular physiology involved, the less plausible these explanations become.” Another warned, “It’s nice when things are simple, but the real world isn’t always so.” Finally, there have been arguments that the mathematics in metabolic scaling theory is incorrect. The authors of metabolic scaling theory have vehemently disagreed with these critiques and in some cases have pointed out what they believed to be fundamental mistakes in the critic’s mathematics.
The authors of metabolic scaling theory have strongly stood by their work and expressed frustration about criticisms of details. As West said, “Part of me doesn’t want to be cowered by these little dogs nipping at our heels.” However, the group also recognizes that a deluge of such criticisms is a good sign—whatever they end up believing, a very large number of people have sat up and taken notice of metabolic scaling theory. And of course, as I have mentioned, skepticism is one of the most important jobs of scientists, and the more prominent the theory and the more ambitious its claims are, the more skepticism is warranted.
The arguments will not end soon; after all, Newton’s theory of gravity was not widely accepted for more than sixty years after it first appeared, and many other of the most important scientific advances have faced similar fates. The main conclusion we can reach is that metabolic scaling theory is an exceptionally interesting idea with a huge scope and some experimental support. As ecologist Helene Müller-Landau predicts: “I suspect that West, Enquist et al. will continue repeating their central arguments and others will continue repeating the same central critiques, for years to come, until the weight of evidence finally leads one way or the other to win out.”
The Unresolved Mystery of Power Laws
We have seen a lot of power laws in this and the previous chapters. In addition to these, power-law distributions have been identified for the size of cities, people’s incomes, earthquakes, variability in heart rate, forest fires, and stock-market volatility, to name just a few phenomena.
As I described in chapter 15, scientists typically assume that most natural phenomena are distributed according to the bell curve or normal distribution. However, power laws are being discovered in such a great number and variety of phenomena that some scientists are calling them “more normal than ‘normal.’” In the words of mathematician Walter Willinger and his colleagues: “The presence of [power-law] distributions in data obtained from complex natural or engineered systems should be considered the norm rather than the exception.”
Scientists have a pretty good handle on what gives rise to bell curve distributions in nature, but power laws are something of a mystery. As we have seen, there are many different explanations for the power laws observed in nature (e.g., preferential attachment, fractal structure, self-organized criticality, highly optimized tolerance, among others), and little agreement on which observed power laws are caused by which mechanisms.
In the early 1930s, a Harvard professor of linguistics, George Kingsley Zipf, published a book in which he included an interesting property of language. First take any large text such as a novel or a newspaper, and list each word in the order of how many times it appears. For example, here is a partial list of words and frequencies from Shakespeare’s “To be or not to be” monologue from the play Hamlet:
Putting this list in order of decreasing frequencies, we can assign a rank of 1 to the most frequent word (here, “the”), a rank of 2 to the second most frequent word, and so on. Some words are tied for