Complexity_ A Guided Tour - Melanie Mitchell [126]
Enter Geoffrey West, who fit the bill perfectly. West, a theoretical physicist then working at Los Alamos National Laboratory, had the ideal mathematical skills to address the scaling problem. Not only had he already worked on the topic of scaling, albeit in the domain of quantum physics, but he himself had been mulling over the biological scaling problem as well, without knowing very much about biology. Brown and Enquist encountered West at the Santa Fe Institute in the mid-1990s, and the three began to meet weekly at the institute to forge a collaboration. I remember seeing them there once a week, in a glass-walled conference room, talking intently while someone (usually Geoffrey) was scrawling reams of complex equations on the white board. (Brian Enquist later described the group’s math results as “pyrotechnics.”) I knew only vaguely what they were up to. But later, when I first heard Geoffrey West give a lecture on their theory, I was awed by its elegance and scope. It seemed to me that this work was at the apex of what the field of complex systems had accomplished.
Brown, Enquist, and West had developed a theory that not only explained Kleiber’s law and other observed biological scaling relationships but also predicted a number of new scaling relationships in living systems. Many of these have since been supported by data. The theory, called metabolic scaling theory (or simply metabolic theory), combines biology and physics in equal parts, and has ignited both fields with equal parts excitement and controversy.
Power Laws and Fractals
Metabolic scaling theory answers two questions: (1) why metabolic scaling follows a power law at all; and (2) why it follows the particular power law with exponent 3/4. Before I describe how it answers these questions, I need to take a brief diversion to describe the relationship between power laws and fractals.
Remember the Koch curve and our discussion of fractals from chapter 7? If so, you might recall the notion of “fractal dimension.” We saw that in the Koch curve, at each level the line segments were one-third the length of the previous level, and the structure at each level was made up of four copies of the structure at the previous level. In analogy with the traditional definition of dimension, we defined the fractal dimension of the Koch curve this way: 3dimension = 4, which yields dimension = 1.26. More generally, if each level is scaled by a factor of x from the previous level and is made up of N copies of the previous level, then xdimension = N. Now, after having read chapter 15, you can recognize that this is a power law, with dimension as the exponent. This illustrates the intimate relationship between power laws and fractals. Power law distributions, as we saw in chapter 15, figure 15.6, are fractals—they are self-similar at all scales of magnification, and a power-law’s exponent gives the dimension of the corresponding fractal (cf. chapter 7), where the dimension quantifies precisely how the distribution’s self-similarity scales with level of magnification. Thus one could say, for example, that the degree distributions of the Web has a fractal structure, since it is self-similar. Similarly one could say that a fractal like the Koch curve gives rise to a power-law—the one that describes precisely how the curve’s self-similarity scales with level of magnification.
The take-home message is that fractal structure is one way to generate a power-law distribution; and if you happen to see that some quantity (such as metabolic rate) follows a power-law distribution, then you can hypothesize that there is something about the underlying system that is self-similar or “fractal-like.”
Metabolic Scaling Theory
Since metabolic rate is the rate at which the body’s cells turn fuel into energy, Brown, Enquist, and West reasoned that metabolic rate must be largely determined by how efficiently that fuel is delivered