Alternative Explanations for Power Laws in Advanced Technology
We have discussed how a simple preferential attachment mechanism produces rich-get-richer dynamics that are accurately modeled over time as a power law distribution. But the distribution appears in many contexts, some of which cannot be explained by this mechanism. For example, the paper I will be discussing finds that the computational power of the largest 500 supercomputers follows a power law distribution, and argues that there is no preferential attachment mechanism to account for this.
Before positing its preferred explanation, it addresses the following result. Randomly spaced samples of an exponentially increasing value form a power law distribution. The samples must be randomly spaced in the sense that at each time step, they are either measured (concluding the experiment) or not (continuing to the next time step). This produces an inverse exponentially distribution of samples over an exponential process, giving the power law. The author points out that the power of supercomputers in general increases exponentially according to Moore’s law, but then claims that the random sample premise is not fulfilled, since we are considering a fixed list of 500 computers.
However, I do not think that this explanation can be so easily dismissed. The critical idea is that we are not sampling the exponentially growing function in one fell, deterministic swoop by recording 500 existing computers. Instead, the time that the computer was built is what actually determines the point along Moore’s curve, and therefore the computational power, of the supercomputer. It is much easier to suppose that these data are essentially randomly sampled, as various organizations decide to invest in creating a new top of the line supercomputer, according to plausibly arbitrary and unrelated timelines.
But nevertheless, the author presents a new idea for the formation of the power law, with a closer tie-in to the study of networks. It uses the first half of the previous argument: that computational power increases exponentially over time according to Moore’s Law. The second exponential, however, is the diffusion of technology. In any suitably uniform graph, diffusion through a graph starting at a certain point behaves exponentially if the graph is unbounded, and logistically (therefore approximately exponentially at the beginning) if the graph is bounded. This is because the branching factor at each node multiplies the affected population by an approximately consistent factor in each time interval. So the diffusion of technological methods causes the normalization or spread of advances, in a way that interacts with the exponentially advancing technological frontier, to produce a power law.
I suspect that this explanation is not altogether unrelated to preferential attachment, nor that the two explanations presented here are unrelated to each other. One thing that we can be sure of is that power laws are extremely robust, arising almost invariably when dealing with exponential, unbounded, and/or optimizing processes, and explainable by a whole collection of related formal justifications.
The original paper can be accessed here: http://martinhilbert.net/Powerlaw_ProgressDiffusion_Hilbert.pdf