We Need to Talk About Scaling

Hello readers!  I’m back in blog mode after a long hiatus, having been completely snowed under by teaching two classes during fall term, among other responsibilities. 

Thanks to everyone who commented on the “Cognitive Computing” post.  This is definitely a topic that warrants further attention on this blog.  In particular, at some point I want to write about Ray Kurzweil’s “singularity” ideas and how seriously we need to take them. 

In partial preparation for that future post, I bring you today’s post:

We Need To Talk About Scaling

The same small set of probability distributions describe the great majority of observed patterns… . These distributions reveal the contours of nature.  We must understand why these distributions are so common and what they tell us… .”
—Steve Frank, The Common Patterns of Nature (J. Evol. Bio., 22(8), 2009).

If you’re interested in understanding complexity, then we need to talk about scaling.

Why?

Because scaling properties—how one variable changes as another is varied— can provide a lot of insight into the mechanisms underlying these properties in a complex system.  The mathematical forms describing scaling properties often can give fundamental clues to how complex systems work, and, in fact, provide a rigorous vocabulary for talking about such systems.  Anyone who wants to understand complexity needs to know about the important mathematical forms that scaling relationships typically exhibit.

Recently, the notion of “power-law scaling” has received a huge amount of attention in the complex systems literature.  Many people have written about the importance of power-law scaling for our understanding of financial crises, electrical blackouts, ecosystem collapse, earthquake prediction, and myriad other phenomena.  One paper called power laws “more normal than ‘normal’”, referring to their importance relative to normal (or “bell-curve”) distributions.  A recent New York Times Bestseller book, The Black Swan by Nassim Nicholas Taleb, is a sort of paean to power laws, and blames many of our country’s economic troubles on ignorance of their implications (and their close link to “fractal geometry”.)

However, even more recently, power-laws have been taking a lot of flak.   There has been a torrent of papers attacking the hype surrounding power laws.   My friend Cosma Shalizi, a very smart scientist, statistician, professor, and prolific blogger (how does he do it all??), has been a prominent voice on this topic (a few years ago he wrote that “our tendency to hallucinate power laws is a disgrace”). 

What to make of all this?  My main motivation for starting this blog was to try to sort out questions like this in my own mind.  For me, writing about scientific controversies—trying to explain the issues to others—is the only way for me to clarify the issues in my own mind. 

In my next several blog posts I want to talk about scaling, especially about the very recent controversies surrounding claims of power-law scaling of particular phenomena—examples include risk in financial systems; metabolic rate in plants and animals; and most recently the surprising results of Geoffrey West and colleagues on the scaling of crime rate, income, technological innovation, and other variables in cities. 

All this is going to require some forays into the wild and unruly land of statistics and data analysis. My goal in the next series of posts is to make sense of the following quite important papers in complex systems, which, taken together, form a kind of mini-course on scaling.  Understanding ideas from these papers is essential in one’s education as a complex-systems scientist or informed “consumer” of this field. You don’t necessarily need to read these papers (they’re rather technical); I’ll try to explain the major ideas and results in this blog.

S. Frank, The common patterns of nature. Journal of Evolutionary Biology, 22 (8), pp. 1563–1585, 2009.

M. E. J. Newman, Power laws, Pareto distributions, and Zipf’s law.   Contemporary Physics, 46, pp. 323–351, 2005.

M Mitzenmacher, A brief history of generative models for power law and lognormal distributions.  Internet Mathematics, 1, pp. 226–251, 2001.

W. Willinger et al., More ‘normal’ than normal:  Scaling distributions and complex systems. In Proceedings of the 2004 Winter Simulation Conference. IEEE Press , Piscataway, NJ, pp. 130–141, 2004.

A. Clauset C. R. Shalizi, and M. Newman, Power law distributions in empirical data. SIAM Review, 51(4), pp. 661–703, 2009. 

L. A. Bettencourt et al., Growth, innovation, scaling, and the pace of life in cities. Proceedings of the National Academy of Sciences, USA, 104 (17), 7301-7306, 2007.

C. Shalizi, Scaling and hierarchies in urban economies. arXiv:1102.4101v2 

More soon – stay tuned.