“Scale”

“Scale” is a fascinating new book (2017) by a physicist using his mathematical and analytic skills to explore the world of biology. The author, theoretical physicist Geoffrey West, is a distinguished  professor at the Santa Fe Institute, famous for studies of complexity. The central theme of the book is the discovery by West and others that underlying many biological systems are surprising regularities known as “scaling laws” or “power laws.” This means that many features of an organism can be predicted simply by knowing the size of the organism.

After describing how these laws were discovered and why they arise in organisms, he extends them to studies of cities and companies. Admittedly more speculative and less precise, the application to cities and companies of power laws reveal surprising regularities. For example, you can roughly predict the number of gas stations in a city simply by knowing its size and the country it is in.

West points out that in physics it is often useful to model a system at the “zeroth order” of approximation, one step below a first order. By ignoring relatively minor variations, you can discover hidden patterns that are obscured by focusing on the details. The general pattern visible at the coarse resolution suggests predictions that can be tested experimentally. Also, developing an explanation of why the general pattern occurs often suggests testable explanations of why the exceptions do not follow the general pattern.

This is the approach he used when examining biological and social systems, which are orders of magnitude more complex than the “simple” phenomena studied by physicists. It was a surprise to almost everyone that simple power law behavior emerged out of such complex systems.

While the book has some editing lapses, it is accessible to the general reader and is full of fascinating and highly topical information. He clearly explains fractals and how they relate to power laws. His discussion of exponential growth is highly pertinent to what I consider the central problem we face: systems based on exponential growth cannot survive in a finite environment (see my heading “Troubling Stuff”). This  book is a must-read.

Some of West’s explanations of the simple math behind the scaling laws puzzled me, so I had to probe deeper to really understand the concepts. The following rather lengthy discussion supplements the discussions in the book.

You are familiar with Cartesian graphs, where one variable is plotted against another on a rectangular grid. Such graphs are ubiquitous, for example the Dow-Jones Average on the vertical Y-axis plotted against time on the horizontal X-axis, a plot that appears in weekday newspapers.

The value of the Dow is not determined by the date. That is, as time goes on, its value does not increase according to some formula or function in the technical, mathematical sense. If the value of the Dow were a mathematical function of time, a lot of people would be out of business!

By contrast, the distance of the moon from the earth is a mathematical function of time. Careful observation coupled with theory has led to a series of equations that define the distance to the moon. Solving the equations for each interval of time produces a curve that can be plotted on a graph. In this case observation has defined the function, which is then plotted on the graph:

Things get more interesting when you go backward from the plot to the function. To take an example from the book, if you plot the average weight of each species of mammal against its average metabolism, does any kind of formula pop out of the data, or do the data points “scatter” randomly all over the graph?

Don’t try this at home! To plot the entire range of mammals from the smallest (Etruscan pygmy shrew) to the largest (blue whale), you need to choose a scale to plot the weight of the mammals on the X-axis. To distinguish the Etruscan shrew (about 2 grams) from a common shrew (about 8 grams), a good scale would be one millimeter per gram. But if you tried to plot the blue whale (165,000 kilograms) on that same graph, the paper would be about 100 miles long. This kind of graph is called a “linear” plot.

In addition, the data points will be scattered around some kind of curve that might not be easily seen. While you might be able to figure out some function whose curve would be close to that of the plotted values, the significance of the function would be far from obvious. We need some kind of graph that suggests relationships more diredctly.

If instead of plotting the actual values you plot the logarithms of the values, you solve both problems at once. Quoting Wikipedia, the logarithm of a number is the exponent to which another fixed number, the “base” must be raised to produce that number.  Commonly used bases are 2, e (2.71828… ), and 10. So for example, in base 10 (the one we use in counting), the exponent of 10 that produces 100 is 2, so 2 is the logarithm of 100 in base 10. Of course a logarithm that is a simple integer is a special case; the logarithm of 3 in base 10 is 0.477121255…. A graph with both the X and Y axis scaled in logarithms is called a “log-log plot.”

You may have seen a logarithmic plot showing the sizes of things from the smallest possible to the largest possible in a line on a single sheet of paper, marked off in powers of 10. The powers of 10 in such a plot range from the Planck length (1.6 x 10 to the -35th power) to the diameter of the universe (8.8 x 10 to the 26th power), thus ranging over almost 62 powers of ten. Logarithms allow us to see relationships that would otherwise escape notice (but see my post at The Wonderful, and Wonderfully Misleading, Powers of Ten , which delves into the downside of logarithmic plots).

Producing useful graphs is hardly the only neat thing about logarithms. To multiply two numbers, you simply add their logarithms to produce the logarithm of the product. Until the advent of electronic calculators, every scientist and engineer consulted tables of logarithms and used slide rules, which are based on logarithms. I still have a couple of slide rules.

A surprising and extremely useful feature of log-log plots is that it converts simple curves into straight lines. This happens when you plot functions where one variable is some power of the other variable. For example, here is a linear plot of the functions Y=X¹ (blue), Y=X² ( red), and Y=X³ (green):

Linear plots of Y = X to the powers 1, 2 and 3.

If you plot the same functions on a plot where the each axis is marked off in logarithms (a log-log plot), the curves magically become straight lines:

Log-log plots of Y = X to the powers 1, 2 and 3.

This happens for any power of x. The only thing that changes when you change the exponent of x is the tilt of the line (its “slope”). Note that x¹ is just x, so the plot of that function is the line where X = Y, which is the same line on both graphs.

The slope of the line is the y value divided by the x value; so the slope of x=y is 1/1 = 1; the slope of y=x² is 2; and the slope of y= x³ is three. Similarly, the slope of y= x to the 1/2 power is .5 and y= x to the 1/3 power is .333…

Log-log plot of Y = X to powers 1, 1/2 and 1/3.

Now the import of all this is that when you plot various properties of a system (mammals, cities and companies are the ones discussed in the book) and compare them on log-log plots, they track remarkably close to a straight line. For organisms, the slopes of these lines are often multiples of 1/4th: 3/4, 1/2, 1/4, 1/8th. For cities, the values .85 and 1.15 come up frequently.

This is what is meant by “power laws.” The major theme of the book is why these systems exhibit these power law behaviors. West uses other terms to describe power laws: scaling laws, allometric data, self-similarity, fractal. They all refer to the cases where two sets of data, when plotted on a log-log graph, fall roughly on a straight line. The scale of the power law is the slope of the line, which is also the power of X.

EXPONENTIAL GROWTH

Exponential functions are quite different from the power functions we have been discussing (where the Y value equals some power of X). In an exponential function, Y is equal to a constant raised to the power X. At small values, the two kinds of functions yield similar values, but exponentials suddenly take off, as shown in this plot:

Parabola (Y – x squared) in red; exponential function (Y – constant raised to the power X) in blue

Again, you can convert an exponential function into a straight line, but this time you use a “log-lin” plot, where the X axis is linear and the Y axis is logarithmic. The slope of the line is the growth rate. Compound interest is the most familiar example of an exponential function.  West discusses the implications of exponential growth (as do I in my posts under the category “Troubling Stuff.”)

ZIPF’S LAW

In discussing the application of power laws to cities and corporations, West discusses Zipf’s Law. Over a surprising range of texts, the words in English follow Zipf’s Law closely. It states that the second most common word occurs about one-half as often as the most common (the), the third one-third as often, the fourth one-fourth as often, etc. This also applies to cities: the second largest (Los Angeles) is about one-half as large as New York, the third largest (Houston) about one-third as large, the fourth (Chicago) about one-fourth as large etc. Many phenomena roughly follow Zipf’s Law.

The function that describes this kind of behavior is a power function, but of the form Y = 1/X, where the power of X is negative. This curve is a hyperbola. I show a linear plot above and a log-log plot below:

Y = 1/X, a hyperbola
Log-log plot of a hyperbola

In these two plots I have shown all four sectors of the function (positive and negative X and Y). All the earlier graphs just show the upper right sector, where X and Y are both positive. In the book, you only see the lower right sector of the graph just above, where X is positive and Y is negative. The slope of the line is negative, as is the power of X.

All the above will become much clearer as you read the book, which I dearly hope you will.