Scale

Did you know that all animals (from the smallest shrew to the largest blue whale) have about 1.5 billion total heartbeats over the course of their lifetime?  We know that rabbits live approximately 3 years, while elephants whose average lifespan is around 80 years.  How can a rabbit and an elephant have the same number of heartbeats during their vastly different lifespans?  The answer is that rabbits have a much higher resting heart rate compared to elephants.  The same is true for most animals (the smaller the animal, the faster the resting heart rate), and there is almost a fairly predictable decrease in heart rate as the body size of an animal increases.  

As it turns out, this mathematical relationship between heart rate and body mass is not unique.  For example, the Swiss biologist Max Kleiber found in the 1930's that an animal's basal metabolic rate is also mathematically related to body mass.  Specifically, metabolic rate scales to the 3/4 power of an animal's body mass, which is now known as Kleiber's Law.  For example, a cat has 100 times the body mass of a mouse, but uses only 31.6 times the same energy as a mouse (the figure below is from an review of the book Scale by Geoffrey West that appeared in the Wall Street Journal).  

 

If you are mathematically savvy, you will note that the scale for both the x-axis (body mass, in kg) and y-axis (metabolic rate, in watts) are logarithmic!  What is remarkable is that all of these different animals fall almost exactly on this line!  The slope of the line is 3/4, which is the exponent in the equation describing the relationship between body mass and metabolic rate.

These relationships between body size and various physiologic variables are known as allometric laws.  More broadly, these relationships have been observed throughout nature and even in disciplines outside of biology.  Geoffrey West (mentioned above) is a theoretical physicist who is now studying complexity theory at the Santa Fe Institute.  West and his team have described a number of what are now called nonlinear scaling relationships (or scaling laws) when studying the characteristics of cities and even organizations!  

Certain characteristics of modern cities, supply chain networks, and transportation can be described by these scaling laws.  So, for example, the number of gas stations per capita, the number of water pipes, or even the average income can all be described by these nonlinear scaling relationships.  Similarly, the average lifespan of a corporation, as well as its market share, number of policies and regulations, and profits are also related to its size.  West gave a TED talk called "The Surprising Mathematics of Cities and Corporations" that is very interesting.

My argument here is that even though our world is incredibly complex, there is a surprisingly vast amount of order in our world too.  As leaders, we should take some time to understand the relatively new fields of complexity theory, chaos, and network science.  As one of my favorite leaders, Winston Churchill, once said, "Out of intense complexities, intense simplicities emerge."