Scaling Law
Tao Ruyi
What is Scaling law
Scientists always expect to summarize the unified laws of the complex world. Physicists have come a long way in search of basic laws. As early as the 17th century, Kepler summarized the three major laws of planetary motion from Tycho’s astronomical observational data. However, for the complexity science, everything is just beginning.
The subject of complexity science is complex systems, and there is currently no perfect unified theory that encompasses all complex systems. However, there exists a theory with such potential, whether it is the richly diverse life system, cities with various forms around the world, or seemingly random economic systems, they all have a unified form of characterization —- this is the scaling law.
The scaling law describes the relative growth relationship between two properties in a system. It can be described with the formula: Y = aX^α. Where X usually represents variables of the system’s size, such as species weight, company employee count, city population, etc.; Y can represent other quantifiable indicators in the system, such as species lifespan, company lifespan, city innovation ability, etc.; α is called the scaling exponent and is the key determinant of the system’s characteristics.
For instance, in the biological system, when we take an organism’s mass as the size variable X and its body length as Y, the fitted scaling exponent α value is 1/3 because volume is the cube of length. Because of this relationship, when an organism’s weight increases from 1 to 2, its length only increases from 1 to 1.26, not reaching 2. This explains why there are no gigantic creatures like Godzilla in the world—they would be crushed by their own weight before they could grow that tall.
Another example is the relationship between animal weight and metabolic rate. Metabolic rate can be seen as the speed of various life activities in a biological body. It is found that an animal’s weight and its metabolic rate satisfy a 3/4 power relationship: when an animal’s weight increases from 1 to 2, its metabolic rate only increases from 1 to 1.68, not reaching 2. This is the “Kleiber’s Law” published by Swiss biologist Max Kleiber in his work in 1930. According to his observational data, this magical rule applies to species weights spanning nearly 30 orders of magnitude, from mammals and reptiles to cells and even the level of mitochondria. Considering that life is known for its diversity, this discovery is equivalent to Kepler’s laws in the field of biology.
Another typical example is urban systems. Although every city has very distinct geographical features, cultural characteristics, population counts, and more, we find that the attributes and sizes of different cities all have very significant quantitative rules. For example, the relationship between city size and city innovation ability follows a power law with a power index of 1.15. Here, innovation includes wages, the number of patents, GDP, etc. This means that when the city size increases from 1 to 2, the city’s output increases from 1 to 2.22, which is a very significant scale effect. However, as with any coin, there are two sides. The relationship between city size and the city’s negative effects also follows a power law of around 1.15, including crime rates and environmental pollution. This means that the dark side of urban development will also expand at a faster rate as city size increases.
In addition to the biological and urban systems mentioned above, many other complex systems exhibit similar patterns, such as the relationship between the size of countries and their GDP, innovation capabilities, and the relationship between the size of companies and their profits, revenues, and debts.
For the formula Y = aX^α, when the power index α = 1, it means Y and X are scaling in proportion, and Y increases linearly with X. A more common situation is α ≠ 1, in which Y and X are not proportionally scaling; we call this phenomenon the scaling law. The difference between α being greater than 1 or less than 1 will result in fundamentally different system properties.