The bigger the city, the more opportunities there are? The law behind this is →

Different systems, common laws of scale | Tuchong Creative

Why is it that an elephant weighs 10,000 times more than a mouse, but only needs 1,000 times the amount of medicine as a mouse? Why do big cities have higher salaries and more opportunities than small cities? Many questions related to scale can be answered in the law of scale. Today, I will introduce to you what the law of scale is.

1. What is the law of scale?

Scientists always hope to summarize the unified laws of things in this complex world. Physicists have gone a long way in finding basic laws. As early as the 17th century, Kepler summarized the three laws of planetary motion from Tycho's astronomical observation data. For complex science, everything has just begun.

The object of complex science research is complex systems . There is no perfect unified theory that encompasses all complex systems. However, there is a theory with such potential. Whether it is a rich and diverse life system, a city system with different forms all over the world, or an economic system that seems to fluctuate randomly, there is a unified form of description - the law of scale.

The law of scale describes the relative growth relationship between two properties in a system . For example, the metabolic rate of an organism follows the law of scale relative to its weight. The metabolic rate can be thought of as the speed of various life activities in an organism. If we take the weight of the organism as the variable X and the metabolic rate of the organism as the variable Y, we can find that the change in metabolic rate with weight (roughly) follows the power law relationship of Y=X^3/4. In other words, when the weight of an organism increases from 1 to 2, the metabolic rate does not increase linearly from 1 to 2, but only increases from 1 to 1.68.

This is the "Kleiber's Law" published by Swiss biologist Max Kleiber in his book in 1930. According to his observation data, from mice to elephants, species with a weight range of nearly 7 orders of magnitude all meet this law. It should be noted that organisms have always been famous for their diversity, and this discovery can be called Kepler's Law in the biological field .

Another typical example is the urban system . Although each city has significant differences in geographical characteristics, cultural features, population size, etc., we found that if we measure the city's innovation ability by indicators such as salary, number of patents, and GDP, then the innovation ability of the city and the population size follow a very significant quantitative law, that is, (roughly) a power law relationship of Y=X^1.15. This means that from a small county with a population of hundreds of thousands, to a city with a population of millions, to a super metropolis with a population of tens of millions, as the size of the city's population expands, the city's innovation ability does not grow linearly, but at a faster superlinear rate. However, just as a coin has two sides, the dark side of urban development, including crime rates, environmental pollution, etc., will also expand at a faster rate as the size of the city expands.

In addition to the biological and urban systems just mentioned, for national systems , there is the relationship between indicators such as GDP and innovation capabilities and the size of the country; for corporate systems , there is the relationship between indicators such as company profits, income and debt and the number of employees, etc. This series of phenomena all follow the universal scale law, which can be described by the power law relationship Y=X^α, where α is called the power exponent, which is the key to determining the properties of the system. When the power exponent α=1, it means that Y and X are scaled in the same proportion, and Y grows linearly with X; the more common situation is α≠1, at which time Y and X are not scaled in the same proportion - we call this nonlinear phenomenon the scale law . Whether α is greater than 1 or less than 1 will bring fundamental differences in the properties of the system.

2. The Law of Scale Explains the Ultimate Life and Death of a System

The key to the law of scale is that it introduces a new perspective of "scale" to look at complex systems. Many questions related to scale, which seem to be unrelated, can be answered in the law of scale. For example, why does an elephant weigh 10,000 times as much as a mouse, but only requires 1,000 times the dose of medicine as a mouse? Why do big cities have higher salaries and more opportunities than small cities? The law of scale can give us the answers.

The more critical reason why the scale law is important is that it can reveal the evolutionary direction of complex systems and point directly to the ultimate life and death of the system. We know that all organisms have an upper limit to their growth and eventually die. This is because the growth of organisms follows a function with a power exponent less than 1, while the loss of growth is a linear function, just like a bucket, the speed of inflow and the speed of outflow are not the same. Therefore, there must be a time point when the growth rate of the organism is equal to the loss rate, and this is when the organism stops growing.

We can also use the same method to analyze the growth of urban systems. Cities are very different from biological systems. The growth of cities follows a superlinear function with a power exponent greater than 1. From this perspective, there is actually no upper limit to the growth of cities. However, along with the superlinear growth of cities, there is also a dark side of superlinear growth. While cities are expanding rapidly, negative effects such as the equally rapid increase in crime rates and uncontrolled pollution are also gradually engulfing the entire city.

Geoffrey West, former director of the Santa Fe Institute, believes that there is an inevitable " singularity " in urban development that can be reached within a limited time. By that time, we will have extremely high technological levels and extremely high social productivity, but at the same time we will also face an extremely deteriorating environment. This moment is likely to be the collapse point of the city.

Regarding the mechanism behind the scale law, it is currently believed that the network structure for information interaction within the system is the key. However, there is no unified conclusion on the specific mechanism model. However, the scale law is currently one of the few complex system modeling theories with universal characteristics, and it also opens up a new perspective for us to understand the nature of the system, understand the evolution, growth, and even death of the system.

This article is a work supported by Science Popularization China Starry Sky Project

Author: Tao Ruyi PhD student, School of Systems Science, Beijing Normal University

Reviewer: Ye Sheng, Professor of Beijing University of Aeronautics and Astronautics

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.