Understanding the mathematics of scaling towards a sustainable future for humanity.

There are some truly pressing problems in our times. These concern the survival of humanity. They can range from environmental to socio-economic such as climate change, world poverty, migration and gender inequality. These are problems which are prevalent in every city around the world. 

Today, the world is urbanizing at an alarming rate. The population is reaching unsustainable levels, especially since resources are finite. Growing population brings with it growing rates of crime, inequality and pollution. Thus, it is now more urgent than ever that we model and understand our cities better. 

While cities can be the problem, they can also point us towards a potential solution. They can be facilitators of social interaction leading to innovation. Due to their multi-variate and highly non-linear nature, cities have evaded a quantitative understanding until now. However, recent insights in the field of complex adaptive systems could be the key to solving these important problems.

A key feature of complex adaptive systems is the scaling properties they follow. These scaling properties are underpinned by predictive mathematical frameworks and can be written as power law equations. Such an equation looks like:

Y = A * X^k, 

The value of the exponent k determines the type of growth in the system. Linear growth — where if X doubles, Y doubles, and so on — results if k = 1.

‘Sub-linear scaling’ is the case where k < 1, in which changes in X result in less-than-proportionate changes in Y: if X doubles, Y will increase by less than double.

In physics, such scaling is observed in things like Kepler’s Laws, which relate the time period of a planet orbiting the sun and its radius. (In this case, k = 2/3.)

Biological systems, on the other hand, are continuously evolving, which makes them difficult to predict. But nonetheless, despite the extraordinary complexity and diversity of life, many of its most fundamental metrics follow similar scaling laws. This can be true of systems as small as cells and as large as ecosystems.

For example, comparing metabolic rates and body mass across species gives k = 3/4. Nature exhibits an inbuilt economy of scaling.

The problem arises when we have k > 1, which is known as ‘super-linear’ scaling. This implies unbounded growth and is unsustainable in the presence of finite resources. Unfortunately, as we will see, cities are systems which follow this kind of growth.

According to studies by complexity expert, Geoffrey West, the origin of such power laws lies in the dynamics of the underlying networks which constitute such complex systems. Every system  needs energy to survive. Networks help deliver energy in a system efficiently. In this way, they enable interaction between seemingly unrelated parts of a system leading to an emergent large scale behaviour. This is the field of complex adaptive systems. 

In the biological world, such a network is the circulatory system. The total (finite) amount of input energy is allocated between maintenance and the growth of a system. As the system grows, a larger volume requires higher allocation of energy towards maintenance which implies less energy for growth. This is what leads to bounded growth and eventually death.

West and his collaborators observed that there is a similarity between growth in the biological world and growth in our cities. Cities, like living systems, are continuously growing and evolving. More importantly, cities, just like biological systems require energy to survive and grow. However, unlike biological systems, cities seem to avoid senescence!

Studying the growth of infrastructural measures in a city, such as numbers of gas stations and lengths of roads and electrical cables, with population size, we see that these follow the same sub linear scaling (k < 1) as in the biological world. However, cities are much more than just infrastructure! In fact, the growth of socioeconomic quantities involving human interaction, such as wages, patents, AIDS cases, and violent crime with respect to increase in population size, follows a super-linear scaling (k >1)! This is a new and surprising class of phenomena separate from anything observed in the biological world. It also implies unbounded growth which explains why cities never stop growing! 

What is even more interesting is that despite their unique histories, cities all over the world exhibit universality by obeying the same scaling laws. These regularities have led to the beginnings of a quantitative framework of cities in terms of their underlying networks. In this case, these are social networks or transportation networks which are universal features of every city. Unfortunately, on earth, resources are finite. And thus, such an unbounded growth is unsustainable and predicts a collapse of the system once resources are depleted. This is known as a finite time singularity. 

Interestingly, human beings have avoided such finite time singularities by making a drastic paradigm shift after regular finite time intervals. Or in other words, we have had to innovate continuously in order to use our resources more efficiently. Examples of such paradigm shifts have been the discovery of coal, the industrial revolution, the discovery of clean energy sources, the IT revolution, digitalization of technology and discovery of automobiles. However, there is a catch. Each subsequent interval is smaller and smaller which implies that in order to survive, humanity needs to innovate faster and faster. Thus, it is important to understand what drives innovation in cities. The answer, perhaps unsurprisingly, lies in diversity. The more diverse a city, the more adaptable and resilient is it. Again, this has parallels in the biological world.

Thus, the need of the hour is this: Can we come up with a recipe for a sustainable future? The answer might lie in combining insights of complexity theory into concrete and smart implementable policies. 

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