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

Complexity, physics

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. 

Diversity in Constructions: Who are our cities built for?


In this blog post, I will touch upon a not often thought of topic. How do architecture and physical details add to the lack of diversity in our workplaces?

I have been thinking about this for a while because for a long time, ever since I have been in cold countries, for example, Canada, UK, US and have been in a shared office, I have had to have an extra of 4-5 layers always on the back of my chair handy in case I needed them. And I inevitably needed them. Sometimes, the overall office temperature was just too low, otherwise my office mates (inevitably males) would get really hot and open the window. Now, this very innocent sounding details can be very very mischievous. Here is how. Every day, I would ask if I could close the window, once and then twice and would be too embarrassed the third time since it is hard to repeat the same things over and over again, and of course, their problem of feeling too hot was also legitimate. However, the age old adage of compromise for females usually clicked into my brain and even though I would try to push hard to stay there and work, the cold temperature would not allow me to after a bare half an hour and I would make some or the other excuse and go home for the day.

I was also very aware that this made me feel/seem like I was working less than others, so I worked weekends and week nights leading to a high level of mental stress. Not being in the office at strategic times also made me feel a little disconnected from many people who would eventually come to matter by the time I needed recommendations. This in a larger picture eventually leads to gender gaps. The gender data gap is both a cause and a consequence of the type of unthinking that conceives of humanity as almost exclusively male.

So, while I was thinking this, I never actively tried to look for data to see what that problem was and how to address this problem which was clearly structural. And a few days ago, I stumbled upon this article which presented the following data:

“The formula to determine standard office temperature was developed in the 1960s around the metabolic resting rate of the average man. But a recent Dutch study found that the metabolic rate of young adult females performing light office work is significantly lower than the standard values for men doing the same activity. In fact, the formula may overestimate female metabolic rate by as much as 35%, meaning that current offices are on average five degrees too cold for women. This leads to the odd sight of female office workers wrapped in blankets in the summer, while their male colleagues wander around in shorts.

Not only is this situation inequitable, it is bad business sense: an uncomfortable workforce is an unproductive workforce. But workplace data gaps lead to a lot worse than simple discomfort and inefficiency. Over the past 100 years, workplaces have, on the whole, got considerably safer. In the early 1900s, about 4,400 people in the UK died at work every year. By 2016, that figure had fallen to 135. But while serious injuries at work have been decreasing for men, there is evidence that they have been increasing among women. The gender data gap is again implicated, with occupational research traditionally focused on male-dominated industries.”

The link to the whole article is here for those of you curious cats who want to know more.

But here you go! Now, this is an example towards a very very important realisation. I was once told by a physicist, “if you are feeling confused about something, it’s because there is something new to learn there. It’s not invalid. ” And it applies to life! If you are uncomfortable, then it is probably because of a reason, and often it might be structural and wide spread. So talking and researching about these things might actually lead us to build more diverse, inclusive and efficient workplaces rather than blindly following data. As always, the question to ask is what are the possible causations towards a correlation?

And a more immediate question. What are the ways we can go towards unbuilding our gendered city and work spaces and work towards more contemporary inclusive architecture?