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. 

Complexity of Gender in Academia and how it adds to the Leaky Pipeline.


Hello everyone. Today, I have been talking to a collaborator of mine regarding the complexity of factors that adds to gender skewness in the workplace. Often I have attended workshops, conferences and events where participants focus on one aspect which leads to the “leaky pipeline” of academia, so to say. Now, just to put it here, I am not a big fan of the phrase “Leaky pipeline” but we will come to the nuances of language in another post.

Last year, I was invited as a Key note speaker to the WomenBeing Conference in Edinburgh where gender studies students presented their work on various topics ranging from how drug abuse rates are different in different genders, the rate of microaggressions faced by various genders, the power distance between supervisor and supervisee relationships, statistics on how race/gender/age determines who are victims of certain kinds of behaviours to domestic abuse and a variety of other factors. This was a wonderful conference full of great ideas and statistics, however, one thing which leapt out to me was how often we miss the intricate complexity of this topics and the interaction they have with each other. For example, a woman who is being abused at home and has a baby, and is also on her periods and in a lot of pain, would be highly prone to having low self esteem. If an interaction happens with an uncooperative boss, that puts her at a risk of aggressions that she would otherwise be able to handle better. This would inevitably lead to depression and anxiety, low performance and then become a self fulfilling prophecy of certain genders low rate of success if she drops out.

I have often believed that the metrics by which we measure success in our workplaces around the world are lacking in their depth of understanding the feedback loops of how each seemingly separate cause feeds into another one and eventually comes back a full circle. This is also the same principle as to how the rich become richer and the poor become poorer, not withstanding how “hardworking” each of these individual people are.

It is common knowledge that once you have one house, you can get loans to build more. And some more. You can rent the rest of the houses out and make more income and so would have a lot more property to show as collateral for your next loan. A tiny difference in the initial amount is often what separates an escalation of wealth from a downward economic spiral. Many other micro-factors add to this. We have all heard the age-old adage of “The First Impression is the Last Impression”. Now, let us compare someone who has money to splurge on “Smart office wear” and is not irritated by not eating a healthy diet, has enough money to spend on child care or has a live-in spouse (often female) at home and so is not distracted at meetings and is able to commit to his work fully while looking dapper to someone who has to juggle multiple shifts, does not have enough money to splurge on a comfortable work wear and has to rush back home to take care of the baby. The strain on their face would show, making people view them as less trustworthy and motivated. This would eventually lead to loss of opportunity. And such examples can be many and varied.

However, in the conference, I decided to take the opportunity to sketch an interaction slide showing the interaction between seemingly disparate topics and how each of them eventually add to the Leaky Pipeline of Academia. The problem is not motivation which can be easily proved by a multitude of statistics. The problem is the lack of proper infrastructure and the delegation of microaggressions and unpaid (often emotional among others) labour onto certain demographics. Here is the slide which provides a (incomplete) list of many tiny details which add to the low percentage of women in high positions.

The conclusion here is that there is an intricate relationship between poverty, comfort, economics, brain power, psychology which overlap with gender, race and privilege. I will pick small topics and go on to talk about these in more detail in my future posts.