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. 

Inclusive STEM, it’s time to stop being Hypocrites.


Last weekend, I was at Amsterdam representing the University of Sussex as a part of our equality and diversity team. My team comprised of Claire, who is the Vice-Chancellor of Equality and Diversity, Nicola, Katy and finally Daniel .. whose project “Grapheel/IRIS” we were showcasing. The team got ready to deploy after a lot of brain storming and a slide show showcasing all the different initiatives that have been applied and have been really successful at Sussex. But Grapheel/IRIS, was a completely different project to be showcased in the conference than any others, because IRIS is a community app designed by Daniel, who is a blind physicist, for other vision impaired researchers like himself to have the resources to understand scientific figures better. How amazing is that?

Claire started her talk talking about how Sussex has been going about building a more inclusive Sussex, not just for people who have everything “normalised” going for them in the world, but for everyone who our ancestors should have taken into account before designing the structures of this world and the essential tangible and intangible structures in the work place. This includes women, LGBTQA people, people with a wide range of disabilities, carers, people who are trying to remake themselves such as ex-prisoners with a record, less-economically able people, racial minorities and so on. Claire focussed on how we can use our research expertise and our brain power to take these steps to completion and ensure their execution and at the right moment, Daniel jumped in to talk about how he lost his vision some five years back and how he did not want to let the various frustrations push him away from his passion to do research and he combined “Be my eyes” and “Citizen Science” to create a love child called “IRIS”. You can check it out here and sign up to become one of their esteemed ambassadors here.

For me, the conference was really really amazing. And in this blog post I would like to summarise the various things I took away from this conference and what I in turn, contributed to it. The conference was amazing in the way it brought together university policy makers, university professors and researchers working on the best way forward for an inclusive atmosphere.

  1. Inclusivity in AI training. We were shown the video clip of the AI humanoid Sophia
    as well as extensively discussed how easy it has been to past and present AI’s to learn from human bias on the internet and perpetuate this much faster. AI safety and AI governance has become a big issue, however I propose that we call it AI education to start from the bottom up approach of educating AI’s on diversity and inclusivity principles, starting them off with the right questions and data sets rather than go back and correct mistakes we see in retrospect.
  2. Redefining the definition of excellence in academia. Excellence is a spectrum which should potentially include normalisation to include challenges specific demographics face. This picture says a thousand words.
  3. Inclusivity in architecture (toilets, temperature) geared towards all genders, orientations and disabilities.
  4. Using our own scientific research to further an inclusive STEM.
  5. How to take into account invisible “disabilities” such as sexual abuse, or family responsibilities, emotional abuse, even pain during menstruation and fertility treatments which certain demographics undergo as opposed to other and how a lot of this and more certainly contributes to a power distance ( along with how it depends on the culture).
  6. How to make sure we go beyond unconscious bias but also consider and address conscious bias.
  7. How to overcome bias against ex-convicts in academia and the responsibility the media has.

There was a lot of debate and discussion on how to make sure we, as human beings can fluidly put ourselves in some one else’s experiences rather than denying them as something “foreign” and “lies” since we did not experience them firsthand. Related to this is the issue of addressing backlash against liberal movements and understanding since the losing of privilege does not come easy to people.

Would an AI government be able to fix these things? Apparently not, since the present day AI technology is spearheaded by straight white men who form only the creamy layer of a certain demographic. And the question remains, how do we go ahead?

If you wanna talk to me, I would very strongly argue that we need to go beyond and above, calling all of these “women”/minority issues since this is just excuse culture. As Claire says, these are things which should have been normal in humanity since time immemorial. We have screwed up and are forced to do error correction and so, let us all call these measures as “Ecosystem management” and make it everyone’s responsibility since we all share the same ecosystem.

Here are some slides from things that sussex has done in strive towards an inclusive STEM policies which I put together and you can find here.