The mathematics of friendship and disease

I was given the chance to give my own lectures. The course is aimed at third year Engineering students, and the idea is to introduce them to some ideas about complexity.

Essentially the course is about complex networks. We look at how many different systems can be fruitfully modelled as graphs, and how this simple mathematical framework can lead to a variety of rich insights. We motivate social networks between dolphins, power grids and even neural networks, and then discuss a unifying set of tools to answer questions about these systems. In one of the introductory slides, I show this network of jazz collaborations, coloring nodes based on the size of their degree (brighter nodes have more connections).

A collaboration network of Jazz musicians and bands from 1912 to 1940.

Actually, showing these sorts of pictures, whilst aesthetic and good for motivating students, are not very useful in practice. Ridiculograms is the term coined by my academic grandfather Mark Newman, to point out the absurdity of these representations. So instead, it is better to deal with these objects purely at the mathematical level.

The big idea is to get students to think about modelling complex systems. For instance, you could simulate the spread of a disease on a network, and ask questions like: when will it stop infecting people? This is analogous to how we model the spread of opinions, friendships or information.

Simulation of a model for disease spread. , and are the proportion of the population at a given time which are susceptible, infected and recovered.

In essence, the big result that I attempt to convey, is about phase transitions. You can think of these as the sudden manifestation of a certain property in a system (if we imagine an infinitely large system). This is literally the same kind of event as when water turns into ice, it is a sudden and immediate transition, there isn’t a progressive flow from liquid into solid.

In this course, the idea is that if you take graphs that are constructed randomly, then there’s a rate at which you add links to these graphs that suddenly gives rise to a really big clump in the graph. The emergence of this big clump (don’t worry there is a nice mathematical way to define this clump), is similar to the emergence of some other phenomena. And what we show is how the mathematics of a seemingly different problem, for say infectious diseases, gives rise to the same equations that describe the sudden appearance of a clump. It’s really cool, and it means that you only have to worry about a handful of results to obtain insights about many different systems.