Connecting with Waterloo

Graphs are structures that help us study objects as well as relationships that arise between them. These topics can be analyzed following two significantly different approaches. On the one hand, you could do studies that are more theoretical, with a focus that is mainly mathematical and, specifically, from the standpoint of combinatory or discrete mathematics. On the other hand, you can perform a research that is more oriented towards application. Graph theory is useful for a multitude of disciplines. In computer science they are used, for example, as a way to create models of networks.

We could say that in my case I have done more work on the theoretical side, with a very subtle mathematical treatment, doing very rigorous proofs and leaving a little aside the part that is more connected to day to day life.

Personally, I think that each person is attracted towards a certain way of working. It seems that this has to be my way. There people with an extraordinary practical ability and people that love mathematical proofs that are beautiful, delicate and precise. I am one of the latter type of people. In fact, I fell that I am in the boundary between the theoretical computer scientist and the applied mathematician.

That been said, the divide between theory and applicability is not something that is perfectly defined. The study of graphs is something very large. Let’s proceed to a more concrete and deep level. Yes, yes... well, I have concentrated on random graphs.

Random graphs are a branch of graph theory that takes into consideration sets of graphs that are very big, and it defines probability distributions. We study classical properties but following a probabilistic model. This has a lot of applications in theoretical Computer Science.
If we zoom into a more detailed level, you could say that I study geometrical random graphs. In other words, given a geometric space, like, for example, a square, a circle or any other, you distribute a set of nodes (points) uniformly on them and you establish certain relationship between them (for example, connections between points), but only within a given range of action. Once you have established these conditions, you study different probabilities: the probabiblity that the whole system is fully connected, the probabiblity that each node is connected to a minimum number n of neighbours, etc.

These studies have very different applications in many different fields, like Medicine or Telecommunications. If we wanted to connect nodes by means of electromagnetic or acoustic waves, the ranges of action would be different depending on the frequency of the waves. In these cases, we work with a variable that is called “chromatic number”.

This has been studied for some years. In my case, I start from this idea and then I add that nodes are mobile. So, in my research I work with nodes that are moving and graphs that change accordingly. That is what we call geometric dynamic random graphs. These conditions increase considerably the level of difficulty of the initial problem, that started with non dynamic graphs.