The invisible mathematics that controls the world

Albert Luszel Barabassi: We live in a very special moment because anything we do is marked by data. This is not only true of us, but also of our biological and global existence.

The more we know about the world, the more we understand that it is a very complex system. Our biological existence is governed by highly complex genetic and molecular networks. How genes and molecules in our cells interact with each other, but society is also not just a collection of individuals. Society is not a phone book. What makes society work is really the interactions between us.

But the question is: How do we understand this complexity? If we want to understand a complex system, the first thing we need to do is define its structure and the network behind it.

We have data on just about everything, and this vast amount of data creates an amazing and unique laboratory for the world; Offering the opportunity to truly understand how our world works.

Graph theory has become a very prominent subject of study for mathematicians, and I am Hungarian, and it turns out that the Hungarian School of Mathematics, thanks to Paul Erdos and Alfred Rennie, had great contributions to this problem. In the mid-1959’60’s, they published eight papers that laid out the “Theory of Random Graphs”.

They looked at some of the complex networks around us and said, you know, “We have no idea how these networks are connected to each other, but for all practical purposes, it looks random.” So their model was pretty simple: pick a pair of nodes and roll a dice. If you get six, you can connect them. If you don’t, move on to another pair of nodes. With this idea, they built what we call the “random network model” today.

What’s interesting from a physicist’s perspective is that to us randomness doesn’t mean unpredictability. In fact, randomness is a form of predictability. And this is exactly what Erdős and Rényi have proven, that in a random network, the mean dominates.

Let me take an example: the average person, according to sociologists, has about a thousand people they know on a first name basis. If the community is random, the most popular person, the person with the most friends, will have about 1150 friends or so. And the least popular one, is around 850. This means that the number of friends we have follows a Poisson distribution that has a large peak around the mean and decays very quickly. Obviously it doesn’t make much sense, right? This was an indication of something wrong with the random network model. Not in the sense that the model is wrong, but it does not capture reality, nor does it capture how networks form.

After years of interest in networks, I realized that I needed to find real data describing real networks. Our first opportunity to study real networks came with a map of the World Wide Web. We know that the World Wide Web is a network. The name says it: it’s a network. Nodes are web pages and links are URLs, which are the things we can click on to go from one page to another. We’re talking about 1998, about six or seven years after the World Wide Web was invented to begin with. The web was very small, containing only a few hundred million pages.

So we set out to map it out, and that really marked the beginning of what we call today, “network science.” Once we got this map of the World Wide Web, we realized that it was very, very different from the random network maps that were creating previous years. When we dig deeper, we realize that the degree distribution, i.e. the number of links per node, did not follow the Poisson we had for the random network, but instead followed what we call the power law distribution. We ended up calling these networks “scaleless networks”.

In a network without scales, we lack averages. Averages are not meaningful. They do not have an intrinsic scale. everything is possible. They are devoid of scales. Most real networks are not formed by connecting pre-existing nodes, but grow, starting with one node, adding other nodes and more nodes.

Think of the World Wide Web: In 1991, there was one web page. How do we get today to more than a trillion? Well, another webpage was created that linked to the first page, and then another page that linked to one of the previous ones. And in the end, every time we put a webpage and connect to other webpages, you add new nodes to the World Wide Web. The network forms one node at a time. Networks are not static objects with a fixed number of nodes that need to connect – networks are growing objects. evolve with growth.

Sometimes it took as many as 20 years as the World Wide Web to reach its current size, or four billion years when it comes to sub-cellular networks to reach the complexity we see today. We know that in the World Wide Web, we don’t randomly communicate. We communicate with what we know. We link to Google, Facebook, and other major web pages that we know of, and we tend to link to the most linked pages. So our connection pattern is biased towards the most connected nodes.

We ended up formalizing this with the concept of “preferential association”. And when we put growth and preferential attachment together, the laws of force suddenly emerge from the paradigm. And suddenly we have hubs, we have the same stats and the same structure that we saw earlier in the World Wide Web. We started looking at the metabolic network within cells, protein interactions within cells, and the way actors communicate with each other in Hollywood. In all of these systems, we saw scale-free networks. We saw non-randomness, and we saw the emergence of hubs. Thus, we realized that the way complex systems build themselves follow the same general structure.

Let’s just be clear that network science is not the answer to all the problems we face in science, but it is a necessary path if we want to understand the complex systems that emerge from the interaction of many components. Today, we don’t have social network theory, biological network theory, and World Wide Web theory—but instead, we have network science, which describes them all in one scientific framework.