# The theory of six degrees of separation: the mathematics that explains social networks | Coffee and theories | Science

Historically, the success of science has been based on the idea of breaking systems down into their fundamental units. However, to understand complex structures it is necessary to adapt another perspective, which allows us to understand the interconnection of the elements that make them up. This is the starting point of the popularization book. *At the mercy of the networks* (Universo de Letras, 2023), by Ernesto Estrada, research professor at the Higher Council for Scientific Research (CSIC) at the Institute of Interdisciplinary Physics and Complex Systems.

The mathematical object that describes – in a simplified way – the relationships between elements is the network, or graph: a set of points – called vertices – and unions – called edges – between them. They serve to capture key information from numerous real-world situations. Estrada presents numerous examples in his book: social relations, epidemics, anatomical structures, gene, metabolic or neuronal networks, social conflicts, transportation networks. The one that offers the greatest mathematical analysis is the first of them, social networks. In this case, the points are people and the vertices can be mutual knowledge, friendship or collaboration.

Estrada talks about different mathematical models that simulate the formation of social networks and that allow us to study, in a simplified way, the structures of a real network. The first, developed by mathematicians Paul Erdös and Alfred Rényi, starts from a number *north* of individuals who do not previously know each other – therefore, at the beginning it has *north* vertices and no edges—and of a number *k* which indicates how conducive the environment is for relationships to be established. In each simulation, a random value is given to each pair of nodes; if this is greater than *k*A vertex is created between those two vertices, if it is smaller, no.

To assess whether the result obtained is similar to what is observed in real-world social networks, one can check whether the main characteristics of real-world networks are maintained. These characteristics allow us to understand the dynamics of the network, that is, how information is transmitted within it. One of them is the density of the network, which corresponds to the number of connections between the elements. It is the percentage of the number of existing connections, over all those that could be on the network. If all the elements are related to the rest, the network is complete.

Another important property is the connectivity of a graph: it will be connected if it is always possible to reach from one node to any other, through the edges of the graph. As Estrada explains in the book, almost all social networks in the world are practically connected. For example, 92.2% of biomedical sciences authors are *related* —in this case, it means that they have a joint publication in the Medline article database—with each other, while in mathematics they are 82% (using the Mathematical Reviews database). This means that information can be transmitted between virtually all members of the network. Furthermore, they are very sparse: none of the previous networks exceeds a density of 0.02%; That is, it is not necessary for everyone to be in communication with everyone. The Erdös and Rényi model also creates connected networks with low density: depending on how conducive the environment is to socialization, but, even for relatively low values of this parameter, the networks that appear are of that type.

In a connected network, the shortest path distance that joins each pair of elements can be calculated: for example, if Ana and Carlos do not collaborate, but Ana collaborates with Beatriz, who does so with Carlos, the distance between Ana and Carlos is of 2. The average of these values—which is called the average length of the simple paths, *l*— relates to how many steps you generally have to take to get from one point to another on the network. In the vast majority of real-world social networks, this number is surprisingly small—for example, 4.6 in the Biomedical Sciences Collaboration Network. This is what is known as the small world effect or the theory of six degrees of separation. In the Erdös and Rényi model, *l* has a value close to the logarithm of the number of starting nodes. For example, starting from five thousand knots, the *l* The average (for different environments) is 8.5 steps and, with five million nodes, 15.4, that is, it is similar to what is observed in reality.

However, there are other characteristics of real-world social networks that are not reflected in Erdös and Rényi’s model. For example, the so-called network transitivity, which indicates how likely it is that in a network, if *TO* is a friend of *b*who is a friend of *c*so *TO* and *c* Also be friends. Faced with this, other models have been proposed, such as that of Steven Strogatz and Duncan Watts or that of Albert-Lazslo Barabási and Réka Albert, which better capture some aspects of real-world social networks. All of them allow us to approach the complexity of these phenomena with mathematical models, which are much easier to study.

**Timon Agate*** She is coordinator of the ICMAT Mathematical Culture Unit.*

**Coffee and Theorems**** ***is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”*

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