Have you ever thought that your friends have more friends than you? This is a very common phenomenon that has a precise name: the friendship paradox. And the incredible thing is that there is a real mathematical explanation to this mechanism, which is called “paradox” by virtue of this observation: but if we all think that others have more friends than us – and each of us is “other” than someone – who has more friends than whom? Surprisingly, the answer lies in probabilistic reasoning that tells us it’s true: probabilistically, our friends on average have more friends than we do. Let’s see it in this article.
What is the friendship paradox and in which fields does it apply
The friendship paradox is that phenomenon whereby it seems to us that our friends have more friends than us. And surprisingly… it’s true! Or at least, that’s generally true. The sociologist Scott L. Feld observed this phenomenon in 1991 and even today there are many sociologists and mathematicians studying this mechanism.
It should be noted that, from a certain point of view, it is a problem of perception. Nowadays, social media tends to give us a perception of others that may be distant from reality. Those who make extensive use of social media tend to post happy moments, moments of togetherness or participation in events or holidays. This could make us perceive the lives of others as “better” than ours, but in this case it is about performativity: social media are often used as a showcase where we can “perform” and therefore what we consider to be the most significant or appreciable moments of our lives are shown and amplified.
That said, the friendship paradox is true! It is true that, probabilistically, our friends have on average more friends than us. This means that if we pick a friend at random from our circle of friends, it is very likely that this individual has more friends or acquaintances than us. This phenomenon applies not only to friendships, but to a wide range of events. If we go to the gym, for example, it seems to us that other people go more often or are fitter than us. If we look at the people we have had relationships with, it seems to us that these people have had more relationships than us. And the same thing goes for social media: if we take a random person among our followers or friends, their number of contacts will probably be greater than ours
But what is the mathematical principle behind this phenomenon?
The mathematical explanation
To explain why it is true that our friends have more friends on average than us we must resort to the so-called graph theory. Without going into too much technicality, a graph is a structure made up of nodes (in our case, friends) and lines (the relationships between these nodes, therefore in our case, the friendship relationships that exist between people), as in the image below.
If we consider a random person in the image, for example Bob, we see that while Bob has 2 friends, his two friends have 2 friends (John) and 5 friends (Jose) respectively. On average, therefore, his friends have 3.5 friends, so more than him! If we repeat this reasoning for all the people present in the graph, the only people who have more friends on average than their own friends are Jose and Jane, who in fact we can notice – on a visual level – are “hubs” of our graph, that is, they connect more parts of the set of people.
What we can conclude is that, probabilistically, it is more common to find a person whose friends have more friends than you, than to find a popular person (like Jose and Jane in our case). The explanation for this fact, beyond its mathematical nature, is actually extremely intuitive: people who have many friends (like Jose or Jane) are more likely to be among your friends, precisely because they have many friends! On the other hand, people who have few friends have few connections, so it is more difficult to be friends with someone who has few friends. In short, it is more likely that in our circle of friends there are few people with few friends (precisely because they have few friends), while it is more likely that there are people with many friends (precisely because they are popular).
And the same thing goes for the gym: the people we meet in the gym are people who go to the gym at least when we go, so in all likelihood they will go more often than us, and that’s why they seem (or are) fitter than us. However, it is much less likely to meet someone in the gym who rarely frequents it, and therefore it is less likely to compare yourself with people who are in “worse” physical shape.
