It is a common observation that individuals often have fewer friends than their acquaintances. This disparity is not a reflection of personal popularity or social standing but rather a consequence of how social networks are structured.
A friendship group can be effectively visualized as a network. In this model, each person is represented as a node, or a dot, and a line connects two nodes if those individuals are friends. Applying this to a group of people, whether interacting in person or online, allows for a clear representation of their interconnected friendships and social links.
This network structure enables us to investigate concepts such as the number of degrees of separation. When someone is a friend of your friend, they are considered to be at degree 2 in relation to you. Their friends then fall into degree 3, and this progression continues outward.
The fundamental question becomes: how many links must be traversed to connect any two individuals within this network? It is observed that social connections tend to form clusters. Consider a group of friends; these might include people from your local area, colleagues from work, or fellow members of a specific hobby group, such as an astrophotography club. It is highly probable that many individuals within such a cluster are friends with each other, meaning a significant portion of your “friends of friends” are also directly connected to you.
However, networks also contain more distant, far-reaching connections. Imagine an old friend who has relocated to another country. This friend likely has their own distinct and close-knit group of friends, perhaps united by a shared interest like a soap-carving club. All members of this group represent your degree-2 connections, even if you have never personally met them.
This principle forms the basis for the well-known “six degrees of separation” theory. By tracing these more extended connections, one can quickly move beyond one’s immediate social circle. For instance, a former colleague who took a job in London might be connected through a shared hobby, like wargaming, to a barista working near Parliament. Suddenly, a path from you to a very prominent figure, such as the prime minister, is reduced to a small number of degrees of separation.
The question of popular individuals in networks is also noteworthy. Within any friendship network, certain people will inevitably possess a greater number of connections than others. To illustrate, consider a group of 20 people where 15 are friends with an individual named Sandy, while only five are friends with Charlie. If a person is chosen at random from this group, there is a 75% probability they are friends with Sandy but only a 25% chance they are friends with Charlie. Your own friends are not a random selection from your social sphere; you are more inclined to be friends with those individuals who are themselves more popular, thus leading to your friends having, on average, more friends than you do.
This phenomenon, identified as the friendship paradox, can prove advantageous in network sampling, particularly when the aim is to identify influential individuals within a population. If you were to select a group of people at random and ask each to name a friend, it is probable that they would name someone with a larger number of connections than themselves. This newly identified group, by its nature, is likely to possess an above-average number of connections.
Therefore, if your friends appear to attend more social gatherings, colleagues at work possess a wider circle of contacts, or fellow members of your art class are involved in more hobby groups than you are, there is no cause for personal inadequacy. This is a predictable outcome of network dynamics.
Peter Rowlett is a mathematics lecturer, podcaster, and author based at Sheffield Hallam University in the UK. He can be followed on Twitter @peterrowlett.