Your friends are more popular than you. In the year 1991, the sociologist Scott L. Feld made an interesting discovery. He realized that on average, most people’s friends have more friends than they have. This phenomenon is called the friendship paradox. At first glance, it seems unintuitive. As it turns out, however, it can be explained using graph theory. Furthermore, we talk about its applications in monitoring disease outbreaks. (Reference 1.)
Friendship
Before we introduce the graph-theoretic elements, we should talk about the definition of friendship. Ideally, friendship is a two-way street. This means that if person 1 is friends with person 2, then person 2 is also friends with person 1. This description of friendship is certainly far from perfect. No one can deny that there are people who think they are friends with others but the feeling is in fact not mutual. Although this is certainly a valid concern, we are not here to philosophize but to model friendships. Therefore, we run with the idealized version of mutual friendships for our model.
Modeling
Let us consider two individuals, denoted with 1 and 2, respectively. We can represent them with circles, called vertices in graph theory.
Figure 1. A graph with two unconnected vertices. (Reference 2.)
At the moment, person 1 and person 2 in Figure 1 are not yet friends. This is represented by the lack of a connection between the two vertices.
Let us now assume that the two have gotten to know each other a little better and have actually become friends. We can represent the blossoming friendship between the two with a connection, called edge in graph theory.
Figure 2. A graph with two connected vertices. (Reference 2.)
Since Scott’s claim is always true for a network containing only two connected vertices, e.g. Figure 2, we are interested in more complex networks.
Let us increase the size of the network by adding more vertices and edges to the graph. We obtain the representation of a social network as seen in Figure 3.
Figure 3. A graph with multiple connected vertices. (Reference 2.)
The claim that in a social network on average, an individual’s friends have more friends than the individual is no longer obvious. We can, however, translate it into graph theory. The social network that we want to model is then called an undirected graph G=(V, E), where V is the set of vertices and E is the set of edges in the graph. We note that V represents the set of all individuals in the social network and E represents the set of all friendships in the social network. In Figure 3,
|V|=5,
since there are exactly 5 vertices, and
|E|=7,
since there are exactly 7 edges connecting the vertices. Additionally, we denote the number of friendships for a given node or vertex v by d(v), called the degree of a vertex in graph theory. If, for example, we consider node 1 in Figure 3, we note that d(1)=4.
Calculations
It is now possible to determine the average number of friends for a random node in the network and the average number of friends of friends in the network. At the end of the calculations, we note that the friendship paradox is in fact true for any social network and on average your friends are more popular than you. If you are interested in the calculations involved, I suggest a visit to the Math Section itself. (Reference 3.)
Applications
The friendship paradox does not only exist to make people feel less popular than others. It does, in fact, have useful applications in medicine, particularly in monitoring disease outbreaks. According to a study conducted by Christakis, friends of a random group of college students got sick earlier than the random group. Thus, by applying the friendship paradox and observing the friends of random people it might be possible to recognize contagion outbreaks earlier. (Reference 4.)
About this article
I am Elias Wirth, owner and main writer of the Math Section. This article is the shortened version of the article Friendship Paradox from said website. The Math Section is a blog that investigates mathematics in everyday life.
References
Feld, S. (1991). Why Your Friends Have More Friends Than You Do. American Journal of Sociology,96(6), 1464-1477. Retrieved from http://www.jstor.org/stable/2781907
All of the graphs are created using the graph editor tool from csacademy: https://csacademy.com/app/graph_editor/
Wirth, E. (2018, September 2). Friendship Paradox[Blog post]. Retrieved from https://mathsection.com/friendship-paradox/
Christakis NA, Fowler JH (2010) Social Network Sensors for Early Detection of Contagious Outbreaks. PLoS ONE 5(9): e12948. https://doi.org/10.1371/journal.pone.0012948