Why does it always seem like your friends have more friends than you do? Whilst this may appear as an insult, or a bleak depiction of one’s lonely life in comparison to the popular lives of others, this feeling is actually a mathematical phenomenon that appears regularly in everyday scenarios. This phenomenon is known as the “Friendship Paradox”.
“On average, our friends have more friends than we do.”
SCOTT L. FELD, 1991
This rather depressing statement comes as a consequence of network structure. For example, consider a network of 4 people, and the network is set up as such, where each line shows who is friends with whom:
We can see that each person has a number of friends related to it. Therefore, we can calculate the average number of friends a person’s friend has, or, put more simply, the popularity of a person’s friend. We can calculate this by taking the average of the number of friends each person has. For example, person A’s friends have an average popularity of (2+3)/2 =2.5. Doing these for all of the people leads to:
| Person | Number of friends | Average friend’s popularity |
|---|---|---|
| A | 2 | 2.5 |
| B | 2 | 2.5 |
| C | 3 | 1.67 |
| D | 1 | 3 |
| Average | 2 |
The average number of friends someone in this network has is 2, but we see individually that in 3/4 cases each friend has friends who are more popular. The only case in which this is not the case is for Friend C, which makes sense as they are the most popular friend. However, delving more deeply, we can also find the average popularity of a friend. To calculate this, we must look at the popularity of each friend from a certain friend’s perspective. Person D would say “Person C has 3 friends”, Person C would say “Person D has 1 friend, Person B has 2 friends, and Person A has 2 friends”, Person B would say “Person A has 2 friends and Person C has 3 friends” and Person A would say “Person B has 2 friends and Person C has 3 friends”. Adding up the popularity of all the friends as perceived by each friend gives 3+1+2+2+2+3+2+3=18. Dividing this by the number of times a friend’s name is mentioned (8) gives the average friend’s popularity to be 2.25. This is greater than the average number of friends a person has, which we found to be 2. As such, the friendship paradox is fulfilled.
Thus, in response to the title: the answer does not necessarily relate to one’s perceived popularity or likeability. Your intuition has proven correct, but why do our instincts tell us that we are less popular than our friends? Well, logically this makes sense. For example, those with 100 friends will crop up in 100 individual friendship networks, whereas those with 5 friends will only show up in 5 individual friendships. This can make those 5 people feel disadvantaged compared to the 100. As such, friends appear disproportionately in sets of friends, and it is this disproportionality that causes our intuition to tell us we are less popular than our friends. To further understand this, we must understand the difference between the distribution of friends of individuals and the distribution of friends of friends. The distribution of friends of individuals means the average number of friends that an individual has (this is the same as the middle column in the table above). When discussing the distribution of friends of friends, we see that the same individuals are repeated. Using the example above, we see that both Person A and B mention Person C in their distribution of friends of friends. In fact, each person contributes to the distribution of friends of friends as many times as they have friends. For example, Person A and B contribute to the distribution of friends of friends twice, because they both have 2 friends, but Person D only contributes to the distribution of friends of friends once, as they only have 1 friend. The relationship between these two friends is the crux of the friendship paradox. The distribution of friends can be defined as the mean number of friends a person may have, written as (x1 + x2 x3…+xn)/n where n is the total number of friends, and xi is the friends each person has. The distribution of friends can be defined as the mean number of friends a person may have, written as (x1 + x2 x3…+xn)/n where n is the total number of friends, and xi is the friends each person has. The distribution of friends amongst friends has (x1 + x2 x3…+xn) cases (for each person) and they have a total of (x12 + x22 +x32…+xn2) friends, as each person is counted as many times as they have friends. Therefore, the mean number of friends of friends (the distribution of friends of friends) is (x12 + x22 +x32…+xn2)= mean(x) + variance(x)/mean(x). This expression shows that the average popularity of friends (the distribution of friends of friends) is at least as large as the average number of friends, which means that the friendship paradox holds.
However, there are some issues with the friendship paradox’s accuracy in the real world. The paradox holds stronger in large general networks, where there is great variation in popularity between friends, but when the variation is small, the friendship paradox may only weakly hold. This makes sense when looking at the above expression; if there were a network in which everyone was friends with everyone, or everyone had the same number of friends, then the friendship paradox would not hold, as there would be 0 variance. However, this is unlikely to occur in day-to-day cases, the friendship paradox appears frequently. This can also be extended to other measures rather than just popularity, for example, attractiveness or wealth, where a similar paradox takes hold. This provides a positive correlation, therefore, between popularity, attractiveness, and wealth, which is also something our intuition tells us.
In short, whilst you may feel as though you are unpopular, ugly or poor in comparison to your friends, the truth is that this is just a result of the friendship paradox, meaning for most people, they feel the same way you do. So, do not fret – you are not as unlikeable as you might think.
Bibliography:
Cantwell, Kirkley, Newman (2021). The friendship paradox in real and model networks, Journal of Complex Networks, Volume 9, Issue 2.
Feld, S. L. (1991). Why Your Friends Have More Friends Than You Do. American Journal of Sociology, 96(6), 1464–1477. http://www.jstor.org/stable/2781907
Santa Fe Institute. (2021, June 7). Applying mathematics takes ‘friendship paradox’ beyond averages. ScienceDaily. Retrieved November 11, 2021 from www.sciencedaily.com/releases/2021/06/210607084636.htm
Saplakoglu, Y., 2021. The ‘friendship paradox’ doesn’t always explain real friendships, mathematicians say. [online] livescience.com. Available at: <https://www.livescience.com/friendship-paradox-math.html>.
This article was written by Lydia Nawas
