How To Really Deal With The Friendship Paradox?

The friendship paradox is a mathematical phenomenon that challenges our intuitive understanding of comparison in relationships. Consider the following question:

“How popular are you compared to your friends?”

For one to be popular, one has to inevitably consider the number of people one is friends with. If we were to associate your popularity with the number of friends you have, the above question transforms into the following:

“How many friends do you have compared to your friends?”

Intuitively, I would say that I am not doing half bad. I might not have the largest number of friends among my friends, but I am certainly not one with the lowest number of friends either. How about you? How would you fare in comparison with your friends?

Well, the friendship paradox is here to ruin our little party by stating the following:

“Most people have fewer friends than their friends have, on average.”

How could that be true? If most people have fewer friends than their friends on average, would it not mean that most of our friends would have fewer friends than you and me? What is going on here? Let us find out!

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The Friendship Paradox — A Thought Experiment

To understand what the friendship paradox is about, let us begin with a thought experiment. Let us consider a hypothetical social environment that features four people: you, Jim, Amy, and Kate.

You happen to have just one friend in this social environment: Jim. Kate is friends with Amy and Jim, whereas Amy is friends with Jim and Kate. Jim enjoys friendships with you, Kate, and Amy.

Now that we have set up the hypothetical environment that we could work with, let us proceed to test the core statement of the friendship paradox.

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Testing the Friendship Paradox

It is quite clear that Jim is the most popular person in this social circle and you are the least popular person. However, the friendship paradox states that most people have fewer friends than their friends have, on average. So, this statement goes beyond just you and Jim. To start testing the statement, let us initially list the number of friends each person has, and compute the average.

So, on average, any random person in this social environment has 2 friends. To test the paradox’s core statement further, we need to list the number of friends each person’s friends have. Then, we need to compute the average of this set. If we do this, we obtain the following result.

When we look at the weighted average of the friends of everyone’s friends, we see that it is higher than the average number of friends the random person in the group has. Even when you consider each individual, except Jim, everyone has a lesser number of friends than the weighted average.

This is essentially what the friendship paradox is trying to convey, and it becomes apparent that it is counter-intuitive.

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A Sceptical Approach to the Friendship Paradox

If you made it this far, it is natural that you are sceptical about the thought experiment that we just conducted. We could easily just manipulate the friendship chain in our hypothetical example such that most people have more friends than the weighted average. If this is the case, why should we trust the example we just saw?

You see, while it is possible for purely arbitrary social chains to exhibit behaviour that doesn’t comply with the friendship paradox, human social chains exhibit exactly the kind of behaviour that complies with the friendship paradox.

This phenomenon was first observed by Scott Feld in 1991. If you wish to understand the mathematics behind this phenomenon, I recommend that you check out his original paper (linked in the references section at the end of this article).

If you are interested in a more contemporary example, Stephen Wolfram conducted a study using data from Facebook users in 2013. Below, you can see how the distribution for the number of Facebook friends for a random user (light yellow) stacked against the distribution for the number of friends for the user’s friends (dark yellow).

The rightward skew explains the friendship paradox. When it comes to comparing popularity among human social relationships (by comparing the number of friends), there seems to be a one-sided inequality in favour of the weighted average of friends of friends over the average number of friends of a random person.

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Applications Beyond Friendship

Applications of the friendship paradox go beyond just social worth and social media. It turns out that mathematical modeling of this phenomenon is extremely useful in tracing and predicting epidemics and pandemics.

Such models were and are being used by scientists to track down the spread of new COVID variants. Superimposing our thought experiment and this context, the term “super-spreader” would be used to denote “Jim”. You can find interesting scientific work in this area linked in the references section at the end of the article.

Similar to the spread of epidemics, the phenomenon highlighted by the friendship paradox can also be used to model the spread of ransomware, computer viruses, and other ill-effects that spread through computer networks.

What the friendship paradox also highlights is a variant of sampling bias. Going back to our thought experiment, one of the reasons why the weighted average of friends of friends is higher is because “Jim” is over-weighted. There could be scenarios where over-weighting data might lead to illogical or false conclusions. In this sense, the friendship paradox provides insights into data treatment and filtering.

Final Thoughts

Toconclude, the friendship paradox appears counter-intuitive and illogical on the surface but is based on mathematical truth. It arises out of inequality between intuitive averages and weighted averages.

In the end, it turns out that most of us indeed have a lower number of friends on average than our friends. But if there is one thing that we all know about relationships, it is that quality matters more than quantity!

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References: Scott Feld (scientific article), Reuven Cohen et al. (scientific article), and Stephen Wolfram (scientific essay).

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Further reading that might interest you: Russell’s Paradox — A Lesson On How To Crash Mathematics and Molyneux’s Problem — Can You Really Solve This Challenge?