Wednesday, 16 September 2015

TANDQ 03: Popular Misconception

In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This third column originally appeared on Thursday, May 29, 2014.

Why is my network below average, compared to all my friends?

It likely isn't - instead, a couple well-connected individuals are skewing our perceptions. Your friends may even be wondering the same thing! The “Friendship Paradox” was first observed by Scott L. Feld in 1991, but it has been gaining more notoriety lately, since it has been found to apply to online social networks too. Not only directed networks (like Twitter, where I can follow you without you doing the same) but also undirected ones (like LinkedIn, where you must confirm me as a connection). How about that, an application of directed graphs!

The paradox boils down to the fact that “on average, most people have fewer friends than their friends”. For instance, consider this set of follower numbers: {2, 4, 6, 8, 100} . Add them, divide by 5, you get 24. Meaning all but one of those people is BELOW average in followers. When George Carlin said “Think of how stupid the average person is, and realize half of them are stupider than that”… he’s assumed a uniform, or perhaps normal distribution. Not a skewed one, as we have here. This outlier issue is also why fully 99% of a population has the potential to be below an average salary, but I digress.

Something a little different is happening with our social networks - after all, the most popular person (let’s call him/her Pop) can’t have a connection of 100 unless there’s 100 other people to connect with in the first place. Two things are going on now. First, if someone has a lot of friends, you are more likely to be friends with them too… that’s simple probability. In other words, if Pop is connected with 90% of the internet, you’re more likely to be connected to Pop, and thus have your perception thrown off. Second, it’s a problem of averages.

Mathematical Means

Imagine a network of “n+1” people where everyone is connected to everyone else. You have “n” friends - you don’t friend yourself - and Pop has “n” friends, and Goofy has “n” friends too. In this network, what is the average number of friend connections? Well, if everyone you know has “n” friends, and we add this up (n+n+n+…) to get n-squared, then divide by the total “n”… average connections is “n”. Makes sense; everyone has this number. Still with me?

Now, what if you and Goofy break your connection? You’ll have “n-1” friends - and so will Goofy, not that you can tell - but everyone you’re connected to still has “n” friends! So from your perspective, the average is STILL “n”. You are now below average. (More rigorously: When we sum again (n+n+n+n…) we’ll only have (n-1) n’s. Divide by your (n-1) friends - to get “n”.) Yet amusingly, from Pop’s perspective, the average becomes (n-squared minus two)/(n), a value LESS than “n” - he’s above average! Balance is preserved.

As we keep chopping out connections (to create something more like an actual network), everyone less connected will - from their perspective - find themselves sinking below average, while a few popular people like Pop will become the big above-average benefactors. Thus while you may believe that everyone else you know is more popular and more connected, guess what? They’re experiencing the same effect. It doesn’t help that you’ll probably see updates from Pop more often than anyone else - the guy had to get popular somehow. (For a slightly more rigorous mathematical proof of why the average connection for friends MUST be less than or equal to the average connect for friends of friends, check out the Mind Your Decisions blog here.)

It turns out this actually has repercussions beyond our own self-esteem. The blog post I just referenced points out a connection to vaccinations (target the more popular individuals), and there’s also a link with the spread of infectious diseases (finding who is most likely to be exposed). Moreover, published this year (the MIT Tech link below) is how this paradox can even apply beyond the numbers - your friends don’t only seem more connected, but also richer! And happier! Yikes! Granted, our own “negativity bias” can also be a factor in how we see the world… but that’s a whole other topic. For the moment, your best bet is simply to stop comparing yourself to whatever you perceive as being "average".

For further viewing:

1. Steven Strogatz in the New York Times

2. MIT Technology Review (2014)

3. “Apple Daily English” Video (< 2 min)

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