This is the assertion that "your friends are more popular than you are."
Why? Simplest way to see it: some people have no friends. But because they appear in nobody's friendship circles, they're not making anyone else feel unpopular.
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The selection effect that applies to these friendless (a.k.a. degree zero) people also applies to other people: the more friends you have, the likelier you are to be represented in people's friendship circles. So popular people are oversampled as friends. Hence the paradox.
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Still, what is the paradox exactly, as a quantitative statement?
Scott Feld, who coined the term and made the paradox famous, had one way of formalizing it. It isn't my favorite way, but it's a classic, and worth meeting first.
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Fix an undirected network. Pick an edge (friendship) uniformly at random and look at one of the people involved (with equal probability). Let D be the degree (number of friends) of that person.
This is the "degree of a random friend" since we picked a random friendship.
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D here is a random variable, since the friendship we selected was random.
The friendship paradox, in this formalization, is simply the statement that the expectation of the random variable D exceeds the average degree in the network.
It's a nice exercise to work through.
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On the other hand, this is kind of a weird way to formalize it. Who ever picks a random friendship? What is this weird exercise?
So here's another way, which you may find more natural.
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For each person i, let's keep track of how popular i's friends are, on average.
Let d(i) be the degree (number of friends) of person i and, for people who have any friends, let f(i) be the arithmetic mean of their friends' d(j)'s.
f(i) is the avg popularity of i's friends.
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Another way to formalize the friendship paradox is: "the mean f(i) -- averaged across the people who have any friends -- is weakly greater than the mean d(i)."
This is a different statement and, curiously, wasn't written down anywhere that I know of until the 21st century.
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This is also a fun exercise to work through!
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PS! Too late to fix this now, but the first tweet should say that the friendless people aren't making anyone else feel POPULAR, since they're not friends and so not "visible."
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The average course size that students experience is bigger than the average course, because by definition the big courses have more students experiencing them ("the class size paradox.")
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When you go to the gym and look around, you feel relatively bad because the very frequent gym-goers are oversampled in your looking around, whereas the never-gym-goers are not sampled at all and don't make you feel (relatively) better.
3/3
A short thread on an obvious selection effect with some big consequences.
The social networks that are huge and very powerful now are the ones that grew the fastest. All else equal, these tend to be those with compelling products, but also another crucial thing:
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Being willing to make most trade-offs in favor of growth during a crucial period, which often was pretty long.
That process isn't pretty: it involves being willing to manipulate users and operate as many viral loops as possible, as long as they don't have a *growth* downside
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There's also a large, and maybe more important, effect on corporate culture: the people who grow most powerful and influential at the company during this period are the ones who were willing to give up a lot of other things for growth.
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Here's what she does: "I consider decision-making constrained by considerations of morality, rationality, or other virtues. The decision maker has a true preference over outcomes, but feels compelled to choose among outcomes that are top-ranked" by a "virtue/duty" preference.
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A question that people ask sometimes is, "What is your favorite paper in topic X written since y?"
It recently struck me that the reason I don't ever give a good answer is that it's a bit like the question, "What is your favorite beam in this building?"
It seeks assessment at the wrong level, both in terms of how most of us experience science, and in terms of what's important for its progress.
But it can take a while to see that!
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This was inspired by this @KevinZollman thread, which I like a lot because it says a similar thing at a different level (and in a different field).