Friends You Can Count On

Concept/Phenomenon: “friendship paradox” in social network

You spend your time tweeting, friending, liking, poking, and in the few minutes left, cultivating friends in the flesh. Yet sadly, despite all your efforts, you probably have fewer friends than most of your friends have. But don’t despair — the same is true for almost all of us. Our friends are typically more popular than we are.
//友谊悖论: 我们的朋友的朋友比我们的朋友更多🤨

in the flesh = in person (rather than via a telephone, a movie, the written word, or other means)当面

• They decided that they should meet Alexander in the flesh.

Don’t believe it? Consider these results from a colossal recent study of Facebook by Johan Ugander, Brian Karrer, Lars Backstrom and Cameron Marlow. (Disclosure: Ugander is a student at Cornell, and I’m on his doctoral committee.) They examined all of Facebook’s active users, which at the time included 721 million people — about 10 percent of the world’s population — with 69 billion friendships among them. First, the researchers looked at how users stacked up againsttheir circle of friends. They found that a user’s friend count was less than the average friend count of his or her friends, 93 percent of the time. Next, they measured averages across Facebook as a whole, and found that users had an average of 190 friends, while their friends averaged 635 friends of their own.
//社交媒体社交网络数据研究论证

stack up against: measure up; compare 比得上

• Our rural schools stack up well against their urban counterparts.

Studies of offline social networks show the same trend. It has nothing to do with personalities; it follows from basic arithmetic. For any network where some people have more friends than others, it’s a theorem that the average number of friends of friends is always greater than the average number of friends of individuals.

This phenomenon has been called the friendship paradox. Its explanation hinges on a numerical pattern — a particular kind of “weighted average” — that comes up in many other situations. Understanding that pattern will help you feel better about some of life’s little annoyances.
//道理论证: 加权平均算法是计算社交网络得出上述悖论的原理

hinge on = depend on

For example, imagine going to the gym. When you look around, does it seem that just about everybody there is in better shape than you are? Well, you’re probably right. But that’s inevitable and nothing to feel ashamed of. If you’re an average gym member, that’s exactly what you should expect to see, because the people sweating and grunting around you are not average. They’re the types who spend time at the gym, which is why you’re seeing them there in the first place. The couch potatoes are snoozing at home where you can’t count them. In other words, your sample of the gym’s membership is not representative. It’s biased toward gym rats.
当我们去健身房看到别人身材都比我们好的时候, 不要自卑!不要气馁!因为你在健身房遇见的很有可能都是gym rats, 而couch potatoes都宅在家里你是不会在健身房遇见的!😄

couch potatoes: A couch potato is someone absorbed in television who vegetates on the couch -- or in simpler words, one lazy individual. 懒汉
<=> gym rats 健身房达人

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In this hypothetical example, Abby, Becca, Chloe and Deb are four middle-school girls. Lines signify reciprocal friendships between them; two girls are connected if they’ve named each other as friends.

Abby’s only friend is Becca, a social butterfly who is friends with everyone. Chloe and Deb are friends with each other and with Becca. So Abby has 1 friend, Becca has 3, Chloe has 2 and Deb has 2. That adds up to 8 friends in total, and since there are 4 girls, the average friend count is 2 friends per girl.

social butterfly🦋 交际花

This average, 2, represents the “average number of friends of individuals” in the statement of the friendship paradox. Remember, the paradox asserts that this number is smaller than the “average number of friends of friends” — but is it? Part of what makes this question so dizzying is its sing-song language. Repeatedly saying, writing, or thinking about “friends of friends” can easily provoke nausea. So to avoid that, I’ll define a friend’s “score” to be the number of friends she has. Then the question becomes: What’s the average score of all the friends in the network?

Imagine each girl calling out the scores of her friends. Meanwhile an accountant waits nearby to compute the average of these scores.

Abby: “Becca has a score of 3.”

Becca: “Abby has a score of 1. Chloe has 2. Deb has 2.”

Chloe: “Becca has 3. Deb has 2.”

Deb: “Becca has 3. Chloe has 2.”

These scores add up to 3 + 1 + 2 + 2 + 3 + 2 + 3 + 2, which equals 18. Since 8 scores were called out, the average score is 18 divided by 8, which equals 2.25.

Notice that 2.25 is greater than 2. The friends on average do have a higher score than the girls themselves. That’s what the friendship paradox said would happen.

The key point is why this happens. It’s because popular friends like Becca contribute disproportionately to the average, since besides having a high score, they’re also named as friends more frequently. Watch how this plays out in the sum that became 18 above: Abby was mentioned once, since she has a score of 1 (there was only 1 friend to call her name) and therefore she contributes a total of 1 x 1 to the sum; Becca was mentioned 3 times because she has a score of 3, so she contributes 3 x 3; Chloe and Deb were each mentioned twice and contribute 2 each time, thus adding 2 x 2 apiece to the sum. Hence the total score of the friends is (1 x 1) + (3 x 3) + (2 x 2) + (2 x 2), and the corresponding average score is

This is a weighted average of the scores 1, 3, 2 and 2, weighted by the scores themselves — the same dual-use pattern as in the class-size problem. You can see that by looking at the numerator above. Each individual’s score is multiplied by itself before being summed. In other words, the scores are squared before they’re added. That squaring operation gives extra weight to the largest numbers (like Becca’s 3 in the example above) and thereby tilts the weighted average upward.
在计算朋友的朋友数量时, 拥有朋友多的人被提名、重复计算的次数多,因此他占的权重大, 而由于他的朋友多,所以最终计算结果大.

So that’s intuitively why friends have more friends, on average, than individuals do. The friends’ average — a weighted average boosted upward by the big squared terms — always beats the individuals’ average, which isn’t weighted in this way.

Like many of math’s beautiful ideas, the friendship paradox has led to exciting practical applications unforeseen by its discoverers. It recently inspired an early-warning system for detecting outbreaks of infectious diseases.
//此悖论的应用: 疾病传播预警系统

In a study conducted at Harvard during the H1N1 flu pandemic of 2009, the network scientists Nicholas Christakis and James Fowler monitored the flu status of a large cohort of random undergraduates and (here’s the clever part) a subset of friends they named. Remarkably, the friends behaved like sentinels — they got sick about two weeks earlier than the random undergraduates, presumably because they were more highly connected within the social network at large, just as one would have expected from the friendship paradox. In other settings, a two-week lead time like this could be very useful to public health officials planning a response to contagion before it strikes the masses.
//社交网广的人先感染

And that’s nothing to sneeze at. 这件事情不容小觑

Nothing to Sneeze At = not to be sneezed at: means something that is not an inconsequential matter, not a trifling thing. 小看, 嗤之以鼻(直译,太形象了🤣)

• When Daniel was chosen to be valedictorian, he was so proud, because the honor of being chosen to represent your entire class is nothing to sneeze at.”

Interesting fact

In the 17th century, sneezing was considered a symbol of status as people believed it cleared their head and stimulated their brain. Soon sneezing at will became a way to show one's disapproval, lack of interest and boredom. The first recorded use of the phrase in its current negative form, was in 1799, in a play by John Till Allingham: 'Fortune's Frolic': "Why, as to his consent I don't value it a button; but then £5000 is a sum not to be sneezed at."