MIT Massachusetts Institute of Technology

The Friendship Paradox: Why We Can’t All Be Popular

Katrina LaCurts
Lecturer
The Department of Electrical Engineering and Computer Science 
Massachusetts Institute of Technology
Cambridge, MA

With Amy Ousterhout
Department of Electrical Engineering and Computer Science
MIT 

 

This lesson explains the Friendship Paradox  --  including the intuition behind the result, how we use graphs to formalize the paradox, and a proof of the paradox. While the primary learning objective is for students to learn how one proves the Friendship Paradox, a secondary objective is to introduce students to graph theory. Students should come away with an understanding of the vocabulary related to graph theory (vertices, edges, etc.) as well as how to use graphs to model simple situations (e.g., social networks). As a prerequisite to this lesson, students should understand how to calculate the mean and variance of a set of numbers, be familiar with summation notation and manipulation of sums, and be able to perform algebraic manipulation. The lesson itself should take between an hour and an hour and a half, and could easily be split over two class periods. The video segments of this lesson run for 25 minutes, interspersed with six classroom activities for students. These activities involve partner discussions, extending algebraic formulas from the video, and working with provided social network datasets. The materials necessary are minimal: students will need the two sets of provided worksheets, their own pencil and paper, and some long pieces of string or yarn for Activity 3 (the number of pieces will vary depending on the size of your class).

Katrina LaCurts is a lecturer in the Electrical Engineering and Computer Science Department at MIT. She spends most of her time teaching undergraduates about communications, networks, operating systems, and security, but also dabbles in math education for high school students. Katrina completed her PhD in computer science at MIT in 2014. Website: http://people.csail.mit.edu/katrina/

A good resource for students and teachers on the Friendship Paradox itself is the following blog post by Presh Talwalker.
http://mindyourdecisions.com/blog/2012/09/04/why-your-friends-have-more-friends-than-you-the-friendship-paradox/

One additional resource for after the lesson - for both students and teachers - is another BLOSSOMS lesson: "Is Bigger Better? A Look at a Selection Bias that is All Around Us".
https://blossoms.mit.edu/videos/lessons/bigger_better_look_selection_bias_all_around_us

For teachers, the original paper on the Friendship Paradox is "Why Your Friends Have More Friends Than You Do" by S. Feld, in The American Journal of Sociology, Vol. 96, No. 6. A copy can be found online here.
http://cs.marlboro.edu/courses/spring2010/statistics/wiki/wiki.attachments/Why_Your_Friends_Have_More_Friends_Than_You_Do.pdf

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This Lesson is in the following clusters: Graph Theory