Social Network Friendship Paradox

Problem / Observation:

The friendship paradox is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average. Prove it mathematically.


Update (23 Oct 2014):
Solution: Posted by Mike Earnest and Taz in comments!

Comments

  1. Mike Earnest8:21 PM, July 08, 2014

    If the friendship graph is (V,E), the average number of friends people have is

    (1/|V|) Sum_{v in V} deg v = 2|E| / |V|

    The average of the average number of friends people's friends have is

    (1/|V|) Sum_{v in V} [ (1/deg v) Sum_{w in v's friends} deg w ]

    = (1/|V|) Sum_{v,w in V, v friends with w} (deg w)/(deg v) (*)

    Consider the summation in (*). For each edge, with endpoints v and w, there are two terms, (deg v)/(deg w) and (deg w)/(deg v). Since a/b + b/a is always at least 2 (to prove this, start with (a - b)^2 >= 0), this implies that (*) is at least (1 / |V|)*2 |E|, completing the proof.

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    Replies
    1. Pratik Poddar4:29 PM, October 23, 2014

      Thanks Mike. This solution is so better than any explanation I have read elsewhere. Thanks.

      Delete
    2. Mike Earnest12:06 AM, October 24, 2014

      :D Thanks!

      Delete
  2. Anonymous12:23 AM, July 09, 2014

    The number of friends people have would not be a normal distribution rather it would be left skewed. So majority of people will have only a few friends while only a small number of people will have lots of friends.

    Thanks to this second group (people with lots of friends) are more likely to number in your friends list. When they are actually part of your friends list, they would significantly skew the average number of friends that your friends have. Hence, on average, your friends will have more friends than you.

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  3. Pratik Poddar5:32 PM, October 25, 2026

    Replying to Anonymous's comment, The number of friends people have is left-skewed, so most have few friends and a small group have many - Your intuition is right; here is the exact argument. If degrees are dᵢ and there are m edges, the average degree seen by following a random friendship is Σdᵢ²/Σdᵢ, because each person with degree dᵢ appears in that many friend lists. This is at least Σdᵢ/n, the population average degree, by Cauchy-Schwarz. Equality holds only if all degrees are equal. (Reply created using AI)

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