Talk:Friendship paradox

This concept needs to define "friendship" in connections/nodes to have any merit at all
It purposes plays of the nebulousness between "friendship" being a one or two way connection. I am a proponent for deletion of this article.

66.116.62.178 (talk) 19:27, 27 August 2013 (UTC)
 * On the contrary, it explicitly assumes (for the purposes of mathematical modeling) that it is only a two way connection and, rather than being nebulous, consists of rigorously proven mathematics. What reason do you have for thinking it should be deleted? —David Eppstein (talk) 20:24, 27 August 2013 (UTC)

Friendship paradox explanation
I would like the following comment regarding the "solution" of the paradox:

The paradox is not explained as described because if one thinks that it is more likely to be friends with someone who has more friends, the same thinking will be also done by that someone! So that sort of thinking is self-contradictory! —Preceding unsigned comment added by Alsims (talk • contribs) 11:11, 18 August 2010 (UTC)


 * It's not a contradiction, because it's only true in general. There will be individuals whose friends have fewer friends than they have, but there will be many more whose friends have more friends than they have. MartinPoulter (talk) 12:42, 18 August 2010 (UTC)

Graphical explanation?
I think it would help to make it clearer if the article had one image (or more) showing how most people in a sample can have less friends than their friends. --TiagoTiago (talk) 00:39, 25 October 2011 (UTC)
 * Couldn't agree more, this is practically screaming for a graph. --94.221.119.210 (talk) 18:55, 7 January 2012 (UTC)

Examples where the law fails?
Right now the article makes the following claim: "there exist undirected graphs (such as the graph formed by removing a single edge from a large complete graph) that are unlikely to arise as social networks but in which most vertices have higher degree than the average of their neighbors' degrees." I don't believe this is true. It certainly contradicts the formula above, because mu and sigma are positive. Can someone provide an example where this statement is true? --keepinitraw, 29 February 2012 —Preceding undated comment added 17:48, 29 February 2012 (UTC).
 * A very explicit example is given in the sentence that you quoted. Let G have n vertices, with one pair st not connected and all other pairs of vertices connected, and with n ≥ 5. Then most vertices have degree n-1 and their neighbors have average degree ((n-3)(n-1)+2(n-2))/(n-1) < n-1. The only two vertices for which this is not true are s and t. —David Eppstein (talk)

I see, however, some holes in here
First of all, this paradox is driven, by the assumption that there is such a thing as "friend of a friend", thus: a friend in the second degree; this is - of course - wishful thinking. Most second degree, or third degree 'friends' we rarely have contact with. The assumption of 'mutual friendship' is also hypothetic; this need not be the case.

Second, this - mathematical - phenomenon is highly influenced by the possibility of one individiual having many friends: this is explained by 'so-called' popularity; this also - of course - a very hypothetic assumption. Most people with large networks have very few real friends, to be realistic.

In short: we are considering mathematical proportions of unbalanced networks; we are talking about nodes, not real friends. So we must skip the philosophical bit; we are not talking about real people, we are talking about nodes, and degrees. We are not talking about real friendships here.

We are talking about the behaviour of networks, formed by Social Media, not Social Behaviour. That is the real paradox, here.

Misuse of phrase "average person"
Does anyone see a difference between the phrases "Average number of friends" and "Average person's number of friends"..? Given: The average number of friends is (1+2+6)/3=3. The average person is Curly, who has 2 friends; thus the average person has 2 friends.
 * Larry has 1 friend
 * Curly has 2 friends
 * Moe has 6 friends

When I saw this on a slate.com article, it read "most people have fewer friends than their friends," which is ambiguous. The whole "paradox" is built up from a misunderstanding of the best metric for the situation, which is median rather than mean.

Who's with me?

A quick google netted this gem:

"In everyday language, we use the word "average" to mean "most people," or the most representative person (as in, "The average person doesn't read classic literature" or "The average Joe can't afford to dress like Prince"). But then when they start using the word "average" to talk about statistics, you get weird results, like the fact that 67 percent of people in the USA make less than the "average" income."


 * Your use of "average" conflates arithmetic mean (as in "average number") and "median" (as in "average person"). Please note that the paradox talks about "most", which refers to mode (statistics). Paradoctor (talk) 05:31, 17 January 2014 (UTC)

"Intuitive explanation" Original Research
I feel the section may contain original research. 92.4.96.96 (talk) 20:02, 28 May 2016 (UTC)

Wrong mathematical explanation
The explanation about the average number of friends that a typical friend has is obscur and wrong.

As described, if we randomly pick an edge, we have more chance to have a "popular" node, and thus the final result is biased by construction. The original paper didn't mentioned any thing like that to explain the formula.

But more important, the formula is wrong. I made a few tests on random graphs and samples from facebook, twitter and google plus. The prediction was far above the actual result. The article was apparently more focused on sociological results and the maths are not reliable.

I made a calculation myself, the average number of friend can be determined by the formula
 * $$\mu_2=\frac{1}{|V|} \sum_{(u,v) \in edges} \frac{d(v)}{d(u)}$$

where edges is the set of all oriented edges.

With non-oriented edges, it becomes
 * $$\mu_2=\frac{1}{|V|} \sum_{(u,v) \in edges} \frac{d(v)^2 + d(u)^2}{d(u)d(v)}$$

And there, because $$x^2+y^2 >= 2xy$$, we have
 * $$\mu_2 \ge \frac{1}{|V|} \sum_{(u,v) \in edges} 2 = \frac{1}{|V|} \sum_{u \in V} d(u) = \mu$$

This proof seems to be correct, and gives experimental results.

So I think we should clean away the wrong part in he mathematical explanation paragraph. The proof I presented is an original content, but any mathematician can easily check.

Kaylier (talk) 03:49, 23 November 2017 (UTC)
 * Your formula for average number of friends is completely bogus. For instance, consider the star network $$K_{1,n}$$ with one degree-$$n$$ vertex and $$n$$ degree-one vertices. Its average number of friends is $$(1\times n + n\times 1)/(n+1)=2n/(n+1)\approx 2$$. Your formula gives $$\tfrac{n}{n+1}(\tfrac{1}{n}+\tfrac{n}{1})=\tfrac{n^2+1}{n+1}\approx n-1$$ (where the first numerator and denominator are the numbers of edges and vertices, and the two fractions 1/n and n/1 are the values d(u)/d(v) and d(v)/d(u) for each edge). —David Eppstein (talk) 00:38, 30 November 2017 (UTC)


 * I may have not been clear in my explanation. The paradox highlight the difference between the average amount of friend of someone, and the average amount of friend of his friends. I agree with the computation of the first one, resulting in $$2|E|/|V|$$ (what you found). I however disagree with the computation of the second one (I should have written "friend the friends have" instead of "friend" in my explanation).
 * In your example, the average number of friend of friend is $$(n\times n + 1 \times 1)/(n+1)$$ (the $$n$$ degree-one vertices have one friend with $$n$$ friends, so a mean of $$n$$, and the one degree-$$n$$ vertex has $$n$$ friends with one friend each, so a mean of $$1$$). The formula in the article predict a value of $$(n^2+n)/(2n+2)$$ and is wrong whether or not my formula is right. Kaylier (talk) 04:21, 1 December 2017 (UTC)
 * The sourced explanation in averaging over random pairs of friends. You are instead doing an average of averages (averaging the number of friends-of-friends over the friends of each person, and then averaging over people). It might be reasonable to analyze it the way you do, but it's a different model, and it's not what's in the source. —David Eppstein (talk) 04:28, 1 December 2017 (UTC)
 * (I've combined the relevant parts of my comments into this single comment) Actually, the original source also considers the quantity "Mean number of friends of her friends" for person $$u$$ is $$ \sum_{v : (u,v) \in edges} d(v)/d(u)$$", and also considers its average over all $$u$$. Hurkyl (talk)


 * We can argue about the definition of 'The average number of friends of friends'. But either way there's an issue with the paragraph that suggests that the direct calculation (sample a node and average his neighbors' degree) is equivalent to the alternative calculation of sampling an edge and take the degree of one of its nodes. IIUC, the paper mentions the alternative calculation as a proxy/intuitive calculation, but doesn't say that they are equivalent. Moreover, he specifically mentions (and shows) they are not equivalent, both in the Table 1 (in the paper) and in its following sections. Calculating these two expectations for almost every non-symmetric graph will give two different results (even for the simple graph of 3 nodes and 2 edges, where the first calculation suggests 'The average number of friends of friends' equals 5/3 and the alternative calculation suggests 3/2) (User: braudeguy) 06:10, 12 June 2020 (UTC)

Removal of NYU article
Hi Fountains of Bryn Mawr, I'm unclear why you removed the NYU citation as "trivia." It also doesn't seem to be primarily associated with a person's assessment of their traits versus those around them, but a person's popularity versus those around them. From the abstract:
 * We report on a survey of undergraduates at the University of Chicago in which respondents were asked to assess their popularity relative to others. Popularity estimates were related to actual popularity, but we also found strong evidence of self-enhancement in self-other comparisons of popularity. In particular, self-enhancement was stronger for self versus friend comparisons than for self versus "typical other" comparisons; this is contrary to the reality demonstrated in Feld's "friendship paradox" and suggests that people are more threatened by the success of friends than of strangers. At the same time, people with relatively popular friends tended to make more self-serving estimates of their own popularity than did people with less popular friends. These results clarify how objective patterns of interpersonal contact work together with cognitive and motivational tendencies to shape perceptions of one's location in the social world.

https://nyuscholars.nyu.edu/en/publications/what-makes-you-think-youre-so-popular-self-evaluation-maintenance

Curious what your thoughts were or what part of the paper led you to that conclusion.

Squatch347 (talk) 18:36, 14 December 2018 (UTC)
 * I think this material should stay. It explains in what sense this is a "paradox": because it contradicts common beliefs. —David Eppstein (talk) 18:58, 14 December 2018 (UTC)
 * The edit was about the article staying on topic. Article leads summarize the content of an article body per WP:MOSLEAD. The body of this article does not cover Zuckerman / Jost / Illusory superiority, nor should it, that is a different topic. Insertion in the lead with the (editor?) noting some interesting "contradiction" makes it more trivia than anything else, its unimportant to the topic and doesn't even show up in the article content. Claiming that this explains the "paradox" is incorrect. The "Friendship paradox" is a mathematical paradox that turns up in statistical analysis. It is not related to the the study of perception or "common beliefs" in the Zuckerman / Jost paper. The authors just use the mathematical results as a basis for their study. Fountains of Bryn Mawr (talk) 21:08, 14 December 2018 (UTC)
 * I think it's highly relevant that people expect the answer to be the opposite of what the paradox states, and I don't understand why you think it should be removed altogether from the article. The reference for that material makes clear that it is explicitly about the friendship paradox, and not just general material on illusory superiority: (1st sentence of 3rd paragraph in intro) "By distinguishing between two targets of comparison (friends and typical others) it is also possible to extend Feld's (1991) paradoxical demonstration that most people must have fewer friends than do their friends". —David Eppstein (talk) 21:25, 14 December 2018 (UTC)
 * As I posted the last time this argument failed to win support, mulitiple sources explicitly tell us that it is a common misconception that people should have about as many friends as their friends. The sources tell us so -- right in the title "What makes you think you're so popular?" It directly references a common misconception. The first sentences says it is a way "in which people are biased and inaccurate in their perceptions". It references "people's estimates of their own popularity". "Against all expectations it turned out...". It's right there in the sources. No sources dispute that it is a common misconception.If Fountains of Bryn Mawr's problem is that it doesn't conform to MOS:LEAD, the correct thing to do is FIX IT. Add the necessary material, fully fleshed out and detailed, in the body where it belongs, and make sure the lead has a succinct and adequate summary of that. MOS:LEAD is not an excuse to delete content entirely from articles; you have a burden to move it to where it belongs if your real beef is that you want the best article layout and organization.  --Dennis Bratland (talk) 21:58, 14 December 2018 (UTC)

"I think it's highly relevant" - that's fine original thought but we are not to reach or imply a conclusion not stated by a secondary source. How the Friendship Paradox Makes Your Friends Better Than You Are is a secondary source but it does not show a specific relevance (or even mention) Zuckerman / Jost. Scholarly papers always cite other papers (example) but you can't claim one is more relevant than the others - a secondary source has to make that claim. WP:FIXIT implies there are secondary sources to follow to "fix it", and you can't hide behind that if you are unwilling, or unable, to supply secondary sources. This has nothing to do with the "common misconception" argument, its basic MOS and WP policy, why was a seeming non sequitur added to the lead. So feel free to fulfill WP:BURDEN on this one. Fountains of Bryn Mawr (talk) 20:09, 15 December 2018 (UTC)
 * Claiming that our opinions on the relevance of the source are "original research" is the last resort of the wikilawyer. It is a bad argument, one that does not address the substance of the issue. It is not the sort of thing our original research policy is actually about, and it is based on a false premise: the connection between the source and the topic of the article is factual, explicit, and directly quoted above, not merely editors' opinion. —David Eppstein (talk) 20:41, 15 December 2018 (UTC)
 * WP:PRIMARY give specific policy on primary sources, re: Any interpretation of primary source material requires a reliable secondary source for that interpretation. So it is exactly what the original research policy is actually about. Crying "wiki-lawyer" when you can't meet a Wikipedia principle is considered a bad faith tactic (see Wikilawyering). Fountains of Bryn Mawr (talk) 21:50, 15 December 2018 (UTC)
 * You keep changing the goalposts. First it was irrelevant trivia; then when that became debunked it became original research; then when that became debunked it became primary rather than secondary. Which is it? —David Eppstein (talk) 22:11, 15 December 2018 (UTC)
 * All the problems have the same root, its a primary source rather than secondary source. If you interpret a primary source without support for that view from a reliable secondary source it comes off as trivia. If you interpret a primary source its original research. Fountains of Bryn Mawr (talk) 01:54, 17 December 2018 (UTC)
 * I don’t think we need an RfC or any other dispute resolution venue. It has been thoroughly discussed and consensus has been reached. What we should do now is request a third opinion on this question: Has consensus been established? If no, then I guess the debate rages on. But I think the answer will be yes, at which point we can put this behind us. Dennis Bratland (talk) 23:11, 15 December 2018 (UTC)
 * "Consensus is ascertained by the quality of the arguments given on the various sides of an issue, as viewed through the lens of Wikipedia policy." Since the relevant policy here is WP:SECONDARY there has not even been an attempt to ascertain consensus. The only argument put forward so far is "my interpretation of a primary source is valid, go away". So we are still as WP:BURDEN. Fountains of Bryn Mawr (talk) 01:54, 17 December 2018 (UTC)


 * Agree with Dennis Bratland and David Eppstein, there seems to be little profit in haggling over the same issue already resolved. The consensus has already been reached on another page with more support for its conclusions coming here. The paper explicitely discusses the Friendship paradox, why it is relevant (rather than just a statistical quirk) and an application.  That would certainly be suitable to an encyclopedic entry.  Squatch347 (talk) 16:10, 17 December 2018 (UTC)
 * "The paper explicitly discusses the Friendship paradox" - there are (9?) papers cited in this article that "explicitly discuss" the "Friendship paradox". We do not pick, chose, or comment on their results, that is the job of secondary sources. You need those first, you can not have an agreement or a vote to ignore Wikipedia policy. Anyway, cleanup is obvious. Fountains of Bryn Mawr (talk) 22:45, 2 January 2019 (UTC)
 * We don't need secondary sources about our secondary sources to tell us which secondary sources to choose. That way lies madness. —David Eppstein (talk) 22:52, 2 January 2019 (UTC)
 * Zuckerman/Jost is a primary source. Any interpretation of primary source material requires a reliable secondary source for that interpretation. It actually doesn't get any clearer than that. Fountains of Bryn Mawr (talk) 00:17, 3 January 2019 (UTC)
 * Because a paper is published in a journal does not mean it is a primary source. Zuckerman/Jost are summarizing and discussing the findings of surveys other researchers conducted.  What's more, the most relevant part of the article is a summary of other papers, thus a secondary source.  Squatch347 (talk) 15:10, 3 January 2019 (UTC)

Its an empirical study and therefor a primary source. Such studies cite other studies but they are not "reviews". Per WP:V you need a "third-party" source (another source looking at these sources) for any type of interpretation. Fountains of Bryn Mawr (talk) 02:42, 4 January 2019 (UTC)
 * You appear to be thinking that sources themselves can be classified as purely primary or purely secondary, independently of how they are used. That is a mistake. Many sources are primary for some material and secondary for other material, depending on what parts of the sources one uses and what claims one is using them for. But that's all irrelevant now that I've added three more indisputably-secondary published book sources for the same material. So why are you still arguing about it? —David Eppstein (talk) 04:22, 4 January 2019 (UTC)
 * You supplied three sources, all confirm Zuckerman/Jost is Illusory superiority, only one ("Interaction in social networks") covers Zuckerman/Jost and the "Friendship Paradox", and it specifically refutes your claim that Zuckerman/Jost is the "explanation for why this should be called a paradox" i.e. the illusion Zuckerman/Jost noted "overcome(s) (the) paradox", its not the paradox. Fountains of Bryn Mawr (talk) 02:24, 5 January 2019 (UTC)
 * Perhaps you could try rewriting that in English? Je ne le comprends pas. And again, they are not intended as sources about the sources about the sources, so whether they "cover Zuckerman/Jost" completely misses the point. They are intended as sources for the claim that people tend to overestimate how popular they are. —David Eppstein (talk) 02:54, 5 January 2019 (UTC)
 * They are intended as sources for the claim that people tend to overestimate how popular they are. .. that part was actually not in question but the sources are useful. Since the article already has a structure to cover this and there is more than ample secondary sourcing Wikipedia recommends editing over endless debate. Fountains of Bryn Mawr (talk) 02:30, 11 January 2019 (UTC)

Lead
The lead doesn't fit the article, as it doesn't mention the mathematical explanation, but rather suggests an alternative conclusion.--Jack Upland (talk) 22:26, 2 January 2019 (UTC)
 * One step (which I was about to do) was move all the alternative application stuff to Applications. The lead should then be rewritten to summarize "Mathematical explanation" and "Applications". Fountains of Bryn Mawr (talk) 22:32, 2 January 2019 (UTC)
 * Per, took a whack at summarizing the paradox itself and getting everything in order. Also expanded the article a bit, related it to sources, and added more sources. Fountains of Bryn Mawr (talk) 02:30, 11 January 2019 (UTC)

There seems to be a back and forth in the edits between Fountains and DavidEpstein. Perhaps to help resolve, Fountains, can you post your recommended text here and we can discuss for a consensus? Squatch347 (talk) 17:18, 13 January 2019 (UTC)
 * Fountains has already made a recommendation for the text. See the article history. It is unreadable, full of run-on repetitive sentences like "The friendship paradox is a statistical phenomenon in social networks first observed by the sociologist Scott L. Feld in 1991 that, instead of everyone having an average number of friends, individuals actually have fewer friends than their friends have, on average". It also removes from the lead the part that most people think they have more friends than their friends do. The contradiction between this and the mathematical fact that most people actually have fewer friends than their friends do is what makes this paradoxical. —David Eppstein (talk) 17:27, 13 January 2019 (UTC)
 * most people think they have more friends than their friends do. The contradiction between this and the mathematical fact that most people actually have fewer friends than their friends do is what makes this paradoxical. Well that's obviously wrong. The Friendship paradox was a paradox in the 1991 Feld paper, 10 years before the 2001 Zuckerman/Jost "people think they have more friends than their friends" paper. The difference between those two findings is not the paradox. Feld named the paradox in his paper: "friends with many friends" "class size paradoxes" i.e. Compare the number of friends one person has to the average number of friends their friends have, and the second number is always bigger. Most of us only have a few friends, but some people that have tons of friends. It’s those super-popular people that creates the paradoxical effect...... all reliable sources say that, the paradox has nothing to do with Zuckerman/Jost work. Fountains of Bryn Mawr (talk) 03:27, 14 January 2019 (UTC)
 * The phenomenon of people overestimating their number of friends did not begin with the ZJ study, that's just one of the references we're using for it. —David Eppstein (talk) 04:44, 14 January 2019 (UTC)
 * The phenomenon of people overestimating their number of friends is also not the friendship paradox per every source currently in this article and . Fountains of Bryn Mawr (talk) 20:51, 14 January 2019 (UTC)
 * Ok, now you're saying things that are true but irrelevant in what appears to be attempt to baffle rather than shed light. The "paradox" is the phenomenon that most people have fewer friends than their friends do. The information that you're trying to suppress or minimize is that most people think they have more friends than their friends do. It is not the paradox itself, but it helps explain why people find the paradox surprising. If you don't see the formal similarity between the two statements, and the relevance of one to the other, then try harder, because it's not a difficult thing to understand. —David Eppstein (talk) 21:02, 14 January 2019 (UTC)
 * As has been said by me and several others in previous iterations of what is essentially the same debate in multiple venues, I agree. The position Fountains of Bryn Mawr has been driving on all these threads has not won over the support of any other editor, while several said they disagree, and have explained -- more than enough times -- why. It's time to drop it. --Dennis Bratland (talk) 21:25, 14 January 2019 (UTC)
 * So lets get some things straight. Agreement that the Feld under-sample is the paradox and that is a mathematical paradox, despite what Dennis Bratland said (reading comprehension?). Looks like we are getting somewhere. Fountains of Bryn Mawr (talk) 22:59, 14 January 2019 (UTC)
 * The list you link to explains it includes informal paradoxes, such as non-intuitive results, not only formal paradoxes when logic contradicts itself. —Dennis Bratland (talk) 23:11, 14 January 2019 (UTC)

Mathematical explanation is incorrect; true explanation is simpler and recently published
Choosing a random node and then a random neighbor of that node does not correspond to choosing a random edge (where "random" here means "uniformly random").

Choosing a random edge in some sense exaggerates the effect, as can be seen from the following simple example: the union of two disconnected components with equal numbers of nodes, one of them a complete subgraph and the other a cycle. Here, a random edge is much more likely to be in the complete component, but a random neighbor of a random node is equally likely to be in either. And indeed here a random node has the same number of neighbors as any of its neighbors do.

The correct proof (assuming friendship is modeled by an undirected graph without self-loops) is extremely elementary and deserves to be highlighted in this article. It was recently published here; see the first two paragraphs of Section 2, which could be copied almost verbatim and cited appropriately. It would be great if someone would like to correct this section. If not, I can do it myself.

216.180.92.59 (talk) 16:36, 30 June 2021 (UTC)
 * It's randomly selecting an edge of the selected person, not any edge, as far as I understand. --mfb (talk) 05:55, 1 July 2021 (UTC)