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Not too long ago - in March 2019 - some changes have been made to the Hot Network Questions (HNQ). Some of the new features are the limit on the number of hot questions from a single site, or limit on how long a question can be in the HNQ list. For more details, see the announcement on Meta Stack Exchange: Updating the Hot Network Questions List - now with a bit more network and a little less “hotness”!

One of the new features was that moderators can remove the question from the HNQ list. I will quote a part of the linked announcement:

We're putting the power in the hands of our moderators to remove questions that don't set a good example for their sites. I recommend each site have a meta discussion with guidance for moderators about when - if ever - a question should be removed.

Once a moderator excludes a question, it can't be selected again, so don't think of this as a temporary "hide this question" option. In general, we recommend that you exclude questions that attract negative attention to your sites, that is, questions that are controversial, start large amounts of debate or arguments or even edit wars.

For example, you can see that many questions have been removed from the HNQ list by moderators on Mathematics Stack Exchange. As far as I can tell, this hasn't happened for any questions on MathOverflow yet.

If you are curious to see what the recent HNQs from MathOverflow look like, you can check the Hot Network Questions chatroom, which has a feed with the HNQs. (Or you can simply search for mathoverflow.net in that room. There is also the room called Listing bounties and HNQs.1) Alternatively, you can use Data Explorer (SEDE), for example, this query. (Keep in mind that the data in SEDE are only updated once a week, so you won't see the questions which entered the HNQ since the last Sunday in this way.)2

Questions.

  1. Should in some cases MO questions be removed from the HNQ list? What types of questions should be treated this way?
  2. If a user spots a MathOverflow question in the HNQ list and they think that it does not belong there, what is the proper action they should take? Is flagging the question for moderators' attention and including a custom explanation a reasonable course of action?

Some additional remarks

  • To some extent, hot questions "represent" MathOverflow to users on other sites in the network, since those are the question that they see unless they visit it regularly. So depending on which questions are chosen for the HNQ, they might give users coming here after clicking on them a wrong impression about what MO is supposed to be. You can see that there are questions which are closed or were closed at some point and yet they were in the HNQ list - if a question got closed, then it's perhaps not the best representative for the site.
  • It seems that there is not that much traffic from the HNQ list compared to total visits of MO: Traffic from the list of hot network questions. In the past discussions about benefits and possible problems with hot questions, it seems that one issue that many users raised was the fact that they might cause some votes from users with insufficient knowledge of the problem at hand. For this reason, I'll mention once again this feature request: Prevent questions on Hot List from being upvoted by casual visitors (only rep is from association bonus).
  • In a way, regular users can also remove a post from HNQ simply by editing the title - since questions containing MathJax in the title are not allowed into the HNQ list. However, there might be some cases when a more natural title would be without any mathematical formulas. And this could possibly lead in some cases to editing wars. So in cases where removal of a question from HNQ is really needed, I would consider an intervention by a moderator as a cleaner solution (unless the title with MathJax is clearly better and more descriptive). This SEDE query lists questions which were in the HNQ list and MathJax was added to the title later - however, we do not know whether removal from the HNQ was the motivation the edit (at least partially).

1This room contains only questions from MO - however, both featured questions and HNQs are posted there. Since April 15 the HNQs can be easily distinguished - those are the questions posted by the feed called HNQ MO.

2I should explicitly mention that entering HNQ is recorded in the revision history/timeline/SEDE only since the end of February 2019, so you won't find information about older questions. The feed in the linked chatroom comes not from Stack Exchange but from the user who maintains it using their own resources. Recording the HNQs started in the first half of February, so you won't find much older posts there, either.

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    $\begingroup$ As a side note, I will mention that I have created the (hot-network-questions) tag here on meta. Perhaps it is useful to have this tag - as you can see there were some discussion related to the HNQs in the past. $\endgroup$
    – Martin Sleziak
    Commented Jun 24, 2019 at 11:54
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    $\begingroup$ Most of the HNQs shown by your query are actually quite reasonable and interesting questions. So I don't think this needs careful curation, just occasional attention. $\endgroup$
    – Joseph O'Rourke
    Commented Jun 24, 2019 at 12:16
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    $\begingroup$ I guess no formal guidelines, agreements, or anything else like that are really needed if the questions are handled on a case by case basis and (subjective, of course, but what opinion is objective?) common sense is exercised. Neither do we need to care too much what the outsiders may possibly think of MO if some controversial question/answer makes it to the list. Everybody knows that humans aren't perfect and, compared to some highly visible public figures (especially in the government), we will, probably, always look like angels with double halo and a spare set of wings. $\endgroup$
    – fedja
    Commented Jun 27, 2019 at 13:54
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    $\begingroup$ In case it's helpful, here is the equivalent thread on Physics Stack Exchange. $\endgroup$
    – Emilio Pisanty
    Commented Jun 27, 2019 at 23:47
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    $\begingroup$ @JosephO'Rourke As you can see from the link above, that's definitely not the case elsewhere on the network. Y'all are some lucky folks on this score. $\endgroup$
    – Emilio Pisanty
    Commented Jun 27, 2019 at 23:50
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    $\begingroup$ This is only a hypothetical case since the community has handled it responsively by closing it in 3 hours (and this is the utility of 8-hour delay for HNQ!), but I believe this "suicidal thought" question should be removed as fast as possible since it gathered many answers and upvotes, if it did enter the HNQ. $\endgroup$
    – Andrew T.
    Commented Jul 8, 2019 at 3:22

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I didn't see this question when it was first posted a few years ago, and am here now because of a comment on a recent MO question that suggests that upvotes from people who are not normally part of the MO community can have the effect of distorting the reputation system. I'm assuming that the appearance of an MO question on the Hot Network Questions list probably contributes to this phenomenon; I have a hard time explaining why (as of this writing) there have been almost 100 upvotes on my answer to a question about the unsolvability of the quintic, other than that something drove an unusual amount of network traffic to the question.

Occasionally we see questions here on meta-MO of the form, "Such-and-such a question was highly upvoted; why was my similar question poorly received?" If the reason for the high number of upvotes was that the question showed up on HNQ, then the high number of upvotes does not necessarily mean that the primary MO community thinks highly of such questions, but this is difficult to explain to people, especially if it's hard to prove that this is what happened.

So, an argument could be made that "widely accessible" but not "research-level" MO questions should not appear on the HNQ list. But I'm not sure that that judgment is easy to make, and especially not before the flood of upvotes starts to materialize.

EDIT 8/7/2022: Here is some anecdotal evidence this effect is getting more and more pronounced: In the past week, the top five questions (1, 2, 3, 4, 5) were all hot network questions. All of them struck me as getting significantly more upvotes than they would have without the hot-network effect. While this is not necessarily a terrible thing, I have mixed feelings about users who don't normally participate much at MO having such a powerful impact on our site. On the one hand, advertising cool math to a wider community seems like a good thing in general; on the other hand, it somewhat interferes with the primary purpose of MO.

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