The explanation of the methodology is quite good, but the discussion in the paper is designed to push the narrative that their algorithm tends to promote conservative / right leaning tweets more than liberal / left tweets.
The raw data is missing, along with data on the political leanings of those engaging with the tweets. From what I gather on the Internet, in the US and UK, the users skew to the left. (In the US, 60% of Twitter users lean Democrat, 35% Republican : https://www.pewresearch.org/internet/2019/04/24/sizing-up-twitter-users/).
As the paper notes: The selection and ranking of Tweets is influenced, in part, by the output of machine learning models which are trained to predict whether the user is likely to engage with the Tweet in various ways (like, reTweet, reply, etc). [SI 1.14]
I think The Economist expressed best when it took a look at Tweet favouring in 2020: The platform’s recommendation engine appears to favour inflammatory tweets https://web.archive.org/web/20200803093134/https://www.economist.com/graphic-detail/2020/08/01/twitters-algorithm-does-not-seem-to-silence-conservatives
Those inflammatory tweets are exactly the ones that are going to get engagement. As the paper notes, the only type of tweets they considered were: We then selected original Tweets authored by the legislators, including any replies and quote Tweets (where they retweet a Tweet while also adding original commentary). We excluded retweets without comment.. [p3. Results]. The rationale for excluding tweets without comment was: attribution is
ambiguous when multiple legislators retweet the same content [ibid]. I think there is an additional problem / bias - (I suspect, but have no data to support this) people will retweet without adding a comment if they agree / support the tweet, but are more likely to attach some editorial comment ("SO STUPID!!!") when they disagree / oppose the tweet. [Ambiguity note: I find the paper ambiguous on whether it is only "legislator" retweets without comment that are ignored, or if that includes any "engaged" retweet of a legislator's tweet / retweet without comment]
Since humans - like all animals - are evolved to watch and attend to (i.e. "engage with") wany real or perceived threat, tweets that are "oppositional" will garner more attention. Since (at least the US and UK) Twitter users are left leaning, they will engage with what they perceive to be threats - which is what the algorithm will serve up to them, which will come from the other side of the political aisle. QED. Or to repeat what The Economist said: The platform’s recommendation engine appears to favour inflammatory tweets
Why the paper is self serving:
In the main body, it argues: With the exception of Germany, we find a statistically significant difference favoring the political right wing. This effect is strongest in Canada (Liberals 43% vs Conservatives 167%) and the United Kingdom (Labour 112% vs Conservatives 176%).
Yet, when you look (in Canada) at the amplifications of individual legislators, you see the Liberals and Conservatives are (almost) perfectly mirror each other (Chart 1C) - i.e. the amplification of individual members of the Liberal or Conservative parties is pretty much the same, yet the group amplifications are very different. The paper explains that this "discrepancy" is explained in SI 1.E.3 (which, I think is meant to be SI 1.5.3).
It is easy to see that if amplification a(G) of a group G were a linear function of the amplification of individuals i ... [then the sum of] individual amplification parity implies equal group amplification" [SI 1.5.3] (substance of the quote, equations didn't come through)
However, our definition of amplification does not satisfy this requirement. To see why, consider the function f (G) = |UTG|, where TG is the set of Tweets authored by members of the group G and UTG is the set of users who registered an impression event with at least one Tweet in TG. The function f is a submodular set function exhibiting a diminishing return property f (G ∪ H) ≤ f (G) + f (H). Equality would hold if Tweets from groups G and H reach completely non-overlapping audiences. [SU 1.5.3] (Again, apologies for the not quite 100% quoting, but ... equation problems).
This means that you have a much wider range of tweets from the Conservatives than the Liberals (remember, I'm looking at the Canada result / conclusion). Recall, from Graph 1C, individual Liberal and Conservative legislators get about the same amplification, but when we aggregate the amplification by group, the Liberals get less amplification than the Conservatives. But, the aggregate is a submodular set function: if the Liberals are all sharing the same tweet ("Conservatives Evil!") then each individual Liberal will get their individual "amplification", but the aggregate tweet amplification will be for that one tweet and consequently lower because of the high overlap for that tweet; if individual Conservatives are tweeting all over the place ("Liberals Evil!" or "Crystal Skulls" or "We're not the Liberals!"), each Conservative will get their individual "amplification" (which, more or less, matches the individual Liberals), but the aggregate group tweet amplification will be higher because there is less overlap with the tweets. This leads to (at least) two different ways to spin: (1) Liberals are focused and on point, Conservatives are all over the place, (2) Liberals share only one voice, Conservatives have many individual voices.
Now, many countries (apart from the US) have multiple parties. The paper focuses on Comparing the amplification of the largest mainstream left- and right-wing parties in each country [SI Figure S1A] and ignores all the other parties. In Canada, there are 2 other parties listed (NDP and BQ, both are leftist - indeed, the BQ has higher amplification in Canada than the Conservatives). Why aren't the Left and Right aggregated together so we can see the Left / Right amplification? Why is the amplification provided for only individual parties, but then generalized as "the right-wing gets more amplification". The Liberals + NDP + BQ are 3 left voices vs the one Conservative voice in Canada.
We should ask what the binary left / right "amplification" was for other countries (as well) and not just the party amplification (and then present that as representative of the left / right amplification):
UK : 3 left + 1 right
Germany : 3 left + 3 right
France : 3 left + 4 right
Japan : 2 left + 3 right