> Also, purchases of followers usually happen in batches, so it is easy to observe the spikes in followers numbers.
I would disagree with this. It is very easy to have a steady stream of followers on instagram, I've run a few campaigns myself.
When an account is private, and promoted, instagram queues up follower requests to 100 at a time, and you can't see how many are queued but you can predict how many users have been attracted to the account. Bot accounts or otherwise.
(Note: this is part of a larger strategy of getting instagram's algorithm to self-promote your posts in the explore section, or as popular in a hashtag. As you can post something new and your new followers will like your post and then you can switch your account public soon after and it is boosted in activity on many feeds, attracting more people to your account)
The user - who is the influencer - account can manually approve the pending followers, or take their account public. When taking the account public, 100 pending followers are added automatically, and it has to be switched back to private to show the next pending followers, only for them to be added when the account is toggled to public yet again. If the account stays public, the other pending followers will stay pending indefinitely and the user can't see that they are pending.
Therefore, this gives the account the ability to control a steady growth of followers, 100 at a time, indefinitely. If they keep the account private during their actual campaign. The authors might be surprised that this strategy obtains more humans than bots.
The main thing here is that there are many ways to "buy followers". And I think the analysis here assumes way too much about a single way that followers are bought. The authors have probably never created their own influencer account, to know about the infrastructure that really exists.