Biggest mistake I did with my Instagram account was to link it to my Facebook. I believe it is impossible to unlink them, even having disconnected all external application access as far as I could go in both.
Why? "Your friend Xxx is now on Instagram". Etc etc. I don't want to know, and I don't want my account visible to everyone there. Please just respect my wishes to make the integration go away!
I recently spent a week out of any mobile signal range and when I got back, I had email from FB (normally I don't get notifications by email, just in the app) for the most banal things like "Friend X has just uploaded a photo!". FB definitely enters panic mode if a regular user doesn't use it for a short while.
Yes, it does this when you stop using it, also it will spam you with "do you know person Foo?" messages. I guess is because they assume you stoped using it because your timeline was boring.
Source: I have the same experience.
I don't use Facebook that much, but I definitely used to associate a notification with something important, like being tagged in a status / photo or invited to an event. Since a year or two ago for me as well, it's usually useless or sometimes a straight-up advertisement. If you admin a group, you'll even get the "boost this post" messages as notifications.
Maybe this is because I stopped using Facebook for so long, there weren't any real notifications to give me. They wanted it to seem like I'd missed something.