@Queen
It's *mostly* random! The cat that initiates the scenario is random, but those who participate in the same scenario are influenced by the initiator's stats (low extroversion, for example, means that cat will primarily have solo scenarios). Cats prefer to hang out with others they already know and are friends/family with.
The best way to influence scenarios is by sending cats traveling as traveling cats can't participate in them. However there is a week cooldown before you can bring them back, so it's up to you whether or not you want to lose the gathering power for some relationship chances.
There's also a few items crafted through Scribe recipes that can make your cats friends, rivals, best friends, or partners, if you're interested in those.