For example, a computer modeling its player's avatar sitting in a vehicle is going to be most interested in the state of the avatar and the vehicle they're in. Computers whose players' avatars are in the vicinity and likely to be affected, will be the next most interested. And so on. We have to continuously determine which computer is most interested in each item of state, and transfer ownership accordingly. Of course, we have to stop this process from thrashing, so the benefits of ownership have to be balanced against the cost of transferring ownership.
Interest is a measure of relevance of an item of state to the player.
Things are more relevant the more they improve the quality of the player's experience, i.e. for good modeling of their avatar's local scenery and its denizens, presenting good images of the avatar's senses, and conveying the player's influence over their avatar. The player will have most interest in controlling the avatar and seeing through its eyes. And the avatar in turn will have an interest in the scenery, in fact anything that is in range of its senses. A computer will express interest in state according to the needs of the player. An interest is a set of criteria designed to match a set of interesting objects as opposed to uninteresting objects.