I think you are speaking of influence. Influence (that is also generated by Culture starbases) can be used or has effects in several ways:
- Asteroid mines can only be mined when in your influenced area.
- You can flip enemy planets to your empire when your influence on them is bigger than that of the original owner. For that you must have very influential own planets nearby or build a couple of culture starbases.
- You get notified if foreign ships enter your influenced area.
- There are certain effects (from ideology when I remember correctly) that give you bonuses in your area of influence.
Did I miss something?
Anyway, apart from that there is no benefit as such from connecting your influence areas.