In my current game I had a cluster of colony worlds bordering another race.  I built a comms starbase to bolster my influence in the region.  Now I know that starbases don't impact the outputs of colonies, however I also built the colonial beacon upgrade on the border colonies.  The influence output of the colonies didn't appear to change.

Also looking through the ideology trees I came across the representation perk which states "all colonies generate influence".

So I suppose how does influence spread/generation works in terms of colonies?  Is the colonial beacon a waste of an upgrade?

Thanks in advance