Relatively old thread, but decided to respond because I was having trouble figuring out the level/adjacent thing too... Especially during a game where, researching Hives, and then giving the Hive the "Coordinated Hives" tech, I wasn't sure it was working right. The Coordinated Hives tech reads "Increases the adjacency bonus of all improvements / Adjacency Level +1". I assumed that this is improvements adjacent to the hive, not all improvements on the planet, since this is a specialization tech pertaining to hives.
The Yellow number on the improvement is the level. It looks like regular improvements start at 0, get a level when you upgrade it, and adjacency bonuses improve that level as well. For instance, I have a mega factory at level 7 (stated on the yellow number 7 when clicking the improvement, and in the tool tip when mousing over it, "Level 7", Manufacturing +35%. The adjacency values (+1 for my mega factory given to others next to it) are what the clicked improvement gives to those around it. To figure out the adjacency bonuses from other things to the factory, you have to do some addition. So, my level 7 mega factory gets +3 from a durantium refinery, and 2 from the two factories next to it. That's only 5, so we can surmise that basic factory is level 0, xeno factory is 1, and mega factory is 2. So 5 adjacency levels plus 2 for being of the mega tier gives level 7.
Back to the hive specialization tech I selected, Coordinated Hives. My assumption that the Adjacency Level +1 was given to any improvements next to the hive seems to be incorrect. Unfortunate, because I planned to drop those suckers in the middle of some important non-manufacturing buildings to boost them all. However, using the logic above with the mega factory, it seems the hive (getting adjacent bonuses only from manufacturing improvements) has an additional level now. It's next to a Solar (+2), and two factories (+2), giving a lvl 6 Hive. Therefore, hives must start at level 1 (not 0), and since I boosted my hives +1, it is level 6.
The hive level 6 gives a 30% boost to production, which is amazing, but the tech text should be updated to be more clear. Or the game should be more clear in general - an adjacency level is a level, and a level (according to the hive specialization text) is an "adjacency bonus". More than a bit confusing.
For someone not clear on Hive placement: get the level of your hive as high as possible with manufacturing improvements. Each hive level gives +5% to production.
I think the text of the tech should be updated to say "Increases the level of all hives +1", because that's what it does. It does not give +2 to adjacent improvements, still just +1. Which is very nice, but not expected based on the tech's explanation.
Now if you'll excuse me, I have some realignment of planet economies to attend to. *mumbles to self* hives in the middle of consulates and farms...