Wow, the influence section of the Wiki could use some help.
First, influence calculation. I cannot confirm that this is actually how the game calculates influence amounts/borders, but as a working theory it, well, works :NOTSURE:
Each planet generates influence based on population, racial bonuses, and planet structures. This influence degrades as distance increases. Starbases generate their own influence, which degrades much faster with distance.
Influence bases: Influence form these degrades far faster than that from planets, as previously stated. The ring around the base is basically the maximum possible effective range. An influence starbase touching an enemy planet is FAR more effective than one which is several spaces from the planet. The influence bonuses on the base do NOT multiply your civ's influence in the ring, it magnifies the influence generated by the base. Influence bases don't precisely stack, but they are more effective in clusters
To generate borders: each source of influence (including bonuses, distance degradation, etc) is added together, generating a numerical value for your influence in each game space. Borders are drawn where a civ's influence is greater than a set value (one, I think). If one or more civs would have overlapping borders, the border is drawn to show where one civ's influence has a greater value than the others.
Flipping planets: A planet can flip if your influence is 4x greater than theirs on the tile the planet occupies. If you have low espionage or better on that race, you can highlight the planet and see how close you are to getting there. Once you get to 4x influence, planet flipping is a random event. Each turn, the planet has a small probability of flipping - I've personally held a planet at 80x influence for most of a game year before it flipped. Others have flipped in the first turn or two. If you see a planet with the skull and crossbones on it, it's ready to flip.
Assorted extras: what percentage of the total influence you hold determines how much of the background tourism income you get. In Dark Avatar and Twilight of the Arnor, there are various bonuses/advantages races can get inside their own influence border - i.e. the Yor limit everyone else's ships to 3 moves/turn in their influence, regardless of the engines on the ship.
As to your Drengin/Altarian thing, were the planets in question in your border, or inside Altarian space? It's possible your influence base was not quite enough to overcome Altarian influence. It's also possible you BOTH had the Drengin planet at 4x, and its just a coin toss as to who it flips to. I've never experimented that way.
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