My understanding of the influence mechanics is that each of your influence generators (colonies and Culture Starbases, for the most part, but also other starbases if you have a certain Benevolent ideology trait) generate some amount of influence per turn. This influence per turn is accumulated on the generator and produces an influence bubble of radius
R = [total accumulated influence]^(4 / 7)
based on a constant defined in GalCiv3GlobalDefs.XML. Your overall zone of influence appears to be the union of these bubbles, and your influence in each tile is equal to the maximum influence reaching it from any nearby influence generator rather than the sum of all the influence reaching it from nearby generators. There is an influence falloff constant defined in GalCiv3GlobalDefs of -0.6, a target bell curve divisor of 3, and a minimum influence to claim a tile of 0.03. The tile claiming constant is probably how the game determines influence radius rounding, and the target bell curve divisor makes me think that influence levels are supposed to form a sort of bell curve around each influence generator.
If your influence bubble is competing with another faction's influence bubble, then the tiles go to whoever has the greater influence in the tile.
A small sample data set, taken starting several turns into the game:
Homeworld influence, influence 1 tile away, 2 tiles away, 3 tiles away, 4 tiles away, 5 tiles away, 6 tiles away. Influence bubble radius
- 16, 12.7, 6.4, 2, 0.4, 0.0, n/a. 5
- 17.2, 13.9, 7.4, 2.6, 0.6, 0.1, n/a. 5
- 18.4, 15.1, 8.4, 3.2, 0.8, 0.1, n/a. 5
- 19.6, 16.3, 9.5, 3.8, 1.1, 0.2, n/a. 5
- 20.8, 17.5, 10.5, 4.5, 1.4, 0.3, 0.0. 6
Colony influence, influence 1 tile away, 2 tiles away, 3 tiles away, 4 tiles away. Influence bubble radius
- 10, 6.8, 2.1, 0.3, n/a. 3
- 11.4, 8.2, 3.0, 0.5, 0.0. 4
When n/a is listed, the tile at that point was not inside the influence bubble. These points have an influence bubble of radius 5 for the first 4 data points and 6 for the last; the influence bubble radii approximately match the expected radius under the assumption that the bubble radius is equal to the fourth power of the seventh root of the accumulated influence at the generator, though a bit of rounding is required and the rounding is not standard rounding or banker's rounding. It can clearly be seen that you are not losing a constant amount of influence from one radius step to the next, and the influence generated at the source is not evenly distributed out to the tiles adjacent to it, nor is this the case for each step out. Instead, it appears that the influence values in each step are following a bell curve of some description, presumably one which is defined by the target bell curve divisor of 3 and possibly by the influence falloff constant of -0.6 (though it is clearly the case that this is not a straight multiplier for determining each ring's influence level from the previous ring's influence level, that this isn't a simple exponent for the decay, and that this is not a constant falloff from the center).
More data would be required to obtain a more accurate picture of the influence mechanics.