Background:

I only have the basic game, without any expansions or DLCs, not even the one that I've heard now comes bundled with the base game.

Ever since I started the game, i wasn't happy about influence. For one I felt that it takes way too little time for it to completely cover even immense galaxies, and it's rather difficult to colonize (and keep) planets once they're inside anothers ZOI. Second, the ZOI defines tourism income, and the entire economy is totally out of whack because of this, unless you stick to very small galaxies.

I searched for information on the game mechanics, but for the  most part it's outdated, and anyway, nobody appears to have figured out the real mechanics in terms of formulas and dependencies from existing XML variables. Plus, apparently, many game mechanics were radically changed over several versions of the game.

There is one XML variable that was explained in some forum post: InfluenceMinToClaim decides which tiles are counted to your ZOI - tiles with influence values greater than InfluenceMinToClaim are assigned to your ZOI, tiles with smaller values are outside. This threshold value is needed because the formula for determining influence at a given range R is designed to always give a (small but) positive value.

The formula

I decided to record some data and try to figure out the formula by myself that governs the amount of influence G that one source with a given value I at its center has on tiles at a range of R. On top of that I tried to figuer out how the XML influence values A=InfluenceRadiusRoot, and B=InfluenceFalloff from GalCiv3GlobalDefs.xml would figure into this formula.

After some trial and error I figured out this formula, which manages to reproduce the observed values from my test game within 1%. (I suspect the differences come from internal rounding effects within the game):

G(I,A,B,R) = I*e^(B*R/(I^(1/A))) , where e is the Euler Number

You can find a Google sheet with the observed values and the calculations using this formula here

Interpretations - what does each variable mean?

The value A=InfluenceRadiusRoot governs the linearity of the influence value at the center with respect to the bell-shaped curve describing the influence value over the distance to the center: a value of 1 would mean that increasing the influence by a factor of K at the center would likewise increase values at a distance of R by the same factor. A factor greater than 1 means that at a distance greater than 0, the influence value would increase by less than K.

The value B=InfluenceFalloff simply describes how quickly the influence drops with the distance from the center. The Google sheet shows some of these curves for the default values for Galciv 3 base game V 3.6. These curves match the values I observed.

You can copy the sheet and change A or B to see how this affects the curves.

P.S.: There's a lot more worth knowing, but the post is already a bit longish. Of course, if there's interest, I can point out a few more things I've found.