What base number does % influence increase modify?

In the colony/planet screen, near where it shows you the morale and the income/expenses, it shows you the Influence.  You can hover over this to find out the exact value that it is increasing.

This exact value is determined based largely on the population, but the planet seems to have something to do with it as well.  Additionally, influence is complicated by the fact that the more influence you have ine one area, the larger its effect is (not entirely sure if the influence value itself rises), such that if you have three planets in a star system your individual influence for each of them is/should be higher than if you only had two of those planets, or for instance if you had 3 planets in 3 separate star systems without overlapping influence spheres.

Willy knows more about it than I think most of us, including myself, but at least you can see what number it is increasing, even if we can't just point you to a formula.

Influence is, however, generally not worthwhile to increase on a planetary basis, as the primary use of it is in flipping enemy worlds, and influence starbases accomplish that goal so much better (so long as they're close to the enemy planets-adjacent tiles is best).  The secondary purpose, UP votes, is easily enough taken care of by either buying up everyone else's IP, or simply colonizing more planets.

In my game when I took a few planets from a wiped out civilization and so have my opponents, my planets were heavily affected by opponent's influence. In this case planetay influence boosting buildings helped me lower opponents influence from 3.91 to about 2.4.

Anyways, when I hover over the influence on the planetary screen where is the breakdown i see: Influence = 54, 25 from planet, +5 from population, +9 from improvements, + 9 from your influence ability, +6 from your civilization's ability.

The first 2 are pretty clear = planetary quality and amount of people. But what about the rest? What do they mean?

 All the planetary improvements give you the value in % e.g. influence +15%. What number of those listed above do 15% apply to?What is my influence ability and my civilization's influence? My race has a bonus to influence but it's about 46% in the stats screen.So i'm lost again.

Well, I'm confused

That's not actually your opponent's influence, it's simply the ratio of yours to theirs, or in that case, theirs to yours.  Since planets tend to flip once you reach 4.0, it's good you were able to decrease it.

I would expect improvements to be your embassies and such, and presumably they derive their % bonus from the planet value (which I haven't studied and can't claim it to be PQ, or anything related),  I'm getting conflicting numbers, so the two possibilities are that it derives it from planet+civ influence, or that it derives it from everything prior to improvements.

Your civilization influence value (the 6 above) is all of your planets as a whole, and this is what makes influence complicated, since it is then added in to each planet (which may affect your civ influence-not positive on that).

Your influence ability (the 9 above) would be the 46% you see in stats and graphs; it looks like this is applied to only the planet value, but that still gives us inconsistent results for your scenario.

Willy knows more than any of us, but even he gave up on understanding it completely, because it's so horrendously complicated.

Thank you! You made it a lot clearer. Well, now we'll wait for Master Willy maybe he'll enlighten me a bit more on this.

Sorry i'm not willy, influence probably is not worth the worry it brings some but not enough tourism revenue to throw alick at does not usually assimilate your planets unless your assimulating others the biggest effect is the galactic council which the krynn torians or thalans will probably dominate anyways if we are going to talk about people john 

I already lost as humans, trying to go for that influence win, beat down by an Altarian+that nasty insectoid race alliance.Couldn't protect myself against their forces, but was doing well influence wise.