I suspect that the game adds up a total of "civil war points" every turn and compares that to a trigger number to see if the war triggers. I'd imagine that several things combine into that total: imperium, family influence, the influence of the other families within the faction, and probably ambition, but I can't say for certain what is most important. Here's a bit of a breakdown on my current game. It is 50 BC, and the civil war has not occurred yet.

Game is normal/normal, as this is my first full campaign and I've been taking it very slowly learning about everything along the way. Most of the territory I've picked up has been from snagging rebellious provinces while the few war declarations have mainly been at the request of military allies or client states. I'm playing as the Julii, my current senate influence is 61% (down from 74%), my two good generals are rank 3 and 2 with gravitas 79 and 58 respectively. The Cornelii are at 11% influence, the Junii are at 10% influence, and the other houses are 17%. The drop in senate influence has come from several adoptions and a couple of promotions, trying to trigger the war. I've got 44 regions at the moment (like I said, taking it very slowly). I am into the fourth imperium level, but I've been there for 20 years plus at this point.

The slow pace of this campaign has had a couple of consequences, one being that nobody has gotten huge amounts of gravitas/influence and no general has been beyond rank 4 to date. Since they've mainly been dying of old age (very old age sometimes, one general was still leading his army in the field at the age of 92) those who do get high gravitas tend to die shortly thereafter. Only now have I been pushing it; my leading family general is making 4 gravitas per turn and is only 37 years old. I've decided to finally conquer Iberia and he's leading the charge, taking all the glory himself. The thing is, he is ambition 2, while my other guy getting 3 gravitas per turn is only ambition 1. I could not adopt any ambition 3 generals who didn't have any negative gravitas per turn traits.