Update: Ahahaha… okay. Thanks to the update which allows me to see exactly how much influence I’m getting and how much it degrades by, I can tell you with some certainty that it degrades WAY too fast.
I did two identical patrol quests for the same faction nearly back to back. I made sure to try to complete the second patrol quest with as little delay as possible. But by the I completed the second patrol quest, the influence I had gained from the first patrol quest had already evaporated into 1 point (51 influence), almost completed gone. The patrol quest only gives 6 faction influence in the first place. Keep in mind I started the second patrol quest very soon after the first one.
So I have to repeat this quest 50 times in order to reach 100 influence with this faction, making sure to stop for nothing and not actually hunt bandits along the way (although having said that, I killed 30 bandits along the road somehow the second time I did the quest. That road is infested with bandits). If I dally a little to hunt bandits who were on the road but run away, I will probably be stuck at 50-ish influence with the faction no matter how many times I do the quest for them.