Here is the difinitive answers.
Influential does absolutely nothing. Nothing at all. Stop saying it does. It does nothing. Nada, Zilch, Zip.
Custom Sov, Custom Faction, No picks whatsoever.
Turn 1 - Build City 0 Citizens - 0 Influence
Turn 2 - 1 Citizen - +1 Influence (1 Influence total)
Turn 13 - 12 Citizens - +1 Influence (2 Influence total)
Turn 27 - 26 Citizens - +1 Influence (3 Influence total)
Turn 31 - 30 Citizens - +1 Influence (4 Influence total)
Turn 102 - 101 Citizens - +1 Influence (5 Influence total)
Turn 106 - 105 Citizens - +1 Influence (6 Influence total)
Turn 110 - 109 Citizens - +1 Influence (7 Influence total)
Turn 402 - 401 Citizens - +4 Influence (11 Influence total)
Custom Sov, Custom Faction, Influential pick only.
Turn 1 - Build City 0 Citizens - 0 Influence
Turn 2 - 1 Citizen - +1 Influence (1 Influence total)
Turn 13 - 12 Citizens - +1 Influence (2 Influence total)
Turn 27 - 26 Citizens - +1 Influence (3 Influence total)
Turn 31 - 30 Citizens - +1 Influence (4 Influence total)
Turn 102 - 101 Citizens - +1 Influence (5 Influence total)
Turn 106 - 105 Citizens - +1 Influence (6 Influence total)
Turn 110 - 109 Citizens - +1 Influence (7 Influence total)
Turn 402 - 401 Citizens - +4 Influence (11 Influence total)
Custom Sov, Custom Faction, Royalty only.
Turn 1 Build City 0 Citizens - 0 Influence
Turn 2 - 2 Citizens - +1 Influence (1 Influence total)
Turn 13 - 24 Citizens - +1 Influence (2 Influence total)
Turn 15 - 28 Citizens - +1 Influence (3 Influence total)
Turn 19 - 36 Citizens - +1 Influence (4 Influence total)
Turn 52 - 102 Citizens - +1 Influence (5 Influence total)
Turn 56 - 110 Citizens - +1 Influence (6 Influence total)
Turn 60 - 118 Citizens - +1 Influence (7 Influence total)
Turn 202 - 402 Citizens - +4 Influence (11 Influence total)
Turn 206 - 410 Citizens - +1 Influence (12 Influence total)
Turn 210 - 418 Citizens - +1 Influence (13 Influence total)
Turn 214 - 426 Citizens - +1 Influence (14 Influence total)
Turn 402 - 802 Citizens - Stopped here. No additional as of that turn.
Custom Sov, Custom Faction, Royalty and Influential picks only.
Turn 1 Build City 0 Citizens - 0 Influence
Turn 2 - 2 Citizens - +1 Influence (1 Influence total)
Turn 13 - 24 Citizens - +1 Influence (2 Influence total)
Turn 15 - 28 Citizens - +1 Influence (3 Influence total)
Turn 19 - 36 Citizens - +1 Influence (4 Influence total)
Turn 52 - 102 Citizens - +1 Influence (5 Influence total)
Turn 56 - 110 Citizens - +1 Influence (6 Influence total)
Turn 60 - 118 Citizens - +1 Influence (7 Influence total)
Turn 202 - 402 Citizens - +4 Influence (11 Influence total)
Turn 206 - 410 Citizens - +1 Influence (12 Influence total)
Turn 210 - 418 Citizens - +1 Influence (13 Influence total)
Turn 214 - 426 Citizens - +1 Influence (14 Influence total)
Turn 402 - 802 Citizens - Stopped here. No additional as of that turn.
As you can see, influential has zero effect in any of the scenarios. You can see that each city gets a set increase at turn 2, and 13. You can also see that population sets the additional influences from then on, however there seems to be a couple of spots that don't match with the faction that gets 1 per turn exactly, but in the end they catch up. (Both have 11 influence at ~400 citizens.
So you can see that influential has no effect at all, and that royalty is what you want to have if your intent is to spam influence.