How does a city's influence ring in size?

City Population. There is typically 2-3 ZoC size per city level. One is at default size (you'll always get it one turn after the city grows), then as the city gets more and more people, the ZoC expands until it's max size for that city level (not completely certain on the formula for this). Once that happens you have to increase the city level again for more...

Erm, this isn't exactly correct.  The only way to expand the influence around a city is to construct improvements that grant bonus influence, or select a faction with the "Influential" perk.

Basically, influence is a hidden stat. It increases at a constant rate and is capped by housing level/population.  At level 1, you get two increases (Radius 1, Radius 2). Generally, you get an influence increase the turn after a level up and about mid-way through though there may be other increases at higher levels.

What influence buildings do is generate their own bubble of influence and probably affects the influence stat in some fashion. As it's a hidden stat, we really can't tell much more than that.

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.

Good work.  And here I was thinking that Elemental was just poorly documented, not that the game outright lies to you.

It's not so much a lie as completely broken/not working. Long list of those.

For example: Watchtower's sight, Assassin sov trait, intimidating sov trait, royalty faction trait and so on and so on.

So influence does not do anything but determines the size of the ring around your city?

I do know that the AI cannot step into your ring unless there is a mutual agreement.

Seems to me that everyone learns the game mechanics through trial and error.

There should be a proper explanation or game tutorial. 

Not a lie...  just such grotesquely bad XML coding that it's broken in too many places to describe (just take a gander at any number of posts in the modding forums about how to do certain "obvious" mods that just don't work due to broken or redundant code).

Ohhhhhhh, don't get me started on the XML interpretation, especially [Calculate].  Do. Not. Get. Me. Started.

So is having Royalty the only way to get your cities to level 5 in any reasonable amount of time?  Does anything else make population grow faster?

The influence trait is broken, it does nothing. Influence itself isn't broken. It's the ZoC (Zone of Control) around your city that allows you to build on resources, this value is defined by the city hub. The higher the level of the hub, the bigger the ZoC, although there are several levels for each Hub, and the population level of the city determines which you get (higher population = bigger) as I tried to explain in my first post. Influence buildings produces a small ZoC around them, for the most part, they are useless, unless you are building some kind of super long town or need to use them to push the ZoC of another player.

Oh and yes, Royalty is broken too, it should add +1 prestige (determines population growth), but instead, it doubles all prestige (like a free great theatre in every city). Thus, for kingdom royalties, it can often add +4 (pub and inn) and can go as high as +11 (pub, inn, & palace). As a result, level 5 cities are actually very easy to get for them.

Actually the influence trait isn't broken, the effect is just too small. I tried upping the value from 1 to 1K and the influence ring goes crazy as it should . Or at least id did in v1.07, maybe the unbelievers can test this.

Kalin> how do you know Royalty is supposed to work that way? I always thought it was about doubling prestige. There's even a mod, that makes royalty for champions work. I think they give +33% prestige on stationed city. Yes there are some worthless history backgrounds, if we compare how they affect the game. Royalty is one of the best ones and I feel it makes sense for it to be.

I'm unsure about ZOC being the same as Influence area, I think maybe it only determines where you can build normal buildings. Upping that number, I was able to place non-adjacent buildings. Still you may be right, god knows I also thought they were same thing, too

I say Royalty is broken because while the effect is okay for empires, it is completely broken for kingdoms. The +100% bonus on a pub, inn, and palace makes a kingdom city grow +24 people a turn. You can get to a level 5 city in about 100 turns easily. If it was just +1 prestige, it would still provide a 100% bonus for the base city bonus (which is still very powerful), but doesn't double the bonus from pub/inn/palace. As for the prestige heroes... that's just something to make the bonus do "something".

No idea if influence is one of those typos that needs a huge multiplier (like the dodge stat), but as it is, the trait is broken and useless. I'm pretty sure ZoC is the city influence. I'm not sure what you mean by being able to place non-adjacent buildings, I've messed with very large ZoC when I made higher city levels (6 and 7), and never encountered that.