I was asked if I could find out how religion and influence spread work, so here is what I have found out.

Religion and influence spread seems to follow the same rules, so there is no reason to separate them, the only difference is that faith is what spread religion and influence is what spread sphere of influence, which I often just call influence spread as that is easier to write and I have problems with my hands and fingers so I don't want to write too much.

When you select the religion view, press on a territory and hold over the icon of a religion you will see how much faith that religion have in that territory. 

And similar you can see the influence breakdown.

Anyway the spread system is worked on a weighted system: strength of a religion/influence = (faith/influence of that religion/influence)/sum of all faith/influence in that territory. 

This system will always lead to the sum of all religion/influence in a territory add up to 100%, in the case of competition you can  have one religion be 75% and another be 25%. This could mean the first religion have15 faith and the second have 5 faith in the territory as 15/20 = 75% and 5/20 = 25%.

A territory wont just convert because a influence or religion have more faith/influence than the current one, it need to control 20% more of the total influence/faith of the territory than the current one. As can be seen in the screenshot below, the territory don't convert even though the current religion only control 41% of the faith and the other one control 59% of the faith because 59 - 41 = 18 which is not enough as 20 is the minimum. 

A territory get faith/influence from two sources, itself and distant territories. Each territory produce influence for its owners sphere of influence. Faith work a bit differently as holy sites produce faith to the religion they belong to, which may not be the controllers state religion, this mean there can be advantage of destroying holy sites of different religions.

Each territory can also spread the dominant religion/sphere of influence. By this I mean if it produce the dominat influence or faith for the dominant religion, it can send it to other territories but it will also act as a pathway for other territories with the same dominant influence/religion to send their faith and influence to distant territories. This mean religions and sphere of influences that fall behind and are dominated by others tend to fall into a downwards spiral and die out.

How spread of sphere of influence/religion work is there need to be a path, between land connected territories it is very simple and easy to see if you have roads. It is possible two territories that are neighbours are actually far appart if for example a mountain chain block a direct path between the territories. Between oversea territories, you need harbors and the placement of the harbors are important as they can impact the distance which impact the power of the influence/faith that distant territories will have. Between different players, importing a resource also import the faith/influence of the territory you import from with full power (it still can only send over the dominant influence/religion) and netural trade ports wont spread influence, so keep that in mind that importing from players with powerful influence or faith can mean your territories start to convert.

As far as I can tell open borders don't impact spread between players, however war close any spread between the players and is thuse one way to recover a dead or dying religion or sphere of influence, war can also be a useful tool to spread your sphere of influence or religion as captured cities tend to quickly convert to your sphere of influence and religion.

 How much influence or faith a territory will send to distant territories depend on the distance in number of territories and is given by this formula:

influence to distant territory = influence produced in territory * 0,42^log2(x)

With x being the shortest distance number of territories you need to travel through from the influencing to reach the target. Increase x by 1 for each territory you need to pass through. Now about trade, trade only spread the exporters influence, the importers influence is not spread by trade. If there is a land trade route going through the influencing territory and the target (even if these are just territories the trade route need to pass through), divide x by 2 and in the case of x = 0.5, set x to 1.Territories connected by a sea or air trade route will have an x of one starting at the territory with sending harbor/airport and use the same rules for land based trade, thus very distant territories can due to sea and air trade have very close influence or religious proximity, which tell how powerful trade can be at spreading religion and sphere of influence.

Here is a graph of how the power of influence/faith diminish when spreading to distant targets, this is particular important for holy sites for obvious reasons.

And here is the same graph but up to 100 territories. 

Here is more images about trade routes:

The green player import a resource from a landlocked territory, the resource travel on a land route (dotted lines) to a port territory and look like it travel around the world to reach the destination, in reality sea trade seems to simply "teleport" the resource from one port to the other.

Here is the impact the territories have on the green player territory the trade ends up in. The port territory that is used to transport the resource get full influence impact and the landlocked territory that resource belong to get to use about 60% of its influence, which is equal to x being 1.5. Other territories don't seems able to take advantage of the trade route however. The green territory don't send any influence back, indicating that trade routes spread influence from exporter to importer and not both ways.

Here is a case when two territories are next to each other but are separated due to large difference in elevation and thus influence, religion and trade wont take place directly between those territories as they have to go through another territory to reach each other.

What this means: 

If you want to spread your influence/religion you want to focus on the territories closest to your neigbours as distant territories have less impact. This however can change if players start to import your resources, especially sea and air trade can be a very powerful way to spread your influence, however you can't control how other players decide to trade with you, but you can take advantage of it when they do. In such case you can focus your influence and faith on these trade hub territories to maximize their spread.

If you want to get another players religion or influence to spread inside your territories for reason such as getting science, you should import their resources, especially from territories that produce alot of influence, use sea routes or air routes if possible.

Anyway I hope this have been of some use.

paxis76 wrote:

I've just calculated the spread impact and I got following results: for distance 2 -> 0,6 and for distance 3 -> 0,3 (see picture below). The city with 9 has base value of 15). Why we have such a difference compared to your graph?

  

Becuase of land trade routes which halves the distances calculation. I can't tell how your trade routes are setup. 12 = 39*x^-1.25 means x = 2.5 and 9 = 15*x^-1.25 means x = 1.5. Only way you those values is due to land trade routes.

My suggestion is to check how your import trade routes look like, those make a huge difference in terms of spreading faith and influence.

paxis76 wrote:

I have only sea import routes with black civ (south of the map) and no one with orange civ. Maybe the reason is the merchant badge i have? Normaly the badge increase the vision but who knows?

We have no idea how orange trade routes look like, but to me it look like someone is importing from orange that go through your city, thus helping spreading orange influence to your city. I can't tell exactly how your city get an distance of 2.5, except it have to do with trade routes, but keep in mind those can be complicated, like the game seems to try to find the shortest path and keep in mind it can use several trade routes to do so, it is complicated.

Like I have had an ally import from me, thus spreading my influence to them, then another player imported from them (my ally) thus helping spreed my influence to them, so you can understand how big impact trade can have on influence spread.

paxis76 wrote:

I mean the exactly calculation of influence spreading taking into acount the trade routes impact.

That is already known.

Trade routes halves the effective distance used for the calculation and airports or docks used for trade is considered to be neighbours. However trade routes only benefit the exporters influence and faith spread. Influence/faith flow with the trade routes from the exporter and towards the importer. 

For influence caculation, a territory use the shortest path to the target possible, using all trade route combinations, however it seems like you can't use half a trade route and then normal travel if that make sense, if there is no direct trade route connection to the target you need to go the normal, slow path. The trade route connection however seems to be able to combine several different trade routes to create a shorter path. Thus it get complicated when alot of trade routes are involved.

We know the math, the hard thing is how exactly how the combination of trade routes interact, I think the best way to view it is like a flow, trade routes flow from point A to point B and with that flow also comes influence and faith. If a new trade route is setup from B to C, I think Influence and faith from A may also flow into C, as long as B share sphere of influence and religion of A which is a requirement for influence and faith to spread through a territory to the next one.

Could you prepare an example calculation based on real game situation when trade routes are involved but the number of them is relatively low means not so complex?

@paxis76 I tested a new format, to make a video as I wanted to test something new. This video should help explain a bit how the influence spread system work: