How does “influence” work ?
Well I will try my best to explain this since there is some important information to know about.
The area that a kingdom have is based on population and treasures.
The kings villages and village size have a matter in this case and if the kingdom have a duke even his villages will matter in this.
So for an example this is a good picture taken from the wiki.
As you se above the population of a village with a treasury in it is increasing the range.
When you get a village with over 1000 population you will reach the maximum area with 4.1 squares in each direction.
If you want to calculate this on your own use this formula