I'm thinking of something like, say, a line diagram. The horizontal axis would be "running speed" and the vertical axis something like "calories per minute". Has this matter ever been scientifically researched?
It would look like this:
It's basically a straight line. Don't take the variations from the straight line too seriously. There are so many other factors involved (wind, distance, running form, etc.) that it doesn't make sense to try to measure it too precisely. The graph above is for a 30 year-old male with 70 kg and 1.75 m, calculated with the corrected MET values taken from the Compendium of Physical Activities. You can make your own graph with your personal parameters with this spreadsheet.
It is well-known that you burn approximately the same calories per km independently of speed. A fortunate numerical coincidence allows us to simplify the formula and simply take the speed in km/h to get how many calories you burn per kg per hour. For example: a 100 kg person running at 12 km/h burns about 1200 kcal per hour and a 50 kg person running at 20 km/h burns about 1000 kcal/h.
Short answer: the line on such a diagram would just be a straight horizontal line. Running faster does not significantly increase calories burned.
Long answer: A "calorie" in the context of food and exercise is actually short for "kilocalorie", which is a unit of energy (or "work", as some people say).
Just like 1 mile is equal to 5,280 feet, 1 kilocalorie is equal to 4,184 Joules, which is the same amount of energy/work required to raise the temperature of 1 kilogram of water by 1° Celsius.
Let's say you dragged a 100lbs boulder 20 feet across the ground. No matter how fast you dragged that boulder, it will always be the same amount of work done: 100lbs moved 20 feet.
So, based on how much you weigh, the amount of calories you burn running a mile will roughly be the same no matter how fast you run. I know it sounds counter-intuitive, since running fast is harder than running slow, but that's science.
You might burn a few more calories in the wind since wind resistance increases the faster you go, but it's not very significant unless you are absolutely sprinting.
That is the reason the calculator on this site will always give you the same number of calories burned, no matter how fast a time you put for the "How long you ran" box:
Caveat: calories burned while running will always be the same, but there might be some increased after-burn of calories if you ran the mile super fast as opposed to normal speed, since your body has to work harder to catch its breath, cool off, etc.