Let’s say you have Nodes which are the basic building blocks (A, B, C). Then you have Links which are connections between Nodes (AB, BC, CD). Then you have Systems which are patterns of connections (ABC). Presume that 1) Nodes can only be linked once to count as a Link 2) Links can only be combined once to count as a System 3) The order of arrangement is irrelevant. E.g.:
With 1 Node, (A) you have 1 Node, 0 Links, 0 Systems because A has no other Node to link to and there are no Links to combine into a System.
2 Nodes (A, B) = 1 Link (AB) = 0 Systems because you only have 1 Link and a System requires 2+ Links.
3 Nodes (A, B, C) = 3 Links (AB, AC, BC) = 1 System (ABC)
4 Nodes (A, B, C, D) = 6 Links (AB, AC, AD BC, BD, CD) = 4 Systems (BCD, ACD, BCD, ABCD)
I started doing 5 Nodes and realized it was going to be unwieldy. Someone else must have worked on this and come up with a way to calculate it.
As for why anyone would care to nerd out on this: It may be relevant to many complex systems. It’s the basis of the network effect in economics* and may be applicable to other fields which we can discuss later. It’s one of the main reasons why Beta eventually had no chance against VHS, Sony’s consoles have been beating Microsoft’s, Youtube/Steam/Facebook are so dominant.
Note how the number of Links increases more than linearly as a function of the number of Nodes. The number of Links starts off lower than the number of Nodes but quickly catches up. When you go from 1000 Nodes to 1001Nodes, you’re only adding 1 Node but you’re adding 1000 (potential) Links. I expect something similar will happen with Systems but I don’t know how to calculate this efficiently.
When answering, please keep in mind that many readers (like me) don’t have a math background.