I'm trying to create/find an index to compare/order systems with multiple entities based on the diversity of the interaction between the entities.
Assume you have few systems of entities that can communicate with each other or with themselves. The volume of communication between each and every entity in the system and also the volume of internal communication (within each and every entity) is recorded.
I would like to have an index, i.e a number assigned to each system that will say how diverse is the communication in each system compared to the other systems while this index has few properties (assume for simplicity that the number of entities within the different systems is the same):
If the communication between entities compared to within entities increases then the value of the index increases.
If the communication between entities become more equal/divers the index increases. For example, let's say there are 3 entities A, B and C. The internal communication that is done within each of A, B and C is 0.1 of the total communication (leaving 0.7 of the communication as communication between entities). And lets say the communication pattern between entities changed from ab- 0.3, bc- 0.3, and ac- 0.1 to ab- 0.3, bc- 0.2, and ac- 0.2 (while keeping the within communication at 0.1 for each entity), the value of the index will increase.
If the communication within countries become more equal (in the same sense of point number 2) once again the value will increase.
I have some ideas but they are not mathematically clean/sleek/tractable. I hope my question is coherent and will be happy to hear ideas/be referred to some existing ideas.
Thank!!