If I choose the vertices h and k in a simple graph uniformly, I know the probability of them being joined by an edge is $ \frac{2e}{n(n-1)} $ , where:
- e is the number of edges in the graph,
- n is the number of vertices,
- n(n-1)/2 is the total number of possible links.
This is not true, in general, when the vertices are chosen randomly but not uniformly.
The question is, does the formula hold true if I choose h, k in a manner that is not directly influenced by their degree or neighbors, but still not uniformly?
Answered below: It really does depend on how I choose the vertices.
It depends on what you mean by "directly influenced by". For instance, I could take the distribution that always picks a particular vertex p and then a particular vertex q. This doesn't seem to directly reference either their neighbors or degrees. The probability that the two vertices I pick "randomly" are connected by an edge is either 0 or 1 depending of whether or not p and q are actually adjacent.