Is there a term in mathematics for Metcalfe's Law? Metcalfe's Law states that the value of a network is proportionate to the square of the number of users. This comes from the idea that there are $N*(N-1)/2$ pairs in a network of size $N$. Does this mathematical rule have a name?

Quadratic scaling/growth.

The Metcalfe law itself is probably better formulated as "value of network is proportional to the number of connections between users" (or volume of interaction between users), without assuming that the quantity scales as $N^2$. The number of connections is reasonably modeled as being on the order of $N^2$ only for small dense networks where most users know some significant fraction of the others. Large real networks are sparse and exhibit "small world" effects where the number of connections scales as a power smaller than $2$.