I would like to generate random graphs that might be geometric graphs in some (unknown) dimension. So I would like every triangle in the graph to satisfy the triangle inequality on its (random) edge lengths/weights. I need something akin to the Erdős/Rényi model such as, "The weighted random graph model," but with the triangle geometric constraint.
The earlier MO question, "Probability that random weights on $K_n$ satisfy triangle inequality," seems quite relevant, but I don't immediately see how it leads to a method for generating the random graphs I need.
So my question is:
Q. How can one generate random Erdős/Rényi weighted graphs that satisfy the triangle inequality for every triangle in the graph?