 If we are interested in metrics that "preserve" extremal properties of graphs,
then we should restrict our attention to metrics that identify random graphs as similar.For example, if we
randomly draw two graphs independently from an Erdős–Rényi model formula_14 for some fixed formula_15, the distance between these two graphs under a "reasonable" metric should be close to zero with high probability for large formula_26.There are two natural metrics that behave well on dense random graphs, in this sense.