Expected Number of Good Pairs in a Randomly Weighted Complete Graph

An undirected complete graph with $$n$$ vertices has one edge for every unordered pair of vertices, giving $$\binom{n}{2}$$ edges in total. Each edge is assigned an independent random weight sampled from a shared continuous distribution. Because the distribution is continuous, the probability that two edge weights are exactly equal is zero. For a vertex, define its best neighbor as the other endpoint of the incident edge with the greatest weight. An unordered pair of…