Successive shortest paths in complete graphs with random edge weights

Consider a complete graph \(K_n\) with edge weights drawn independently from a uniform distribution \(U(0,1)\). The weight of the shortest (minimum-weight) path \(P_1\) between two given vertices is known to be \(\ln n / n\), asymptotically. Define a second-shortest path \(P_2\) to be the shortest path edge-disjoint from \(P_1\), and consider more generally the shortest path \(P_k\) edge-disjoint from all earlier paths. We show that the cost \(X_k\) of \(P_k\) converges in probability to \(2k/n+\ln n/n\) uniformly for all \(k \leq n-1\). We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterise the collectively cheapest \(k\) edge-disjoint paths, i.e., a minimum-cost \(k\)-flow. We also obtain the expectation of \(X_k\) conditioned on the existence of \(P_k\).