Successive shortest paths in complete graphs with random edge weights

TL;DR

This paper analyzes the asymptotic behavior of successive shortest edge-disjoint paths in complete graphs with random edge weights, revealing convergence patterns for their costs.

Contribution

It provides new asymptotic results for the costs of multiple edge-disjoint shortest paths in complete graphs with uniform and exponential weights.

Findings

Cost of the k-th shortest path converges to 2k/n + ln n/n

Results hold uniformly for all k ≤ n-1

Characterizes minimum-cost k-flow in such graphs

Abstract

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 $P_k$.