A Small-Gain Analysis for an All-Pairs Shortest-Path Algorithm: Global Exponential Input-to-State Stability

In this article, we propose a distributed all-pairs shortest-path algorithm that finds the shortest path between every pair of nodes in an undirected graph and analyzes its robust stability from a control perspective. The proposed algorithm and the derived stability results outperform the previous related works in two aspects. First, while previous works consider the additive noise on the measured edge weight as the perturbation, in this article, we consider a general perturbation such that the edge weight is bounded and time-varying. Moreover, we consider quantization effects on the communication channels. Second, by using a Lyapunov-based small-gain theorem, we prove the global exponential input-to-state stability of the proposed algorithm under perturbations, implying its global exponential stability without perturbations and quantization effects while previous works only give global asymptotic stability or regional exponential stability results of theirs. Simulations are provided to verify the validity of our results.