arXiv:2610.08489v1 Announce Type: new Abstract: This paper focuses on relay-assisted topology optimization to accelerate the convergence of distributed consensus algorithms over weakly connected networks. Instead of permanently adding a fixed set of relay links, we introduce a time-sharing framework where multiple relay configurations are activated in a probabilistic manner. An ActiveSet-based algorithm incrementally constructs candidate relay edge sets and jointly optimizes the mixing matrices and the associated relay set activation probabilities, allowing adaptive relay selection in the optimization process. Simulation results demonstrate a substantially improved performance-cost trade-off compared with fixed-cardinality relay selection strategies.