Re-Rooting-Assisted Edge-Minimum Runtime Repair for Node and Link Failures in Dense Gaussian Broadcast Networks

Dense Gaussian networks are degree-four algebraic interconnection networks with compact diameter and coordinatebased routing. Their diameter-level broadcast trees are efficient but fragile under node faults, link faults, and faults iscovered during propagation. This paper develops a self-contained runtime recovery framework for dense Gaussian broadcast networks under static node/link faults, mixed static faults, and runtimediscovered single-link faults. The method first re-roots the effective source so known node faults become boundary leaves whenever possible, then filters failed links and repairs residual fragmentation by connecting the healthy components of the pruned tree. For a selected root with connected healthy component graph, we prove that exactly c − 1 external component-crossing repair edges are necessary and sufficient. We also prove deterministic single-link repair, give a constant-size shifted boundaryintersection primitive for two-node source selection, derive a linkavoidance exclusion test, and add a local-obstruction probability bound explaining why fixed-size high-order cuts rapidly vanish as k grows. Experiments over k ∈ 10, 25, 50, 100, 200, up to 80, 401 nodes, 280, 000 static trials, and 15, 000 transient trials show 100% recovery for all deterministic and bounded mixed regimes, 99. 998% recovery for multi-link faults, and 99. 963% recovery for higher-order heuristic regimes; every non-recovered static trial is explained by disconnected component graphs or relocation failure. Compared with fixed-source component repair, re-rooting reduces average external repair edges by 80–100%. Patched native Gaussian-link Noxim scheduled replays at k = 25 and audited k = 50 confirm packet-complete router-level execution and show that hybrid re-rooting sharply reduces repair edges, components, and repaired depth. A completion-cycle audit separates structural repair benefit from latency: zero-setup and latency-weighted ablations confirm that completion time depends on relocation/setup, replay scheduling, delivery tail, and selector objective, so the paper claims edge-minimum repair and depth reduction rather than universal completion-cycle dominance.