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

Abstract

Dense Gaussian networks are degree-four algebraic networks with compact diameter and coordinate-based routing. Their diameter-level broadcast trees are efficient but fragile under node, link, and runtime-discovered faults. This paper develops a runtime recovery framework for dense Gaussian broadcast networks under static node/link faults and mixed faults, plus single-link faults discovered live. The method re-roots the source so known node faults become boundary leaves whenever possible, then filters failed links and repairs gaps by connecting healthy components of the pruned tree. For a selected root with connected healthy component graph, we prove exactly $c-1$ external repair edges are necessary and sufficient. We also prove deterministic single-link repair, give a constant-size boundary-intersection primitive for source selection, derive a link-avoidance exclusion test, and add a local-obstruction bound explaining why high-order cuts vanish as $k$ grows. Experiments over $k\in\{10,25,50,100,200\}$, up to $80{,}401$ nodes, $280{,}000$ static trials, and $15{,}000$ transient trials show 100\% recovery for deterministic and bounded regimes, $99.998\%$ for multi-link faults, and $99.963\%$ for heuristic regimes; non-recovered trials are explained by disconnected components or relocation failure. Re-rooting reduces average repair edges by 80--100\% versus fixed-source repair. Patched Gaussian-link Noxim replays confirm packet-complete execution and show re-rooting reduces repair edges, components, and depth. A completion-cycle audit separates repair benefit from latency: ablations confirm completion time depends on relocation, scheduling, delivery tail, and selector objective, so the paper claims edge-minimum repair rather than universal completion-cycle dominance.