Heaven & Hell: One-Step Hub Consensus

Abstract

Many networked systems require a central authority to enforce a global configuration against local peer influence. We study influence dynamics on finite weighted directed graphs with a distinguished hub node and binary vertex states ('Glory' or 'Gnash'). We give a sharp, local, and efficiently checkable criterion that guarantees global convergence to Glory in a single synchronous update from any initial state. At each non-hub vertex, the incoming weight from the hub must at least match the total incoming weight from all other nodes. Specialising in uniform hub broadcasts, the exact threshold equals the maximum non-hub incoming weight over all vertices, and we prove this threshold is tight. We extend the result to a tau-biased update rule and to asynchronous (Gauss-Seidel) schedules, where a single pass still suffices under the same domination hypothesis. Machine-checked proofs in Coq accompany all theorems.