Scalar Unit-Gauge Consensus over Structurally Balanced Quaternion Edge-Weight Graphs

Author preprint of a manuscript submitted to IEEE Transactions on Automatic Control. This is the author-submitted version, not the version of record. The scientific content is unchanged from the submitted manuscript; the preprint notice and running headers identify this repository version. Abstract: This note studies scalar quaternion single-integrator protocols over directed graphs with quaternion edge-weights. The edge action is one-sided: the state received from a neighbor is premultiplied by the corresponding quaternion edge-weight. We formulate a structural-balance condition for such graphs, show it to be equivalent to consistency of every weak cycle, and prove that under it a diagonal gauge of unit quaternions maps the quaternion adjacency matrix onto the matrix of its entrywise norms, and the quaternion Laplacian onto an ordinary nonnegative real edge-weight graph Laplacian matrix. The proposed continuous-time and discrete-time protocols therefore reduce to standard real consensus in the transformed coordinates. Under the corresponding graph-connectivity conditions, the resulting behavior is unit-gauge consensus: the transformed states agree, while the original states agree in norm and retain prescribed unit-quaternion offsets. Signed bipartite and complex multiparty consensus are recovered as special cases. The same reduction yields pinning protocols with gauge-compatible pinning gains, and extends to agents sharing a common purely imaginary right pole. The structurally balanced graph convergence results follow from gauge similarity and classical nonnegative-matrix theory; our arguments do not require general quaternion spectral theory. Simulations validate the proposed approach. Funding acknowledgment: This work was supported by the European Union under the project ROBOPROX, registration no. CZ.02.01.01/00/22_008/0004590. Version: Version 1.0 of the Zenodo preprint, prepared on 27 October 2026. The date identifies preparation of this repository version, not the date of submission to IEEE Transactions on Automatic Control.