Attracting Rotations Obstruct Almost-Global Consensus on Spheres over Strongly Connected Graphs

Galina Sidorenko, Johan Thunberg · 2026 · arXiv

DOI: 10.48550/arXiv.2610.28341 · open-access PDF

Abstract

We study discrete-time consensus on the Euclidean unit sphere, where weighted combinations of neighboring states are normalized to unit length. For symmetric weights on connected undirected graphs and ambient dimension $d\ge3$, earlier results establish instability of non-consensus fixed points under suitable weight conditions. For directed graphs on the unit circle, earlier results show that fixed points other than consensus and antipodal configurations occur only for a measure-zero set of weights. We show that these conclusions do not extend directly to general strongly connected directed gr

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