Consensus and balancing on the three-sphere

摘要

We study consensus and anti-consensus on the 3-sphere as the global optimization problems. The corresponding gradient descent algorithm is a dynamical systems on S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$S<^>3$$end{document}, that is known in Physics as non-Abelian Kuramoto model. This observation opens a slightly different insight into some previous results and also enables us to prove some novel results concerning consensus and balancing over the complete graph. In this way we fill some gaps in the existing theory. In particular, we prove that the anti-consensus algorithm over the complete graph on S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$S<^>3$$end{document} converges towards a balanced configuration if a certain mild condition on initial positions of agents is satisfied. The form of this condition indicates an unexpected relation with some important constructions from Complex Analysis.