Simplicial Kuramoto models have emerged as a diverse and intriguing class of
models describing oscillators on simplices rather than nodes. In this paper, we
present a unified framework to describe different variants of these models,
categorized into three main groups: "simple" models, "Hodge-coupled" models,
and "order-coupled" (Dirac) models. Our framework is based on topology,
discrete differential geometry as well as gradient flows and frustrations, and
permits a systematic analysis of their properties. We establish an equivalence
between the simple simplicial Kuramoto model and the standard Kuramoto model on
pairwise networks under the condition of manifoldness of the simplicial
complex. Then, starting from simple models, we describe the notion of
simplicial synchronization and derive bounds on the coupling strength necessary
or sufficient for achieving it. For some variants, we generalize these results
and provide new ones, such as the controllability of equilibrium solutions.
Finally, we explore a potential application in the reconstruction of brain
functional connectivity from structural connectomes and find that simple
edge-based Kuramoto models perform competitively or even outperform complex
extensions of node-based models.Comment: 36 pages, 11 figure