We formulate a general Kuramoto model on weighted simplicial complexes where phases oscillators are supported on simplices of any order $k$. Crucially, we introduce linear and non-linear frustration terms that are independent of the orientation of the $k+1$ simplices, providing a natural generalization of the Sakaguchi-Kuramoto model. In turn, this provides a generalized formulation of the Kuramoto higher-order parameter as a potential function to write the dynamics as a gradient flow. We study the properties of the dynamics of the simplicial Sakaguchi-Kuramoto model with oscillators on edges using a selection of simplicial complexes of increasingly complex structure, to highlight the complexity of dynamical behaviors emerging from even simple simplicial complexes. In particular, using the Hodge decomposition of the solution, we understand how the nonlinear frustration couples the dynamics in orthogonal subspaces. We discover various dynamical phenomena, such as the partial loss of synchronization in subspaces aligned with the Hodge subspaces and the emergence of simplicial phase re-locking in regimes of high frustration.

Connecting Hodge and Sakaguchi-Kuramoto: a mathematical framework for coupled oscillators on simplicial complexes

Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing attention in network theory, dynamical systems, signal processing and machine learning. Topological signals defined on the nodes are typically studied in network dynamics, while topological signals defined on links are much less explored. Here we investigate topological synchronization describing locally coupled topological signals defined on the nodes and on the links of a network. The dynamics of signals defined on the nodes is affected by a phase lag depending on the dynamical state of nearby links and vice versa, the dynamics of topological signals defined on the links is affected by a phase lag depending on the dynamical state of nearby nodes. We show that topological synchronization on a fully connected network is explosive and leads to a discontinuous forward transition and a continuous backward transition. The analytical investigation of the phase diagram provides an analytical expression for the critical threshold of the discontinuous explosive synchronization. The model also displays an exotic coherent synchronized phase, also called rhythmic phase, characterized by having non-stationary order parameters which can shed light on topological mechanisms for the emergence of brain rhythms.

Lucille Calmon

Papers

Topological synchronization: explosive transition and rhythmic phase