Lucille Calmon, Juan G. Restrepo, Joaquín J. Torres, Ginestra Bianconi
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.
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