Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex . The dynamics ofsignals defined on the nodes is affected by a phase lag depending on the state of nearby nodes . We show that topological synchronization on afully connected network is explosive and leads to a discontinuous forwardtransition and a continuous backward transition . The model alsodisplays an exotic coherent coherent synchronized phase, also called rhythmic phase,characterized by having non-stationary order parameters which can shed light ontopological mechanisms for the emergence of brain rhythms . The analytical investigationof the phase diagram provides an analytical expression for the criticalthreshold of the discontinuous explosive synchronization. The model was used as a reference to the analysis of the critical threshold of the model . The study concludes that the model has been published in the journal The Open Science Press Press Presses Presses and

Author(s) : Lucille Calmon, Juan G. Restrepo, JoaquĆ­n J. Torres, Ginestra Bianconi

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Keywords : phase - model - topological - nodes - explosive -

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