Topological signals defined on nodes, links, and higher-dimensional simplices describe the dynamic state of a network or simplicial complex. This paper investigates Dirac synchronization, a model that locally couples topological signals on nodes and links using the topological Dirac operator. The dynamics includes a phase lag dependent on the state of nearby links (and vice-versa). On a fully connected network, Dirac synchronization exhibits explosive behavior with a hysteresis loop, featuring a discontinuous forward transition and a continuous backward transition. Analysis of the phase diagram reveals a theoretical understanding of this explosive synchronization. The model also displays a rhythmic phase with non-stationary order parameters, potentially explaining topological mechanisms for brain rhythms.

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