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We consider a nonlinear flow on simplicial complexes related to thesimplicial Laplacian, and show that it is a generalization of various consensusand synchronization models commonly studied on networks. In particular, ourmodel allows us to formulate flows on simplices of any dimension, so that itincludes edge flows, triangle flows, etc. We show that the system can berepresented as the gradient flow of an energy functional, and use this todeduce the stability of various steady states of the model. Finally, wedemonstrate that our model contains higher-dimensional analogues of structuresseen in related network models.

Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian