Consensus on simplicial complexes: Results on stability and synchronization
Lee DeVille
Abstract
We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.
Topics & Concepts
Synchronization (alternating current)GeneralizationFlow (mathematics)Stability (learning theory)Laplace operatorMathematicsSimplicial complexDimension (graph theory)Nonlinear systemTopology (electrical circuits)Computer sciencePure mathematicsCombinatoricsMathematical analysisGeometryPhysicsQuantum mechanicsMachine learningSlime Mold and Myxomycetes ResearchNonlinear Dynamics and Pattern FormationTopological and Geometric Data Analysis