Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian

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 analogues of structures seen in related network models.

References in corpus (7)

• Synchronization in complex networks
• Higher-order simplicial synchronization of coupled topological signals
• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Spectrum of extensive multiclusters in the Kuramoto model with higher-order interactions
• Multistability of Phase-Locking in Equal-Frequency Kuramoto Models on Planar Graphs
• Nonlinear consensus on networks: equilibria, effective resistance and trees of motifs
• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points

Cited by in corpus (19)

• Dynamics on higher-order networks: A review
• Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes
• Weighted simplicial complexes and their representation power of higher-order network data and topology
• Hypergraph reconstruction from dynamics
• Dirac synchronization is rhythmic and explosive
• Balanced Hodge Laplacians Optimize Consensus Dynamics over Simplicial Complexes
• Turing patterns on discrete topologies: from networks to higher-order structures
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Local Dirac Synchronization on Networks
• Topology and dynamics of higher-order multiplex networks
• Multistability in coupled oscillator systems with higher-order interactions and community structure
• Stochastic epidemic model on a simplicial complex
• Global topological synchronization of weighted simplicial complexes
• Higher-order synchronization on the sphere
• Analyzing states beyond full synchronization on hypergraphs requires methods beyond projected networks
• Higher order interactions lead to "reluctant" synchrony breaking
• On the convergence of nonlinear averaging dynamics with three-body interactions on hypergraphs
• Finite-time consensus in a compromise process
• On the efficiency of pairwise Hamiltonian control to desynchronize the higher-order Kuramoto model