Balanced Hodge Laplacians Optimize Consensus Dynamics over Simplicial Complexes

Abstract

Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, $k$-dimensional "simplices") and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide the building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence, and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when 2-simplices are well dispersed, as opposed to clustered together.

arXiv comment10 pages, 7 figures, submitted to AIP Chaos

References in corpus (20)

• Simplicial models of social contagion
• Explosive higher-order Kuramoto dynamics on simplicial complexes
• Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplices
• Persistent Homology of Complex Networks
• Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
• Abrupt phase transition of epidemic spreading in simplicial complexes
• Multi-body Interactions and Non-Linear Consensus Dynamics on Networked Systems
• Accuracy of Mean-Field Theory for Dynamics on Real-World Networks
• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• The topological Dirac equation of networks and simplicial complexes
• Topological exploration of artificial neuronal network dynamics
• Hodge Laplacians on graphs
• Scaling-up Distributed Processing of Data Streams for Machine Learning
• Simplicial Homology for Future Cellular Networks
• Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes
• Non-Convergence and Limit Cycles in the Adam optimizer
• PowerGossip: Practical Low-Rank Communication Compression in Decentralized Deep Learning
• The Master Stability Function for Synchronization in Simplicial Complexes
• Simplicial branching random walks and their applications
• Impact of Community Structure on Consensus Machine Learning

Cited by in corpus (19)

• Dynamics on higher-order networks: A review
• Weighted simplicial complexes and their representation power of higher-order network data and topology
• Global topological synchronization on simplicial and cell complexes
• Dirac signal processing of higher-order topological signals
• Dirac synchronization is rhythmic and explosive
• Turing patterns on discrete topologies: from networks to higher-order structures
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Topology and dynamics of higher-order multiplex networks
• Global topological synchronization of weighted simplicial complexes
• Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions
• Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes
• Higher-order Connection Laplacians for Directed Simplicial Complexes
• Filtering higher-order datasets
• Quantum entropy couples matter with geometry
• Hyper-diffusion on multiplex networks
• Global Topological Dirac Synchronization
• Hypergraphon Mean Field Games
• Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
• Designing topological cluster synchronization patterns with the Dirac operator