Balanced Hodge Laplacians Optimize Consensus Dynamics over Simplicial Complexes

TL;DR

This paper investigates how higher-order interactions in simplicial complexes influence consensus dynamics, revealing that balanced Hodge Laplacians optimize convergence speed and are affected by network topology.

Contribution

It introduces a generalized Hodge Laplacian framework with adjustable interaction strengths, demonstrating optimal balance for accelerated consensus convergence using algebraic topology techniques.

Findings

Balanced higher- and lower-order interactions maximize convergence speed.

Optimal balance aligns with curl and gradient subspace dynamics.

Dispersed 2-simplices improve consensus acceleration.

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.