  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.
