Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

Area of Science:

• Network science
• Algebraic topology
• Computational neuroscience

Background:

• Understanding network dynamics on higher-order structures (simplices) is crucial but underexplored.
• Neuroscience suggests neural computations may arise from groups of neurons, highlighting the importance of higher-order interactions.
• Existing models often focus on pairwise interactions, neglecting complex network structures.

Purpose of the Study:

• To investigate consensus dynamics on edges within simplicial complexes.
• To analyze the influence of higher- and lower-order interactions on convergence speed.
• To explore the role of network topology in collective dynamics.

Main Methods:

• Utilized a generalized Hodge Laplacian for simplicial complexes.
• Applied techniques from algebraic topology to analyze dynamics.
• Employed Hodge decomposition to study interaction balancing and convergence acceleration.

Main Results:

• Collective dynamics converge to a subspace corresponding to the simplicial complex's homology space.
• Optimal balancing of higher- and lower-order interactions maximally accelerates convergence, aligning with curl and gradient subspace dynamics.
• Consensus over edges is faster when two-simplices (triangles) are dispersed rather than clustered.

Conclusions:

• Higher-order interactions significantly influence network dynamics, particularly consensus processes.
• The Hodge Laplacian and algebraic topology provide powerful tools for analyzing complex network dynamics.
• Optimizing interaction strengths and considering network topology are key to controlling and accelerating collective behavior in higher-order networks.