Hodge Decomposition Guides the Optimization of Synchronization over Simplicial Complexes

TL;DR

This work addresses optimizing synchronization in higher-order networks by formulating HLK dynamics on simplicial complexes and extending the Synchrony Alignment Function (SAF) to both lower- and upper-projected HLK dynamics. A balancing parameter $oldsymbol{ abla}$ controls the relative influence of lower- and upper-dimensional interactions, and two SAFs $J^{[-]}$ and $J^{[+]}$ are derived to guide frequency allocation for edges. The authors develop MCMC-based and spectrally constrained optimization schemes, reveal how optimal frequencies align with the Hodge subspaces (gradient, curl, harmonic), and show the harmonic subspace can serve as a 'safe haven' for heterogeneity; they also discuss a correction term $oldsymbol{ ext{chi}}$ that is typically small. Together, these results extend synchronization optimization to topological higher-order networks and provide practical algorithms linking algebraic topology, combinatorial optimization, and dynamical systems.

Abstract

Despite growing interest in synchronization dynamics over "higher-order" network models, optimization theory for such systems is limited. Here, we study a family of Kuramoto models inspired by algebraic topology in which oscillators are coupled over simplicial complexes (SCs) using their associated Hodge Laplacian matrices. We optimize such systems by extending the synchrony alignment function -- an optimization framework for synchronizing graph-coupled heterogeneous oscillators. Computational experiments are given to illustrate how this approach can effectively solve a variety of combinatorial problems including the joint optimization of projected synchronization dynamics onto lower- and upper-dimensional simplices within SCs. We also investigate the role of SC homology and develop bifurcation theory to characterize the extent to which optimal solutions are contained within (or spread across) the three Hodge subspaces. Our work extends optimization theory to the setting of higher-order networks, provides practical algorithms for Hodge-Laplacian-related dynamics including (but not limited to) Kuramoto oscillators, and paves the way for an emerging field that interfaces algebraic topology, combinatorial optimization, and dynamical systems.

Figures (12)

• Figure 1: Hodge Laplacian Kuramoto (HLK) phase-oscillator dynamics give rise to two simultaneous dynamical systems obtained through lower- and upper-dimensional projections. We study HLK dynamics (see Sec. \ref{['sec:HodgeLaplacian']}) in which phase oscillators are assigned to 1-simplices (i.e., edges) within a simplicial complex (SC). The associated dynamics can be projected onto their neighboring 0-simplices (i.e., nodes) and 2-simplices (i.e., filled-in triangles), both of which can exhibit synchronization and phase locking.
• Figure 1: Phase transitions for the Kuramoto model. As one increases a coupling strength $\sigma$, Eq. \ref{['eq:GraphKuramoto']} exhibits three distinct phases: (i) $\sigma\in[0,\sigma_{c1}]$ yields an "incoherent state" in which no synchronization occurs as measured by $r\approx 0$; (ii) $\sigma\in(\sigma_{c1},\sigma_{c2}]$ yields a state with "partial synchronization" in which $r$ diverges from 0 due to some phases clustering near $\theta_i\approx \psi(t)$; and (iii) $\sigma>\sigma_{c2}$ yields a "phase locked" state in which all oscillators approach fixed-point limits in the rotating frame $\theta_i\mapsto\theta_i-\psi(t)$.
• Figure 1: Balancing the projected HLK dynamics. We plot order parameters $r^{[\pm]}$ versus $\sigma$ for the lower- and upper-projected HLK dynamics for choices of the balancing parameter $\delta$. Varying $\delta$ tunes the relative onset of synchronization for these two projected systems, and the inset shows that all curves effectively collapse onto a single curve if one plots the order parameters versus $\sigma_{[\pm]}$ rather than $\sigma$.
• Figure 1: Optimization of lower- and upper-projected HLK model. We plot both synchronization order parameters $r^{[-]}$ (left) and $r^{[+]}$ (right) while aiming to maximize each of them separately, or maximize them simultaneously through their mean $\overline{r}$. These $r^{[\pm]}$ values are plotted versus coupling strength $\sigma_{[\pm]}=\sigma$ (or equivalently, $\delta=0$) and compared to $r^{[\pm]}$ for random systems that have not been optimized. Observe that optimizing $\overline{r}$ significantly improves synchronization for both projections.
• Figure 1: Simplicial Complexes studied throughout our paper. (a) SC 1 was used for Figures \ref{['fig:SynchSimulations']}-\ref{['fig:Bifurcation']} and \ref{['fig:Noise Plot']} (b)--(d) The remaining SCs were used for Fig. \ref{['fig:Noiseless examples']}.