Cycle holonomy induces higher-order constraints and controls remote synchronization transitions via twisted Laplacian spectra

TL;DR

This paper reveals how topological structures in networks induce higher-order constraints affecting synchronization, using twisted Laplacian spectra to predict stability and phase transitions.

Contribution

It introduces a spectral framework linking topological holonomies to dynamical constraints and synchronization transitions in higher-order networks.

Findings

Twisted Laplacian spectrum depends on the cohomology class of the connection.

Synchronization is hindered by topological frustration on cycles.

Spectral transitions predict loss of stability as holonomy increases.

Abstract

Higher-order interaction networks are typically modeled using hypergraphs or simplicial complexes, where interactions explicitly involve more than two nodes. Here we demonstrate that effective higher-order dynamical constraints emerge naturally on the 1-skeleton of a graph, provided the interaction carries nontrivial topological structure. We study phase-oscillator dynamics with edge phase lags modeled as a $U(1)$-valued connection. This structure induces a gradient Sakaguchi--Kuramoto-type flow and an associated twisted Laplacian whose spectrum depends on the cohomology class of the connection. We prove that the associated twisted Laplacian admits a zero mode if and only if the connection is cohomologically trivial, that is, when all cycle holonomies vanish. Consequently, synchronization is obstructed not by local pairwise mismatches, but by intrinsic topological frustration on cycles. We derive that the smallest eigenvalue of the twisted Laplacian scales with the magnitude of the holonomy, and its spectral transitions accurately predict the loss of stability of the phase-locked state as frustration is increased. For the specific case of constant phase lag, we analytically derive the critical transition point, $\alpha_c = \pi/3$ for a pentagonal cycle, which is in quantitative agreement with previously reported numerical thresholds. Our results establish a spectral framework linking dynamical frustration to network cohomology, and show that transitions in remote synchronization are shaped by cycle-level topological constraints.