Operational Emergence of a Global Phase under Time-Dependent Coupling in Oscillator Networks

Abstract

Collective synchronization is often summarized by a complex order parameter $R e^{i\Psi}$, implicitly treating the global phase $\Psi$ as a meaningful macroscopic coordinate. Here we ask when $\Psi$ becomes \emph{operationally well-defined} in oscillator networks whose coupling varies in time. We study damped (and optionally inertial) phase-oscillator models on graphs with time-dependent coupling $K(t)$, covering standard Kuramoto dynamics as a limit and including network and spatial topologies relevant to engineered settings. We propose an operational emergence criterion: a macroscopic phase is emergent only when it is robustly estimable, which we quantify via gauge-fixed phase-lag fluctuations under weak noise and finite sampling. This yields a quantitative threshold controlled by $NR^2$ and makes explicit why $\Psi$ is ill-posed in incoherent states even when formally definable. Nonautonomous coupling introduces a ramp timescale that competes with relaxation. Using a Laplacian-mode reduction near coherence, we derive a graph-spectral rate criterion: ordering tracks the protocol when $K(t)\lambda_2$ dominates the ramp rate, while faster ramps induce freeze-out. Numerically, we extract an operational freeze-out time from an energy-based tracking diagnostic and show that, for non-spatial networks, the residual incoherence at freeze-out collapses when plotted against the spectral protocol parameter $\lambda_2\tau$ across Erd\H{o}s--R\'enyi and small-world graph families. Finally, on periodic lattices we show that topological sectors and defect-mediated ordering obstruct complete alignment, leading to protocol-dependent, long-lived partially synchronized states and systematic deviations from spectral collapse.