Large and small fluctuations in oscillator networks from heterogeneous and correlated noise

Technical Summary: Large and Small Fluctuations in Oscillator Networks from Heterogeneous and Correlated Noise

Problem Statement
Coupled oscillator networks, such as those modeling power grids and neuronal dynamics, exhibit collective behaviors like synchronization. These systems often possess multiple stable and unstable states. While external noise inevitably induces fluctuations around stable synchronized states, most existing analyses assume noise is homogeneous in intensity and uncorrelated across network components. However, real-world systems frequently exhibit heterogeneous noise intensities (e.g., varying input power fluctuations) and correlations (due to spatial embedding or unmodeled interactions). This manuscript addresses the gap in understanding how such realistic noise characteristics affect two distinct regimes: small fluctuations (SFs) near synchronized states and large fluctuations (LFs) that drive the system to saddle instabilities, causing phase slips and basin escapes. A central question investigated is whether the magnitude of small fluctuations (network variance) reliably predicts the rate of large fluctuations (escape times).

Methodology
The authors analyze a network of Kuramoto oscillators subject to noise, governed by the differential equation:
$$\dot{\theta}_i = \omega_i + K \sum_j A_{ij} \sin(\theta_j - \theta_i) + \sum_m G_{im}\xi_m(t)$$
where $\xi_m(t)$ are independent white Gaussian processes and $G$ is a feed-in matrix encoding noise intensity and correlation.

1. Small Fluctuations (SFs): The authors perform a linear stability analysis around a synchronized fixed point $\theta^*$. By expanding the dynamics to the first order, they derive an expression for the total network variance $\langle \delta\theta^T \delta\theta \rangle$. This variance is shown to be proportional to an effective noise amplitude $q$, which depends on the noise covariance matrix $GG^T$ and the eigenvalues/eigenvectors of the network's weighted Laplacian.
2. Large Fluctuations (LFs): To analyze rare events leading to saddle instabilities, the authors introduce the Indicator Mode Approximation (IMA). This technique projects the high-dimensional network dynamics onto a single amplitude dimension $a(t)$, representing the distance between the current state and a saddle point. This reduction transforms the problem into a one-dimensional nonlinear system with additive noise.
3. Rate Estimation: Using the IMA, the authors construct the Fokker-Planck equation and apply the WKB approximation to derive the "action" $S$, which determines the exponential rate of large fluctuations. The rate is governed by an effective noise amplitude $Q$, distinct from the SF amplitude $q$.
4. Noise Models: The study evaluates four specific noise configurations:
   • Heterogeneous and uncorrelated node noise.
   • Edge-correlated and edge-anti-correlated noise.
   • Collective-mode noise (noise aligned with specific network eigenmodes).
   • Targeted node noise (noise concentrated on specific oscillators).

Key Contributions and Results

• Quantification of Small Fluctuations: The authors derive an analytical formula (Eq. 5) linking network variance to the overlap between noise covariance and the slow modes of the synchronized state.

  • For heterogeneous uncorrelated noise, variance is amplified in weakly connected networks, networks with central hubs, and those with large diameters. In dense networks, heterogeneity effects average out.
  • For anti-correlated edge noise, the total network variance in the strong coupling limit depends only on the network size $N$, not the topology.
  • For correlated edge noise, variance is significantly larger than in the anti-correlated case for sparse networks with large diameters.
  • For collective-mode noise, variance is maximized when noise is injected into the slowest stable mode (the Fiedler mode).

• Quantification of Large Fluctuations: The IMA provides a tractable method to estimate escape rates. The escape rate is determined by an effective noise amplitude $Q$ (Eq. 17).

  • For heterogeneous node noise, LFs are maximized when noise targets nodes that must travel the farthest to reach the saddle, rather than necessarily targeting the Fiedler mode.
  • For edge-correlated noise, LFs are maximized when the correlation sign between connected nodes aligns with the product of their displacements to the saddle.
  • For collective-mode noise, the IMA captures the trend of slip times, though with moderate spread due to the high correlation.

• Decoupling of SF and LF Regimes: A primary finding is that the total network variance (SF regime) does not necessarily predict the rate of large fluctuations (LF regime).

  • In the case of randomly heterogeneous noise, there is a strong correlation ($R \approx 0.83-0.84$) between SF variance and LF rates.
  • However, for targeted noise and edge-correlated noise, the correlation is significantly weaker ($R \approx 0.21-0.41$). This demonstrates that a network with high small-fluctuation variance does not automatically imply a high rate of basin escape, and vice versa.

Significance and Claims
The paper claims to bridge the gap between small and large fluctuation analyses in noisy oscillator networks by introducing distinct effective noise amplitudes ($q$ for SFs and $Q$ for LFs) for heterogeneous and correlated noise environments. The authors demonstrate that while small fluctuations are well-described by linearized network variance, large fluctuations depend on a different projection of the noise onto the specific path to instability.

The significance lies in the finding that larger network variance in the small fluctuation regime does not necessarily lead to higher rates of large fluctuations. This challenges the assumption that monitoring small-scale variance is sufficient for predicting the stability or escape rates of complex networks. The Indicator Mode Approximation (IMA) is presented as a novel analytical tool that reduces the dimensionality of the escape problem while preserving the exact boundary conditions (stable state and saddle), offering a more efficient alternative to numerical solutions of high-dimensional Hamiltonian systems. The work suggests that network design strategies aimed at noise mitigation must consider the specific regime of interest (small excursions vs. basin escape) and the specific structure of noise correlations, as optimizing for one does not guarantee optimization for the other.