Modules as effective nodes in coarse-grained networks of Kuramoto oscillators

TL;DR

This work tackles reducing modular networks of Kuramoto oscillators by coarse-graining each module into an effective node, inspired by EEG measurements that average activity over large neuron groups. It derives reduced module dynamics $\dot{\theta}_{\sigma} = \omega_{\sigma} + \sum_{\sigma'} \lambda_{\sigma\sigma'} \sin(\theta_{\sigma'} - \theta_{\sigma})$ and introduces modular synchrony weights $q_{\sigma}$ to bound the global order parameter when intra-module coherence is imperfect. For two modules, the global synchronization transition occurs near $\lambda_c \approx \omega$, with bounds $r \ge \frac{1}{2}\sqrt{q_1^2+q_2^2+2q_1q_2\cos\phi}$ and a two-oscillator closed form $r(\lambda,\omega) = \frac{1}{\sqrt{2}}\sqrt{1 + \sqrt{1 - (\omega/\lambda)^2}}$. For three modules, constant-$r$ contours approximate ellipses given by $\Omega_1^2 - \Omega_1\Omega_3 + \Omega_3^2 = \alpha$, with $\lambda_c^2 = \omega_1^2 - \omega_1\omega_3 + \omega_3^2$, offering a practical bound for synchronization. When applied to real networks (Karate club and C. elegans), the coarse-graining provides useful insights and a lower-bound predictor, but accuracy declines with weak modularity and structural heterogeneity, though overshooting the inner coupling can recover two-oscillator-like behavior. Overall, the method offers a principled, EEG-inspired framework for reducing modular oscillator networks while clarifying the conditions under which such reductions preserve essential synchronization dynamics.

Abstract

Most real-world networks exhibit a significant degree of modularity. Understanding the effects of such topology on dynamical processes is pivotal for advances in social and natural sciences. In this work we consider the dynamics of Kuramoto oscillators on modular networks and propose a simple coarse-graining procedure where modules are replaced by effective single oscillators. The method is inspired by EEG measurements, where very large groups of neurons under each electrode are interpreted as single nodes in a correlation network. We expose the interplay between intra-module and inter-module coupling strengths in keeping the coarse-graining process meaningful and show that its accuracy depends on the degree of intra-module synchronization. We show that, when modules are well synchronized, the phase transition from asynchronous to synchronous motion in networks with 2 and 3 modules is very well described by their respective reduced systems, regardless of the network structure connecting the modules. Application of the method to real networks with small modularity coefficients, on the other hand, reveals that the approximation is not accurate, although it still allows for the computation of the critical coupling and the qualitative behavior of the order parameter if the inter-module coupling is large enough.

Figures (5)

• Figure 1: (a): Numerical integration of Eq. \ref{['multi pop II/ku']} for $\lambda=2$. Green, yellow and purple curves are the order parameters $r_1$, $r_2$ and $r$ and dashed line represents $q_\sigma=0.9$ synchrony threshold (all referring to left $y$ axis). Black ladder function shows $\lambda_{\text{in}}$ values (referring to the right $y$ axis). (b)-(c): Final $\lambda_{\text{in}}$ and its respective $\langle r \rangle$ with one standard deviation $\sigma$ above and bellow in purple. In (b) magenta and turquoise lines marks the average of critical $\lambda$ and $\lambda_{\text{in}}$ over 3 simulations. In (c) triangles represent $\sigma>0.01$ and circles otherwise, green ribbon marks the 2 oscillators limits where the top green curve is Eq. \ref{['2 osc r(w,l)']} and bottom boundary is Eq. \ref{['multi pop II/2 pop/r']}. Theoretical mean and standard deviation for the asynchronized region are shown with dashed green lines. Panel (d) shows the analogous of panel (c) for minimal modular synchrony of $q_\sigma=0.7$, lower ribbon boundary is given by Eq. \ref{['r q1q2']} with $q_\sigma=0.7$. Simulation values: $N_1=200$, $N_2=100$ and $p=0.5$. Gaussian frequency distributions with $\Delta=1$ and $(\omega_1,\omega_2)=(1.5,0)$.
• Figure 2: Dots represents the numerical integration of system \ref{['multi pop II/ku']} for 3 different connection probabilities $p$ and frequency offsets $\omega$. (a): Critical inter-module coupling strength. Lines represent a 2 oscillators system. (b): Minimum inner-module coupling strength at phase transition. Lines simply connect the points. Global values: $N_1=200$ and $N_2=100$. Gaussian frequency distributions with $\Delta=1$ and $(\omega_1,\omega_2)=(1.5,0)$. Error bars made over 3 simulations.
• Figure 3: (a): Eq. \ref{['3 osc r of phi']} in the phase space $\phi_{12}\times\phi_{23}$. Red curve divides stable from unstable points, stable within. (b): Stable points reparameterized to the parameter space $\Omega_1\times\Omega_3$ via Eq. \ref{['pts fixos w1 w3']}. (c): How stable $r$ values changes through the ellipses. Dashed lines are the empirical values from Eq. \ref{['3 osc r proposed w12 w23']}. (d): Contrast between the ellipses \ref{['3 osc ellipse regions w12w23']} (dashed) and constant $r$ values (continuous).
• Figure 4: (a)-(c): Final $\lambda_{\text{in}}$, global order parameter $r$ and standard deviation $\sigma$ in the frequency parameter space. Simulation values: $N_1=90$, $N_2=100$, $N_3=110$, $p=0.01$ and $\lambda=3$. Gaussian frequency distributions with $\Delta=1$ and means $\omega_1$, $\omega_2=0$ and $\omega_3$.
• Figure 5: Top: Dynamics of Eq. \ref{['multi pop II/ku']} for Zachary's Karate social network. (a): Final $\lambda_{\text{in}}$. Magenta and turquoise lines marks the average of critical $\lambda$ and $\lambda_{\text{in}}$. (b): $\langle r \rangle$ with $\lambda_{\text{in}}$ relative to (a) in purple and with fixed $\lambda_{\text{in}}=20$ in red, green ribbon represents Eq. \ref{['r q1q2']} for $0.9\leq q_\sigma\leq1$. Simulation values: $\Delta=0$, $\omega_1=1.5$ and $\omega_2=0.0$. Bottom: Dynamics of Eq. \ref{['multi pop II/ku']} for C. Elegans gap junctions network. (c)-(e): $\langle r \rangle$ for the network with fixed $\lambda_{\text{in}}=20$ in red, equivalent coarsed-grained system in green. Simulation values: $\Delta=0$, (a): $(\omega_1,\omega_2,\omega_3)=(0,1,6)$; (b): $(\omega_1, \omega_2, \omega_3, \omega_4, \omega_5)=(0,1,1,-2,7)$; (c): $(\omega_1,\omega_2,\omega_3,\omega_4,\omega_5,\omega_6,\omega_7,\omega_8,\omega_9,\omega_{10})=(0,1,-3.5,-2,2,0.6,0.4,1.2,1.5,-1.5)$.