Model reduction for Kuramoto models with complex topologies

Synchronization of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics to social networks and technology. A fundamental and long-time goal in the study of synchronization has been to find low-order descriptions of complex oscillator networks and their collective dynamics. However, for the Kuramoto model, the most widely used model of coupled oscillators, this goal has remained surprisingly challenging, in particular for finite-size networks. Here, we propose a model reduction framework that effectively captures synchronization behavior in complex network topologies. This framework generalizes a collective coordinates approach for all-to-all networks [G. A. Gottwald, Chaos 25, 053111 (2015)] by incorporating the graph Laplacian matrix in the collective coordinates. We first derive low dimensional evolution equations for both clustered and nonclustered oscillator networks. We then demonstrate in numerical simulations for Erdős-Rényi networks that the collective coordinates capture the synchronization behavior in both finite-size networks as well as in the thermodynamic limit, even in the presence of interacting clusters.

Figure 1

Snapshot of the phases ${\phi }_i$ obtained from simulating the full Kuramoto model (1) against the collective coordinate ansatz ${\mathrm{\Phi }}_i$ (8) for $K=180$, with $\alpha =1.03$ obtained as the stationary solution of (10), for a small-world topology with $N=200$ oscillators, each with two neighbors and a rewiring probability $p=0.3$, and native frequencies drawn from a normal distribution $\mathcal{N}(0,0.1)$. The corresponding value of the order parameter is $\overline{r}=0.92$. The continuous line indicates perfect correspondence between the ansatz and the observed phases.

Figure 2

Snapshot of the phases ${\phi }_i$ obtained from simulating the full Kuramoto model (1) against the collective coordinate ansatz ${\mathrm{\Phi }}_i^{(1,2)}$ (16), with ${\alpha }_1=1.003,{\alpha }_2=1.006,f_1=-0.095$, and $f_2=0.11$ obtained from (18) and (19). The network is an ER network with $N=500$ which consists of two topological clusters with only a few intercluster connections. The two clusters are labeled with triangles (online red) and open circles (online blue), respectively. The native frequencies are drawn from a normal distribution $\mathcal{N}(0,0.02)$. The snapshot is taken at $K=160$ and the corresponding value of the order parameter is $\overline{r}=0.83$. The continuous line indicates perfect correspondence between the ansatz and the observed phases.

Figure 3

Right-hand side $\mathcal{F}\left(\alpha \right)$ of the evolution equation (11) for several values of $K$ for the $N=500$ ER network with uniformly distributed native frequencies with a subcritical coupling strength $K=20<K_c$, near critical coupling strength $K=28\approx K_c$, and supercritical coupling strength $K=35>K_c$.

Figure 4

Order parameter $\overline{r}$ as a function of the coupling strength $K$ for an ER network with uniformly distributed native frequencies. Depicted are results from a direct numerical integration of the Kuramoto model (1) (continuous line, online blue) and from the collective coordinate approach (8) using (23) (crosses, online red). (a) ER network with $N=2000$ nodes. (b) ER network with $N=500$ nodes.

Figure 5

Equilibrium solution ${\alpha }^{★}$ for the $N=500$ ER network with uniformly distributed native frequencies for the same network as in Fig. 4. The solution was obtained solving (11) for a set of nodes ${\mathcal{C}}_l$ consisting of linearly stable nodes according to (13). We only plot the equilibrium solution which is closest to $\alpha =1$ from the two stationary solutions of (11).

Figure 6

Order parameter $\overline{r}$ as a function of the coupling strength $K$ for an ER network with normally distributed native frequencies. Depicted are results from a direct numerical integration of the Kuramoto model (1) (continuous line, online blue) and from the collective coordinate approach (8) using (23) (crosses, online red). (a) ER network with $N=2000$ nodes. (b) ER network with $N=500$ nodes.

Figure 7

Normalized domain length $L_{\mathrm{domain}}$ as a function of the coupling strength $K$ for the ER networks depicted in Fig. 6 with $N=2000$ nodes and normally distributed native frequencies.

Figure 8

Order parameter $\overline{r}$ as a function of the coupling strength $K$ for an ER network consisting of two coupled topological clusters with normally distributed native frequencies. The network is the same as that used for Fig. 2. Depicted are results from a direct numerical integration of the Kuramoto model (1) (continuous line, online blue) and from the collective coordinate approach (8) using (23) (crosses, online red).

Figure 9

Normalized eigenvector $\widehat{v}$, corresponding to the (single) positive eigenvalue of $L_{\mathrm{lin}}$ for an ER network consisting of two coupled topological clusters with normally distributed native frequencies. The network is the same as that used for Fig. 2 and the coupling strength is $K=142$. The eigenvector clearly separates into the two clusters with $N=270$ and 230 nodes, respectively. For visual aid, only every sixth component is shown.