Grass-roots optimization of coupled oscillator networks

Despite the prevalence of biological and physical systems for which synchronization is critical, existing theory for optimizing synchrony depends on global information and does not sufficiently explore local mechanisms that enhance synchronization. Thus, there is a lack of understanding for the self-organized, collective processes that aim to optimize or repair synchronous systems, e.g., the dynamics of paracrine signaling within cardiac cells. Here we present “grass-roots” optimization of synchronization, which is a multiscale mechanism in which local optimizations of smaller subsystems cooperate to collectively optimize an entire system. Considering models of cardiac tissue and a power grid, we show that grass-roots-optimized systems are comparable to globally optimized systems, but they also have the added benefit of being robust to targeted attacks or subsystem islanding. Our findings motivate and support further investigation into the physics of local mechanisms that can support self-optimization for complex systems.

Figure 1

Grass-roots optimization for a network of heterogenous phase oscillators. (a) Visualization of Kuramoto order parameter $r\in [0,1]$ and mean field $\psi \in [0,2\pi )$ for a set $\left\{{\theta }_j\right\}$ of oscillator phases with ${\theta }_j\in [0,2\pi )$. Strong phase synchronization occurs when ${\theta }_i\approx {\theta }_j$ for any $i$ and $j$, which yields $r\approx 1$. (b) Synchrony-optimized networks that maximize $r$ can be obtained using the synchrony alignment function (SAF) [22], which reveals two microscale, intuitive mechanisms that promote synchronization: positive correlations between an oscillator's natural frequency ${\omega }_i$ and its associated node degree $d_i$; and negative correlations among the frequencies ${\omega }_i$ and ${\omega }_j$ of neighboring oscillators $i$ and $j$. (c) Here, we develop grass-roots optimization to reveal a multiscale mechanism for optimization with two steps: the mean frequency $\langle {\omega }^{\left(s\right)}\rangle$ within each subsystem $s$ is balanced across the subsystems; and subsystems are separately optimized.

Figure 2

Grass-roots optimization. Illustrations of (a) a random network with two communities, (b) the IEEE RTS 96 power grid, and (c) a random geometric network. (d)–(f) The degree of synchronization $r$ and (g)–(i) synchronization error $1-r$ as a function of coupling strength $K$ for the three respective network types with either randomly allocated frequencies (green triangles), globally optimized frequencies (blue circles), or grass-roots-optimized frequencies (red crosses).

Figure 3

Robustness to islanding and target attacks. (a) Example of local (subsystem) order parameters for the RTS 96 power grid before and after islanding at $t=0$ for global (solid blue) and grass-roots (dashed red) optimization. (b) Density of local (subsystem) SAFs after islanding for global (solid blue) and grass-roots (dashed red) optimization. (c) Illustration of the islanded subsystems.

Figure 4

Grass-roots optimization of cardiac pacemakers. For a geometric network of $N=100$ cardiac pacemakers with two subsystems, the time series of non-dimensional voltage for (a) random, (b) globally optimized, and (c) grass-roots-optimized allocations.