On the location and the strength of controllers to desynchronize coupled Kuramoto oscillators

TL;DR

The paper tackles desynchronization of coupled Kuramoto oscillators via non-invasive pinning control, comparing random, degree-based, and functionability-based node selection on networks with heterogeneous degree distributions. By embedding Kuramoto dynamics in a Hamiltonian framework and applying a small control term to a subset of controllers, it shows that targeting high-degree hubs most effectively desynchronizes scale-free and core-periphery networks, while concentrating pins in the core yields benefits in core-periphery structures. However, overly many controllers or strong control can backfire, and in small-world networks functionability underperforms relative to degree-based strategies. Real brain connectomes largely follow degree-based pinning, with degree distribution emerging as a key determinant; overall, the results point to a possible universal principle: efficient control is achieved by minimizing distances between controllers and the rest of the network and by focusing on highly connected or strategically central core nodes.

Abstract

Synchronization is an ubiquitous phenomenon in dynamical systems of networked oscillators. While it is often a goal to achieve, in some context one would like to decrease it, e.g., although synchronization is essential to the good functioning of brain dynamics, hyper-synchronization can induce problems like epilepsy seizures. Motivated by this problem, scholars have developed pinning control schemes able to decrease synchronization in a system. Focusing on one of these methods, the goal of the present work is to analyse which is the best way to select the controlled nodes, i.e. the one that guarantees the lower synchronization rate. We show that hubs are generally the most advantageous nodes to control, especially when the degree distribution is heterogeneous. Nevertheless, pinning a too large number of hubs is in general not an appropriate choice. Our results are in line with previous works that studied pinning control aimed to increase synchronization. These observations shed light on an interesting universality of good practice of node selection disregarding the actual goal of the control scheme.

Figures (19)

• Figure 1: Representation of different synchronization states. Dots (red online) denote the Kuramoto oscillators at a fixed time, the (blue online) dot represents $R(t)e^{i\psi(t)}$ and the black segment its module. On the left panel the phases exhibit an asynchronous behavior and thus $R\approx0$; on the right panel the oscillators are almost perfectly synchronized and thus $R\approx1$. In the middle panel we report an intermediate case.
• Figure 2: Compare node centrality, betweenness centrality and functionability. We show two simple networks where nodes size is proportional to their degree (left), the inverse of their betweenness (middle) and their functionability (right).
• Figure 3: Evolution of $R(t)$ on a star network composed by $N=10$ oscillators modeled with the original KM, i.e., without controlled nodes (blue curve), with the controlled KM where the hub and one leaf have been pinned (red curve) and where two leaves act as controllers (black curve). Each curve has been obtained by averaging the results of $50$ independent simulations, where natural frequencies are distributed according to a normal distribution $\mathcal{N}(1,0.1)$ and initial conditions for angles are uniformly distributed in $[0,2\pi]$. We can observe that the original KM quickly synchronizes, the blue curve rapidly tends to a number close to $1$, while the controlled KM do not synchronize as testified by the low values of $R(t)$. Moreover the choice of controlling the hub seems to have the better desynchronization feature, indeed the red curve is well below the black one.
• Figure 4: Normalized order parameter, $\hat{R}$, of the controlled KM defined on a scale free network, $\gamma=-3$, as a function of the number of controllers, $M$, and the control signal strength. By using a color code we show the value $\hat{R}$ averaged over $100$ scale free networks. Pinned nodes have been selected uniformly at random (left panel), on a degree-based selection (middle panel) and by using the functionability criterion (right panel). The red lines indicate the level contours at the threshold $\hat{R}=0.15$.
• Figure 5: We present the size, $\delta$, of parameters region resulting into a well desynchronized state, $\hat{R}\leq 0.15$, as a function of $\gamma$ for scale free networks and the three used controllers selection methods. The darker lines correspond to the average of $\delta$ obtained by using $100$ independent networks realization, while the shaded areas about the average denote one standard deviation.