Synchronization and Control Theory

Synchronization is relevant in a wide range of disciplines. This phenomenon is commonly studied using the Kuramoto model, which describes the dynamics of a system of coupled oscillators with varying natural frequencies and sinusoidal coupling. In electric power systems, the synchronization of generators is essential for ensuring a reliable supply of electricity. The generator's dynamics is described by a second order Kuramoto model where the sinusoidal coupling term gives the flow of real power across transmission lines. Stable operation requires that phases are locked and phase differences across lines must be below a certain bound $\gamma$, a condition referred to as phase cohesiveness. Hence, it is of utmost interest to understand when a network supports a phase cohesive state and when it does not.In this contribution we introduce a novel approach to the synchronization and phase cohesiveness problem based on mapping the fixed point equations to a convex optimization problem. This approach allows us to systematically compute all synchronized states where the phase difference across an edge does not exceed $\pi/2$, including exotic states with loop flows. Furthermore, our approach sheds light onto the commonly used ``DC'' approximation, where the sinusoidal coupling function is linearized. We derive rigorous bounds on the error introduced by the linear approximation, obtaining a region of trust for individual lines. Based on these results, we find a rigorous sufficient condition on the existence of phase cohesive states that holds regardless of the underlying network topology. We demonstrate our results for an adapted Matpower 30-bus test case.

Identifying the driver nodes of a network has crucial implications in complex systems from unveiling causal interactions to informing effective intervention strategies. Despite recent advances in network control theory, results remain inaccurate as the number of drivers becomes too small compared to the network size, thus limiting the concrete usability in many real-life applications. To overcome this issue, we introduced a framework that integrates principles from spectral graph theory and output controllability to project the network state into a smaller topological space formed by the Laplacian network structure. Through extensive simulations on synthetic and real networks, we showed that reducing the dimension of the original network state to a lower number of projected components improves the control accuracy, here measured as the capacity of the driver nodes to effectively steer the system towards desired target configurations. Taken together, our results offer a theoretically-grounded solution to effectively translate network controllability into concrete applications.

Collective synchronization of the Kuramoto model has been extensively studied in diverse scientific disciplines, and since then, it has been generalized in various perspectives. As the Kuramoto model itself is defined on the unit circle as a one-dimensional object, it however would not be suitable to model high-dimensional behaviors appropriately. Moreover, another limitation is that it is restricted to two-body interactions. In this presentation based on [1], we focus on the high-dimensional Kuramoto model particularly together with three-body interactions. For this model, {we find a critical threshold for complete synchronizability in terms of interaction strengths. Precisely, by denoting $\kappa_1$ and $\kappa_2$ for two- and three-body interaction strengths, respectively, we show that if $\kappa_1+\kappa_2>0$, then complete synchronization can emerge, whereas if $\kappa_1+\kappa_2<0$, then complete synchronization cannot occur. For the critical case $\kappa_1+\kappa_2=0$,} we show that the emergence of complete synchronization crucially depends on the sign of $\kappa_1$ and particularly that critical slowing down is observed at critical transition. It should be mentioned that the line $\kappa_1+\kappa_2=0$ was mentioned in [2] where the readers can easily notice that the line seems critical. Our theoretical results are supplemented by numerical experiments which also provide qualitative insight not captured in the theoretical analysis.

We recently presented (doi:10.1073/pnas.2303332120) an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, as a function of the underlying directed network structure. We consider both continuous-time and discrete-time linear dynamics on networks, which can represent linearization around an attractor for non-linear dynamics such as the Kuramoto model. These are a common starting point, leading to well-known bounds on eigenvalues for whether a given network structure will synchronise or not.Going beyond "sync-or-not", our new approach is able to fully and generally relate the structure of a network to how well it will synchronize. We quantify the quality of synchronization in terms of total expected deviation from synchronization in response to a single perturbation, or equivalently steady-state divergence from synchronization (or variance across nodes) under driving noise. Our approach represents a substantial advance over existing methods: we handle directed, weighted connections, and non-diagonalizable networks.More importantly, our approach can be expanded to fully interpret the complete impact of local network structure on whole of network synchronizability. Our results show that the deviation from synchronization is fully determined by the proportion of paired walks (or process motifs) on the network which converge rather than diverge: the more convergent paired walks, the worse the quality of synchronization. A simple corollary is that more clustered structure, which induces more convergent walks, leads to deviations away from synchronization. This explains many known results, for example that regular and small-world networks exhibit worse quality synchronization than random networks. Our results also reveal subtle differences in the relevant walks involved in determining deviation from synchronization in discrete-time versus continuous-time dynamics.

Network controllability integrates classical control theory with structural network insights from large-scale biological systems, such as intracellular molecular and brain neuronal networks[1-3]. While the set of minimum driver nodes is non-unique, critical nodes, those present in every solution, are crucial for network control. A key challenge in network control is dealing with edge failures and uncertainties in molecular interactions, especially in directed probabilistic networks [4]. To address this, we introduce a probabilistic control framework based on the minimum dominating set (MDS), which incorporates the probabilistic nature of directed molecular interactions to identify critical control nodes. Our algorithm, enhanced by novel mathematical propositions, offers efficient identification of critical nodes in large probabilistic networks [5]. As shown in Figure 1ab, the human intracellular signal transduction network is represented with links having varying failure probabilities. To efficiently identify control categories, we use a preprocessing step based on four mathematical propositions that capture a large fraction of control nodes (Fig. 1c). The remaining nodes are identified using an integer linear programming (ILP)-based algorithm (Fig. 1d). Our analysis shows that a small fraction of nodes are classified as critical node (Fig. 1e). Further exhaustive analysis of multiple diseases reveals that critical control nodes are associated with key biological features and perturbed gene sets, including SARS-CoV-2 targets, Type 1 Diabetes Mellitus (Fig. 1f), and rare diseases like cytokine receptor deficiency. This methodology can be applied to study biological systems where directed edges are probabilistic, whether in natural systems or those modeled with high uncertainties in silico.