Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis

Generalized N-rotor problems, synchronized subsystems, and associated solitons

We consider systems of N particles interacting on the unit circle through 2π-periodic potentials. An example is the N-rotor problem that arises as the classical limit of coupled Josephson junctions and for various energies is known to have a wide range of behaviors such as global chaos and ergodicity, together with families of periodic solutions and transitions from order to chaos. We focus here on selected initial values for generalized systems in which the second order Euler-Lagrange equations reduce to first order equations, which we show by example can describe an ensemble of oscillators with associated emergent phenomena such as synchronization. A specific case is that of the Kuramoto model with well-known synchronization properties. We further demonstrate the relation of these models to field theories in 1+1 dimensions that allow static kink solitons satisfying first order Bogomolny equations, well-known in soliton physics, which correspond to the first order equations of the generalized N-rotor models. For the nonlinear pendulum, for example, the first order equations define the separatrix in the phase portrait of the system and correspond to kink solitons in the sine-Gordon equation.

The Loss Surface of Deep Linear Networks Viewed Through the Algebraic Geometry Lens

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of deep linear neural network models. After providing clarification on the various definitions of "flat" minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized L₂-regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Combining these upper bounds with a method in numerical algebraic geometry, we find all stationary points for modest depth and matrix size. We demonstrate that, in the presence of the non-zero regularization, deep linear networks can indeed possess local minima which are not global minima. Finally, we show that even though the number of stationary points increases as the number of neurons (regularization parameters) increases (decreases), higher index saddles are surprisingly rare.

Functional control of oscillator networks

Oscillatory activity is ubiquitous in natural and engineered network systems. The interaction scheme underlying interdependent oscillatory components governs the emergence of network-wide patterns of synchrony that regulate and enable complex functions. Yet, understanding, and ultimately harnessing, the structure-function relationship in oscillator networks remains an outstanding challenge of modern science. Here, we address this challenge by presenting a principled method to prescribe exact and robust functional configurations from local network interactions through optimal tuning of the oscillators’ parameters. To quantify the behavioral synchrony between coupled oscillators, we introduce the notion of functional pattern, which encodes the pairwise relationships between the oscillators’ phases. Our procedure is computationally efficient and provably correct, accounts for constrained interaction types, and allows to concurrently assign multiple desired functional patterns. Further, we derive algebraic and graph-theoretic conditions to guarantee the feasibility and stability of target functional patterns. These conditions provide an interpretable mapping between the structural constraints and their functional implications in oscillator networks. As a proof of concept, we apply the proposed method to replicate empirically recorded functional relationships from cortical oscillations in a human brain, and to redistribute the active power flow in different models of electrical grids.

In network systems governed by oscillatory activity, such as brain networks or power grids, configurations of synchrony may define network functions. The authors introduce a control approach for the formation of desired synchrony patterns through optimal interventions on the network parameters.

Sufficiently dense Kuramoto networks are globally synchronizing

Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least μ(n-1) other oscillators. There is a critical value of the connectivity, μ_c, such that whenever μ>μ_c, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when μ<μ_c, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be μ_c=0.75. In 2020, Lu and Steinerberger proved that μ_c≤0.7889, and Yoneda, Tatsukawa, and Teramae proved in 2021 that μ_c>0.6838. This paper proves that μ_c≤0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.

Data-driven control of complex networks

Our ability to manipulate the behavior of complex networks depends on the design of efficient control algorithms and, critically, on the availability of an accurate and tractable model of the network dynamics. While the design of control algorithms for network systems has seen notable advances in the past few years, knowledge of the network dynamics is a ubiquitous assumption that is difficult to satisfy in practice. In this paper we overcome this limitation, and develop a data-driven framework to control a complex network optimally and without any knowledge of the network dynamics. Our optimal controls are constructed using a finite set of data, where the unknown network is stimulated with arbitrary and possibly random inputs. Although our controls are provably correct for networks with linear dynamics, we also characterize their performance against noisy data and in the presence of nonlinear dynamics, as they arise in power grid and brain networks.

Consensus on simplicial complexes: Results on stability and synchronization

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.

Clusterization and phase diagram of the bimodal Kuramoto model with bounded confidence

Inspired by the Deffuant and Hegselmann-Krause models of opinion dynamics, we extend the Kuramoto model to account for confidence bounds, i.e., vanishing interactions between pairs of oscillators when their phases differ by more than a certain value. We focus on Kuramoto oscillators with peaked, bimodal distribution of natural frequencies. We show that, in this case, the fixed-points for the extended model are made of certain numbers of independent clusters of oscillators, depending on the length of the confidence bound-the interaction range-and the distance between the two peaks of the bimodal distribution of natural frequencies. This allows us to construct the phase diagram of attractive fixed-points for the bimodal Kuramoto model with bounded confidence and to analytically explain clusterization in dynamical systems with bounded confidence.

Dense networks that do not synchronize and sparse ones that do

Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least μ(n-1) other oscillators. Then, there is a critical value of μ above which the system is guaranteed to converge to the in-phase synchronous state for almost all initial conditions. The precise value of μ remains unknown. In 2018, Ling, Xu, and Bandeira proved that if each oscillator is coupled to at least 79.29% of all the others, global synchrony is ensured. In 2019, Lu and Steinerberger improved this bound to 78.89%. Here, we find clues that the critical connectivity may be exactly 75%. Our methods yield a slight improvement on the best known lower bound on the critical connectivity from 68.18% to 68.28%. We also consider the opposite end of the connectivity spectrum, where the networks are sparse rather than dense. In this regime, we ask how few edges one needs to add to a ring of n oscillators to turn it into a globally synchronizing network. We prove a partial result: all the twisted states in a ring of size n=2^m can be destabilized by adding just O(nlog₂⁡n) edges. To finish the proof, one needs to rule out all other candidate attractors. We have done this for n≤8 but the problem remains open for larger n. Thus, even for systems as simple as Kuramoto oscillators, much remains to be learned about dense networks that do not globally synchronize and sparse ones that do.

Multistability in lossy power grids and oscillator networks

Networks of phase oscillators are studied in various contexts, in particular, in the modeling of the electric power grid. A functional grid corresponds to a stable steady state such that any bifurcation can have catastrophic consequences up to a blackout. Also, the existence of multiple steady states is undesirable as it can lead to transitions or circulatory flows. Despite the high practical importance there is still no general theory of the existence and uniqueness of steady states in such systems. Analytic results are mostly limited to grids without Ohmic losses. In this article, we introduce a method to systematically construct the solutions of the real power load-flow equations in the presence of Ohmic losses and explicitly compute them for tree and ring networks. We investigate different mechanisms leading to multistability and discuss the impact of Ohmic losses on the existence of solutions.

Directed acyclic decomposition of Kuramoto equations

The Kuramoto model is one of the most widely studied models for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general nonuniform, heterogeneous, and sparse network is important yet challenging due to complicated nonlinear interactions. From the view point of homotopy deformation, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks, which lays the foundation for a divide-and-conquer approach to studying the configurations of frequency synchronization of large Kuramoto networks.

Power grid stability under perturbation of single nodes: Effects of heterogeneity and internal nodes

Power flow dynamics in electricity grids can be described by equations resembling a Kuramoto model of non-linearly coupled oscillators with inertia. The coupling of the oscillators or nodes in a power grid generally exhibits pronounced heterogeneities due to varying features of transmission lines, generators, and loads. In studies aiming at uncovering mechanisms related to failures or malfunction of power systems, these grid heterogeneities are often neglected. However, over-simplification can lead to different results away from reality. We investigate the influence of heterogeneities in power grids on stable grid functioning and show their impact on estimating grid stability. Our conclusions are drawn by comparing the stability of an Institute of Electrical and Electronics Engineers test grid with a homogenized version of this grid.

Mean-field approach for frequency synchronization in complex networks of two oscillator types

Oscillator networks with an asymmetric bipolar distribution of natural frequencies are useful representations of power grids. We propose a mean-field model that captures the onset, form, and linear stability of frequency synchronization in such oscillator networks. The model takes into account a broad class of heterogeneous connection structures and identifies a functional form as well as basic properties that synchronized regimes possess classwide. The framework also captures synchronized regimes with large phase differences that commonly appear just above the critical threshold. Additionally, the accuracy of mean-field assumptions can be gauged internally through two model quantities. With our framework, the impact of local grid structure on frequency synchronization can be systematically explored.

Cycle flows and multistability in oscillatory networks

We study multistability in phase locked states in networks of phase oscillators under both Kuramoto dynamics and swing equation dynamics-a popular model for studying coarse-scale dynamics of an electrical AC power grid. We first establish the existence of geometrically frustrated states in such systems-where although a steady state flow pattern exists, no fixed point exists in the dynamical variables of phases due to geometrical constraints. We then describe the stable fixed points of the system with phase differences along each edge not exceeding π/2 in terms of cycle flows-constant flows along each simple cycle-as opposed to phase angles or flows. The cycle flow formalism allows us to compute tight upper and lower bounds to the number of fixed points in ring networks. We show that long elementary cycles, strong edge weights, and spatially homogeneous distribution of natural frequencies (for the Kuramoto model) or power injections (for the oscillator model for power grids) cause such networks to have more fixed points. We generalize some of these bounds to arbitrary planar topologies and derive scaling relations in the limit of large capacity and large cycle lengths, which we show to be quite accurate by numerical computation. Finally, we present an algorithm to compute all phase locked states-both stable and unstable-for planar networks.

Synchronization of cyclic power grids: Equilibria and stability of the synchronous state

Synchronization is essential for the proper functioning of power grids; we investigate the synchronous states and their stability for cyclic power grids. We calculate the number of stable equilibria and investigate both the linear and nonlinear stabilities of the synchronous state. The linear stability analysis shows that the stability of the state, determined by the smallest nonzero eigenvalue, is inversely proportional to the size of the network. We use the energy barrier to measure the nonlinear stability and calculate it by comparing the potential energy of the type-1 saddles with that of the stable synchronous state. We find that the energy barrier depends on the network size (N) in a more complicated fashion compared to the linear stability. In particular, when the generators and consumers are evenly distributed in an alternating way, the energy barrier decreases to a constant when N approaches infinity. For a heterogeneous distribution of generators and consumers, the energy barrier decreases with N. The more heterogeneous the distribution is, the stronger the energy barrier depends on N. Finally, we find that by comparing situations with equal line loads in cyclic and tree networks, tree networks exhibit reduced stability. This difference disappears in the limit of N→∞. This finding corroborates previous results reported in the literature and suggests that cyclic (sub)networks may be applied to enhance power transfer while maintaining stable synchronous operation.

Frequency spirals

We study the dynamics of coupled phase oscillators on a two-dimensional Kuramoto lattice with periodic boundary conditions. For coupling strengths just below the transition to global phase-locking, we find localized spatiotemporal patterns that we call "frequency spirals." These patterns cannot be seen under time averaging; they become visible only when we examine the spatial variation of the oscillators' instantaneous frequencies, where they manifest themselves as two-armed rotating spirals. In the more familiar phase representation, they appear as wobbly periodic patterns surrounding a phase vortex. Unlike the stationary phase vortices seen in magnetic spin systems, or the rotating spiral waves seen in reaction-diffusion systems, frequency spirals librate: the phases of the oscillators surrounding the central vortex move forward and then backward, executing a periodic motion with zero winding number. We construct the simplest frequency spiral and characterize its properties using analytical and numerical methods. Simulations show that frequency spirals in large lattices behave much like this simple prototype.

Phase-locked patterns of the Kuramoto model on 3-regular graphs

We consider the existence of non-synchronized fixed points to the Kuramoto model defined on sparse networks: specifically, networks where each vertex has degree exactly three. We show that "most" such networks support multiple attracting phase-locked solutions that are not synchronized and study the depth and width of the basins of attraction of these phase-locked solutions. We also show that it is common in "large enough" graphs to find phase-locked solutions where one or more of the links have angle difference greater than π/2.

Linear stability and the Braess paradox in coupled-oscillator networks and electric power grids

We investigate the influence that adding a new coupling has on the linear stability of the synchronous state in coupled-oscillator networks. Using a simple model, we show that, depending on its location, the new coupling can lead to enhanced or reduced stability. We extend these results to electric power grids where a new line can lead to four different scenarios corresponding to enhanced or reduced grid stability as well as increased or decreased power flows. Our analysis shows that the Braess paradox may occur in any complex coupled system, where the synchronous state may be weakened and sometimes even destroyed by additional couplings.

Index statistical properties of sparse random graphs

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P(N)(K,λ) that a large N×N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P(N)(K,λ), from which the typical sample-to-sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N≫1 for |λ|>0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a nonmonotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as lnN, with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.