We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure. PACS numbers: 05.45.Xt, 89.75.Hc A central goal of complexity theory is to understand the emergence of collective behavior in large ensembles of interacting dynamical systems. Synchronization of networkcoupled oscillators has served as a paradigm for understanding emergence [1][2][3][4], where examples arise in nature (e.g., flashing of fireflies [5] and cardiac pacemaker cells [6]), engineering (e.g., power grid [7] and bridge oscillations [8]), and at their intersection (e.g., synthetic cell engineering [9]). We consider the dynamics of N network-coupled phase oscillators θ i for i = 1, . . . , N , whose evolution is governed bẏHere ω i is the natural frequency of oscillator i, K > 0 is the coupling strength, [A ij ] is a symmetric network adjacency matrix, and H is a 2π-periodic coupling function [10]. We treat H(θ) with full generality so long as H ′ (0) > 0. The choices H(θ) = sin(θ) and H(θ) = sin(θ − α) with the phase-lag parameter α ∈ (−π/2, π/2) yield the classical Kuramoto [10] and Sakaguchi-Kuramoto models [11].Considerable research has shown that the underlying structure of a network plays a crucial role in determining synchronization [12][13][14][15][16][17][18][19][20][21], yet the precise relationship between the dynamical and structural properties of a network and its synchronization remains not fully understood. One unanswered question is, given an objective measure of synchronization, how can synchronization be optimized? One application lies in synchronizing the power grid [22], where sources and loads can be modeled as oscillators with different frequencies. To this end, we ask: what structural and/or dynamical properties should be present to optimize synchronization?We measure the degree of synchronization of an ensemble of oscillators using the Kuramoto order parameterHere re iψ denotes the phases' centroid on the complex unit circle, with the magnitude r ranging from 0 (incoherence) to 1 (perfect synchronization) [10]. In general, the question of optimization (maximizing r) is challenging due to the fact that the macroscopic dynamics depend on both the natural frequencies and the network structure. To quantify the interplay between node dynamics and network structure, we derive directly from Eqs. (1) and (2) a synchrony alignment function which is an objective measure of synchronization and can be used to systematically optimize a network's synchronization. We highlight this result by addressing two classes of op...

Collective behavior in large ensembles of dynamical units with non-pairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.PACS numbers: 05.45.Xt, 89.75.Hc Research into the macroscopic dynamics of large ensembles of coupled oscillators have extended our understanding of natural and engineered systems ranging from cell cycles to power grids [1][2][3][4][5]. However, with few exceptions (including [6-8]), little attention has been paid to the synchronization dynamics of coupled oscillator systems where interactions are not pair-wise, but rather n-way, with n ≥ 3. Such interactions are called "simplicial", where an n-simplex represents an interaction between n + 1 units, so 2-simplices describe three-way interactions, etc [9]. Recent advances suggest that simplicial interactions may be vital in general oscillator systems [10-12] and may play an important role in brain dynamics [13][14][15] and other complex systems phenomena such as, the dynamics of collaborations [16] or social contagion [17]. In particular, interactions in 2-simplices (named holes or cavities) are important because they can describe correlations in neuronal spiking activity (that can be mapped to phase oscillators [18]) in the brain [19] providing a missing link between structure and function. In fact, coupled oscillator systems that display clustering and multi-branch entrainment have been shown to be useful models for memory and information storage [20][21][22][23][24][25][26]. Despite these findings, the general collective dynamics of coupled oscillator simplices and their utility in storing information are poorly understood.In this work we study large coupled oscillator simplicial complexes, considering the impact of 2-simplices, i.e., threeway interactions, on collective behavior. Specifically, we consider the 2-and 1-simplex multilayer system given bẏThe dynamics in the θ-layer are the natural generalization of the classical Kuramoto model [27] with 2-simplex interactions (namely, coupling is sinusoidal and diffusive), where θ i represents the phase of oscillator i with i = 1, . . . , N , ω i is its natural frequency which is assumed to be drawn from the distribution g(ω), an...

Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems. Recent work in physics and neuroscience highlights the importance of higher-order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior. Here we show that higher-order interactions between coupled phase oscillators, encoded microscopically in a simplicial complex, give rise to added nonlinearity in the macroscopic system dynamics that induces abrupt synchronization transitions via hysteresis and bistability of synchronized and incoherent states. Moreover, these higher-order interactions can stabilize strongly synchronized states even when the pairwise coupling is repulsive. These findings reveal a self-organized phenomenon that may be responsible for the rapid switching to synchronization in many biological and other systems that exhibit synchronization without the need of particular correlation mechanisms between the oscillators and the topological structure.