Collective dynamics on higher-order networks

TL;DR

This work surveys how higher-order interactions reshape collective dynamics in networks, introducing generalized Kuramoto models with triplet and higher-order couplings described by tensors and multiorder Laplacians. It connects node-dynamics with higher-order structure through dimensionality-reduction tools such as the master stability function and phase reduction, and discusses when nonpairwise terms drive phenomena like explosive transitions, multistability, and complex synchronization patterns. The review further covers data-driven reduction and reconstruction of higher-order structures, including order minimization, cross-order diffusion renormalization, and both model-based and model-free inference, as well as dynamics on edges and hyperedges via simplicial Kuramoto topologies and Dirac-type couplings. It highlights open questions about optimal hypergraph designs for synchronization, coupling-function selection, and scalable inference, and provides practical tooling (hypersync) to study these systems, underscoring the potential impact for neuroscience, ecology, and beyond.

Abstract

Higher-order interactions that nonlinearly couple more than two nodes are ubiquitous in networked systems. Here we provide an overview of the rapidly growing field of dynamical systems with higher-order interactions, and of the techniques which can be used to describe and analyze them. We focus in particular on new phenomena that emerge when nonpairwise interactions are considered. We conclude by discussing open questions and promising future directions on the collective dynamics of higher-order networks.

Figures (6)

• Figure 1: Dynamics and higher-order networks. Coupled dynamical units such as Kuramoto oscillators can exhibit ordered and disordered states such as (a-b) synchronization, and (c-d) incoherence. (e) Typically, in networks there is a continuous transition from incoherence to synchronization as the coupling strength is increased. Adding higher-order (f) structure, and (g) coupling to the network can fundamentally change the dynamics, which is the focus of this review.
• Figure 2: Effect of higher-order structure on dynamics. Various structural properties affect dynamics such as (a) degree homogeneity, (b) cross-order degree correlation, defined as the correlation between degree sequences from different orders, and (c) intra-order overlap, measuring the overlap between hyperedges of the same order. For example, (d) simpliciality (how close is a hypergraph from becoming a simplicial complex), which affects both degree homogeneity and cross-order degree correlation, can change whether higher-order interactions stabilize or destabilize synchronization.
• Figure 3: Higher-order interactions can induce explosive transitions and multistability. (a) Hysteresis loop can develop between order parameter and dyadic coupling strength when sufficiently strong higher-order interactions are present. (b--c) The choice of nonpairwise coupling function, for example, asymmetric (\ref{['eq:explosive:01']}, panel b) or symmetric (\ref{['eq:explosive:03']}, panel c), can lead to distinct stable states and bifurcation diagrams.
• Figure 4: BOX PANEL: Phase descriptions of coupled oscillators.
• Figure 5: One can decrease the complexity of a higher-order network by either (a) reducing the maximum order of interactions, or (b) coarse grain it into fewer nodes. Both reduction and renormalization utilize information from observing the dynamics unfolding on the nodes. Complementarily, when the coupling structure is unknown to begin with, there are methods to reconstruct it from observed time series (c).