https://doi.org/10.1016/j.physrep.2020.05.004

Networks beyond pairwise interactions: Structure and dynamics

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.

Dynamics on higher-order networks: a review

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. While first order phase transitions between incoherence and synchronization are critical for collective behavior in various oscillator system application, e.g., the brain and power grids, such transitions typically require finely tuned properties. In this work the authors show that first order phase transitions and bistability can emerge naturally as a consequence of the presence of higher-order interactions between oscillators.

/pdf/higher-order-interactions-in-complex-networks-of-phase-4z7dyu72ga.pdf

Higher-order interactions in complex networks of phase oscillators promote abrupt synchronization switching

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.

/pdf/higher-order-interactions-in-complex-networks-of-phase-126gjckhc9.pdf

The structure and dynamics of networks with higher order interactions

In the study of dynamical systems on networks or graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that, instead of microscopic details of the individual nodes or vertices, rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here, we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As, for instance, in the theory of coupled map lattices, we study Laplace-type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of different dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.

https://link.aps.org/pdf/10.1103/PhysRevE.101.062313

Coupled dynamics on hypergraphs: Master stability of steady states and synchronization.

Hypergraph Laplace operators for chemical reaction networks

Spectral Theory of Laplace Operators on Oriented Hypergraphs

We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $$N/(N-1)\leq \lambda_N\leq 2$$
 , to the case of chemical hypergraphs.

Sharp Bounds for the Largest Eigenvalue

We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $\frac{N}{N-1}\leq \lambda_N\leq 2$, to the case of chemical hypergraphs.

Sharp bounds for the largest eigenvalue of the normalized hypergraph Laplace Operator

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