Finance

Stability (learning theory)

Combinatorics

Physics

Topology (electrical circuits)

Invariant (physics)

Pure mathematics

Living systems

Mathematics

Dynamical systems theory

Statistical physics

Machine learning

Order (exchange)

Pairwise comparison

Artificial intelligence

Economics

Simplicial complex

Synchronization (alternating current)

Computer science

Complex system

Quantum mechanics

Theoretical computer science

Nature Communications

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Stability of synchronization in simplicial complexes