Geometry, Topology and Simplicial Synchronization

TL;DR

This paper explores how the geometry and topology of simplicial complexes influence their spectral properties and, consequently, the synchronization behavior of the Kuramoto model, revealing phase transition phenomena and the importance of spectral dimension.

Contribution

It demonstrates the impact of simplicial complex geometry and topology on spectral dimensions and synchronization phases, introducing a higher-order Kuramoto model with novel transition behaviors.

Findings

Spectral dimension determines the ability of simplicial complexes to sustain synchronization.

Non-trivial simplicial network geometry prevents infinite network synchronization if spectral dimension ≤ 4.

Higher-order topology influences the dynamical properties of topological signals and their synchronization transitions.

Abstract

Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the graph Laplacian and of the higher-order Laplacians of simplicial complexes. Here we show how the geometry of simplicial complexes induces spectral dimensions of the simplicial complex Laplacians that are responsible for changing the phase diagram of the Kuramoto model. In particular, simplicial complexes displaying a non-trivial simplicial network geometry cannot sustain a synchronized state in the infinite network limit if their spectral dimension is smaller or equal to four. This theoretical result is here verified on the Network Geometry with Flavor simplicial complex generative model displaying emergent hyperbolic geometry. On its turn simplicial topology is shown to determine the dynamical properties of the higher-order Kuramoto model. The higher-orderKuramoto model describes synchronization of topological signals, i.e. phases not only associated to the nodes of a simplicial complexes but associated also to higher-order simplices, including links, triangles and so on. This model displays discontinuous synchronization transitions when topological signals of different dimension and/or their solenoidal and irrotational projections are coupled in an adaptive way.