Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with the competing simplex-encoded couplings

Abstract

Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviours on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here, we investigate the phase synchronisation dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous 4-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds $d_s=4$. We determine the time-averaged system's order parameter to characterise the synchronisation level. In the absence of higher interactions, the complete synchrony is continuously reached with the increasing positive pairwise interactions ($K_1>0$), and a partial synchronisation for the negative couplings ($K_1<0$) with no apparent hysteresis. Similar behaviour occurs in the degree-preserved randomised network. In contrast, the synchronisation is absent for the negative pairwise coupling in the entirely random graph and simple scale-free networks. Increasing the strength $K_2\ne 0$ of the triangle-based interactions gradually hinders the synchronisation promoted by pairwise couplings, and the non-symmetric hysteresis loop opens with an abrupt desynchronisation transition towards the $K_1<0$ branch. However, for substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronisation, and the abrupt transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronisation processes.

9 pages, 6 figures included

References in corpus (8)

• Synchronization in complex networks
• Networks beyond pairwise interactions: structure and dynamics
• Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplices
• Higher-order simplicial synchronization of coupled topological signals
• Growing scale-free simplices
• D-dimensional oscillators in simplicial structures: odd and even dimensions display different synchronization scenarios
• Altered structural balance of resting-state networks in autism
• Hidden geometry and dynamics of complex networks: Spin reversal in nanoassemblies with pairwise and triangle-based interactions

Cited by in corpus (8)

• Dynamics on higher-order networks: A review
• Weighted simplicial complexes and their representation power of higher-order network data and topology
• Higher-Order Components Dictate Higher-Order Contagion Dynamics in Hypergraphs
• Turing patterns on discrete topologies: from networks to higher-order structures
• Diffusion-driven instability of topological signals coupled by the Dirac operator
• Fundamental interactions in self-organized critical dynamics on higher-order networks
• Hyper-diffusion on multiplex networks
• Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behaviour of phase oscillators on 5-cliques simplicial assemblies