Fibration Symmetries and Cluster Synchronization in Multi-Body Systems

TL;DR

The paper develops a framework connecting fibration symmetries in hypergraphs to cluster synchronization in higher-order Kuramoto models with frustration. By modeling HOI via incidence bipartite graphs, it shows that fibre partitions—derived from identical input structure—coincide with Kuramoto synchrony partitions under homogeneous conditions, and provides algorithms to compute and manipulate fibres. It analyzes both synthetic and real-world HOI datasets, demonstrating how representation choices affect observed synchronization patterns. The work also offers topology-modification strategies, including sparsification and structured edge additions, to recover or enforce desired symmetries in noisy or incomplete data, highlighting practical pathways for symmetry-guided clustering and control in complex HOI systems.

Abstract

Based on recent advances in fibration symmetry theory, we investigate how structural symmetries influence synchronization in systems with higher-order interactions (HOI). Using bipartite graph representations, we identify a node partition in fibres, based on equivalent incidence relations in hypergraphs. We study how identical nodes with an isomorphic input set can synchronize due to structural properties under our specific model assumptions, examining the dynamical model of Kuramoto with higher-order interactions and frustration parameters. Recent works established for directed hypergraphs that balanced partitions characterize robust synchrony, invariant under all admissible dynamics, whereas our contribution isolates the case of Kuramoto dynamics and shows that synchrony under homogeneous initial conditions and natural frequencies necessarily coincides with the fibration partition. As a conclusion, let us examine situations that require adjustments to the hypergraph topology to handle redundancy or to align with a target cluster configuration, especially in the presence of noise or incomplete information. These considerations open up new questions for future investigations. Our methodology combines theoretical modeling and simulations with applications to real-world data topologies, highlighting how representational choices and local input equivalence influence synchronization behavior.

Figures (15)

• Figure 1: Example of graph fibration $\varphi: G \rightarrow B$. Nodes with the same color belong to the same fibre in the total graph $G$ and are mapped by $\varphi$ to the unique node with the same color in the base graph $B$, which is the quotient graph.
• Figure 2: Starting from an undirected hypergraph, we can associate its incidence bipartite projection. Considering it as an inhomogeneous graph, we can apply standard graph algorithms, as the version of kamei_computation_2013 revised by morone_fibration_2020, to obtain the hypergraph node partition, here depicted with different colors.
• Figure 3: Hypergraph quotient via fibration symmetry. The left hypergraph shows the original structure where nodes with identical shapes and colors represent distinct vertices belonging to the same fibre under the fibration map $\varphi$ (i.e., nodes receiving equivalent input information). The right side depicts the quotient hypergraph, where nodes with matching shapes and colors represent a single supernode corresponding to an entire fibre. In the quotient representation, slightly overlapping identical nodes within hyperedges indicate multiple instances of the same supernode participating in that hyperedge, preserving the structural information while reducing redundancy through the fibration symmetry $\varphi$. Edges without arrows are undirected (equivalent to having arrows in both directions).
• Figure 4: Example of fibres for a hypergraph, its projection to a simple graph, and its projections to a graph with multiple edges. In each representation, nodes belonging to the same fibres are pictured with the same color and icon.
• Figure 5: Example of different fibres between the hypergraph representation and its multigraph projection in a real-world case for collaboration on publications for the ICML conference (MAG-10 dataset filtered for ICML conference). Nodes represent authors, and paper collaborations are depicted with a hyperedge among the authors, stressed with a paper icon. Nodes painted with the same color and mark belong to the same fibre. For the sake of the example, in this image, we represent only the papers related to authors labeled with codes 7979 and 4508. Nodes 4509 and 7980 are in different fibres in both cases as they are related (illustrated with dashed lines) to different other nodes not present in the picture.

Theorems & Definitions (33)

• Definition 2.1: Directed graph
• Definition 2.2: Undirected graph
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6: Quotient graph
• Definition 3.1: Directed hypergraph
• Remark 3.2
• Definition 3.3: Hypergraph
• Remark 3.4