Analyzing states beyond full synchronization on hypergraphs requires methods beyond projected networks

TL;DR

This paper demonstrates that analyzing hypergraphs requires methods beyond simple projections to accurately understand their structure and dynamics, especially for complex synchronization patterns.

Contribution

It introduces a framework that considers full hypergraph structure and edge clusters, improving analysis of cluster synchronization and stability beyond projected network methods.

Findings

Projected networks can misrepresent hypergraph dynamics.

Symmetry groups of hypergraphs differ from those of their projections.

Full hypergraph analysis is essential for studying complex dynamics.

Abstract

A common approach for analyzing hypergraphs is to consider the projected adjacency or Laplacian matrices for each order of interactions (e.g., dyadic, triadic, etc.). However, this method can lose information about the hypergraph structure and is not universally applicable for studying dynamical processes on hypergraphs, which we demonstrate through the framework of cluster synchronization. Specifically, we show that the projected network does not always correspond to a unique hypergraph structure. This means the projection does not always properly predict the true dynamics unfolding on the hypergraph. Additionally, we show that the symmetry group consisting of permutations that preserve the hypergraph structure can be distinct from the symmetry group of its projected matrix. Thus, considering the full hypergraph is required for analyzing the most general types of dynamics on hypergraphs. We show that a formulation based on node clusters and the corresponding edge clusters induced by the node partitioning, enables the analysis of admissible patterns of cluster synchronization and their effective dynamics. Additionally, we show that the coupling matrix projections corresponding to each edge cluster synchronization pattern, and not just to each order of interactions, are necessary for understanding the structure of the Jacobian matrix and performing the linear stability calculations efficiently.