Fast and exhaustive algorithms for equitable partitions in multiplexes and hypergraphs

Develop fast and exhaustive algorithms to find externally equitable partitions of the node set in arbitrary multiplex networks and hypergraphs (higher-order networks), computed directly from the per-layer adjacency matrices or the higher-order adjacency tensors, respectively, and satisfying the simultaneous equitability constraints required for independent cluster synchronisation.

Background

The paper proves that, for systems of identical units with multilayer and higher-order interactions, layer-by-layer (multiplex) or interaction-order-by-order (hypergraph) structural equitability is a necessary and sufficient condition for the existence of independent cluster-synchronised solutions. Consequently, identifying all equitable partitions of a given multiplex or hypergraph is fundamental to predicting which cluster synchronisation patterns can exist.

While equitable partitions and symmetry-based methods are well studied in monolayer networks, the simultaneous constraints across layers or interaction orders in multiplexes and hypergraphs are significantly more restrictive and algorithmically challenging. The authors therefore highlight the need for fast and exhaustive procedures to compute all such partitions in these richer settings.