Realisation and ordering of synchronising equitable partitions as coupling increases

Determine, for a given multiplex network or hypergraph and associated coupling strengths, which externally equitable partitions are realised as cluster-synchronised solutions and establish the order in which these cluster states emerge as the coupling parameters increase.

Background

The main results characterise when cluster-synchronised solutions can exist via structural equitability. However, whether a particular equitable partition is dynamically realised and in what sequence clusters synchronise as coupling strengths vary depends on stability and bifurcation structure, which are not addressed by the existence results.

The authors identify the need to generalise the realisation and ordering analysis—previously studied in the monolayer network case—to multiplex and hypergraph dynamics, where simultaneous equitability across layers or interaction orders adds complexity.