Equitability and explosive synchronisation in multiplex and higher-order networks

Cluster synchronisation is a key phenomenon observed in networks of coupled dynamical units. Its presence has been linked to symmetry and, more generally, to equability of the underlying pattern of interactions between dynamical units. However, it is not known under which conditions equitability-induced synchronisation is the only cluster synchronisation that can occur on a particular system. Here, we reveal a natural linear independent condition such that equitability becomes necessary, and sufficient, for the existence of cluster synchronised solutions on a very general dynamical system which allows multiplex or higher-order, arbitrarily weighted interactions. Our results explain the ubiquity of explosive synchronisation, as opposed to cluster synchronisation, in multiplex and higher-order networks: equitability imposes additional constraints that must be simultaneously satisfied on the same set of nodes. Our results have significant implications for the design of complex dynamical systems of coupled dynamical units with arbitrary cluster synchronisation patterns and coupling functions.

💡 Research Summary

The paper investigates the fundamental conditions under which cluster synchronization can arise in very general coupled dynamical systems that feature multiplex (multilayer) and higher‑order (hypergraph) interactions. The authors consider N identical D‑dimensional units whose intrinsic dynamics are described by a smooth vector field f(x) and whose coupling is captured by a set of functions h_i that aggregate inputs from all other units. In the multiplex case, each of the M layers has its own adjacency matrix A^{(m)} and coupling strength σ_m, leading to a total coupling term that is a sum over layers (Eq. 2). In the hypergraph case, interactions of order m involve an (m+1)-dimensional adjacency tensor A^{(m)} and a corresponding coupling function g^{(m)} (Eq. 3).

A central assumption throughout most of the analysis is that all coupling functions are “non‑invasive”: g^{(m)}(x,…,x)=0 for any state x. This guarantees that when a set of nodes is synchronized, the internal coupling among them vanishes, so the only requirement for a synchronized cluster is that each node receives the same external input from the rest of the network. The authors formalize this as dynamical equitability (Eq. 8): for any two nodes i, j belonging to the same cluster C, the external input h_ext,i equals h_ext,j.

They then introduce the purely structural notion of external equitability (Eq. 10). A partition P of the node set is externally equitable if, for every pair of distinct clusters C and C′ and for every layer (or interaction order) m, the aggregated structural input from C′ to any node in C is identical. In other words, the adjacency matrices (or tensors) must induce the same row sums for all nodes within a cluster when restricted to connections from any other cluster.

The key technical contribution is the identification of a linear independence condition for the inter‑cluster coupling functions (Eq. 11). A cluster‑synchronised solution is called independent if, for each ordered pair of distinct clusters (C, C′), the collection of functions {g^{(m)}(x_C, x_{C′}) | m = 1,…,M} is linearly independent as functions of time along the trajectories of the synchronized clusters. Under this condition the authors prove that dynamical equitability and external equitability are equivalent: a P‑synchronised, independent solution exists if and only if the partition P is externally equitable. Consequently, external equitability becomes a necessary and sufficient structural condition for the existence of cluster‑synchronised states in these general systems.

Because external equitability must hold simultaneously on every layer (or every interaction order), the condition becomes dramatically more restrictive as the number of layers or the order of hyperedges increases. In multiplex networks with a few layers it may be satisfied by many partitions, but in hypergraphs with higher‑order interactions the same partition must balance inputs across many distinct tensors, which is often impossible. This structural bottleneck explains why explosive synchronization—a sudden, global transition to synchrony—appears far more frequently in higher‑order settings: the network cannot meet the external equitability constraints required for partial (cluster) synchrony, so the only viable synchronized state is the fully synchronized one, which emerges abruptly as coupling strength crosses a threshold.

The authors also discuss extensions beyond non‑invasive couplings. For generic coupling functions they define a broader notion of general equitability, and show that the equivalence with dynamical equitability still holds, albeit with additional algebraic constraints that depend on the specific form of the g^{(m)}. In the non‑invasive case the analysis is clean because internal inputs vanish automatically; in the generic case internal inputs must be accounted for, and stability analysis becomes essential.

To illustrate the theory, the paper presents simple toy examples: a three‑node, two‑layer multiplex and a three‑node hypergraph with pairwise and triadic interactions. By explicitly constructing adjacency matrices/tensors and choosing coupling functions, they demonstrate how a partition can be externally equitable in the multiplex case but fail in the hypergraph case, leading to cluster synchrony in the former and explosive synchrony in the latter.

Finally, the paper outlines algorithmic approaches for detecting externally equitable partitions in large networks, noting that the problem reduces to solving linear equations derived from the row‑sum constraints of each layer/tensor. The authors argue that these tools enable systematic design of networks that either promote desired cluster synchrony (by enforcing equitability) or deliberately avoid it (to induce explosive transitions).

In summary, the work provides a rigorous, unified framework linking structural equitability to the existence of cluster‑synchronised solutions across multiplex and higher‑order networks, clarifies why explosive synchronization is prevalent in high‑order settings, and offers practical guidelines for engineering complex dynamical systems with prescribed synchronization patterns.