Equitability and explosive synchronisation in multiplex and higher-order networks

TL;DR

The paper develops a general theory for cluster synchronisation in multiplex and higher-order networks, revealing that equitability of the inter-cluster interaction pattern is a necessary and, under a linear-independence assumption, sufficient condition for the existence of independent cluster-synchronised solutions via quotient dynamics. It shows that explosive synchronisation naturally arises when no common equitable partition across layers exists, explaining the prevalence of abrupt global synchronisation in higher-order systems. The authors extend the framework to general coupling, provide quotient-dynamics constructions, and validate the theory with Lorenz-oscillator simulations on biplex networks and hypergraphs. They also discuss practical methods to find equitable partitions (e.g., WL refinement) and outline open questions on stability, algorithm efficiency, and non-identical units.

Abstract

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.

Figures (6)

• Figure 1: Multiplex and hypergraph dynamical systems. Toy example of $\mathbf{a)}$ a biplex (multilayer network with two layers with three identical nodes), and $\mathbf{b)}$ a hypergraph with pairwise and triadic interactions among three nodes, along with their respective adjacency matrices/tensors and dynamical equations. The shorthand $(123)$ indicates any permutation of the three indices. To illustrate equitability, consider the partition into clusters $\{1, 3\}$ and $\{2\}$: it is externally equitable for the multiplex $\mathbf{(a)}$ if $w_{12}^{(1)}=w_{23}^{(1)}$ (layer 1) and $w_{12}^{(2)}=w_{23}^{(2)}$ (layer 2); and for the hypergraph $\mathbf{(b)}$ if $w_{12}=w_{23}$.
• Figure 2: Equitability on multiplex networks. Two biplex networks $G_A=(G_A^{(1)}, G_A^{(2)})$ on the left and $G_B=(G_B^{(1)}, G_B^{(2)})$ on the right that exhibit very different dynamical behaviour: cluster synchronization on $G_A$, and explosive synchronization on $G_B$ (see Fig. \ref{['fig:multilayer-condition-fulfilled-and-not']} for the accompanying numerical results). This can be easily explained in terms of equitability: the green nodes form an equitable cluster in their respective layers, but only the common nodes $\{7,8,9,10\}$ are equitable overall on $G_A$, with no similar equitable partition on $G_B$.
• Figure 3: Equitability and synchronisation on two multiplex networks. Synchronization error Eq. \ref{['eq:syncerror']} of coupled Lorenz oscillators in the two biplex networks shown in Fig. \ref{['fig:multilayer-condition-fulfilled-and-not-graphs']} with respect to the cluster $\{7,8,9,10\}$. Panels A1 and A2 (respectively B1 and B2) show the cluster synchronization errors in $G_A$ (respectively $G_B$) when only layer 2, or layer 1, is active. Panels A3 and B3 show the $10^{-5}$ contours of the synchronization errors (i.e. the regions whose error is below or above $10^{-5}$) for $(\sigma_1,\sigma_2)\in [0,10]\times[0,10]$ for networks $G_A$ and $G_B$, respectively. See the SI for full details.
• Figure 4: Equitability and synchronisation on a hypergraph. Synchronization error of coupled Lorenz oscillators for four cluster on a hypergraph (fully described in the SI) with 20 nodes and both pairwise and triadic interactions. The clusters satisfy only pairwise respectively triadic structural equitability (clusters $C_1$ respectively $C_2$), none (cluster $C_3$) or simultaneously both (cluster $C_4$). The coupling parameter $\sigma_2$ is fixed at 0.2 while $\sigma_1$ is increased. Only cluster $C_4$ synchronises before the whole system does, as shown by the synchronisation error, which agrees with our prediction, as the only simultaneously equitable cluster for all types of interactions. Cluster $C_1$ becomes almost synchronous due to the triadic interactions becoming small with respect to the pairwise interactions for which $C_1$ does have external equitability.
• Figure 5: Multilayer network $G_A$ with the same non-trivial external equitable (or structurally equitable) partition in both layers.

Theorems & Definitions (33)

• Remark 1
• Remark 2: Multiplex and multilayer dynamics
• Definition 3
• Remark 4
• Remark 5
• Remark 6
• Theorem 7
• proof
• Definition 8
• Definition 9