Hypernetworks: cluster synchronisation is a higher-order effect

TL;DR

This paper explores how cluster synchronization in hypernetworks is a higher-order nonlinear effect, determined by high-order polynomial admissible maps, leading to complex bifurcations.

Contribution

It introduces a classification of robust cluster synchronization patterns on hypernetworks using polynomial admissible maps and extends graph fibration concepts to hypernetworks.

Findings

Synchronization patterns are governed by high-order polynomial maps.

Cluster synchronization is a nonlinear, higher-order effect in hypernetworks.

High polynomial degree can cause bifurcations and synchronization breaking.

Abstract

Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronisation patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronisation patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronisation on hypernetworks is a higher-order, i.e., nonlinear effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronisation patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by displaying how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.