Network dynamics with higher-order interactions: coupled cell hypernetworks for identical cells and synchrony

Abstract

Network interactions that are nonlinear in the state of more than two nodes—also known as higher-order interactions—can have a profound impact on the collective network dynamics. Here we develop a coupled cell hypernetwork formalism to elucidate the existence and stability of (cluster) synchronization patterns in network dynamical systems with higher-order interactions. More specifically, we define robust synchrony subspace for coupled cell hypernetworks whose coupling structure is determined by an underlying hypergraph and describe those spaces for general such hypernetworks. Since a hypergraph can be equivalently represented as a bipartite graph between its nodes and hyperedges, we relate the synchrony subspaces of a hypernetwork to balanced colourings of the corresponding incidence digraph.

Hypernetworks: Cluster Synchronization Is a Higher-Order Effect

“…Higher order interaction networks have found their way into various recent mathematical studies as well. We mention in particular the theoretical papers [4,5,6,7,8,9,10], which investigate synchronisation in classes of networks with non-pairwise, nonlinear interaction in their equation of motion. We also refer to the excellent surveys [11,12,13,14,15] and references therein, for an in-depth discussion of higher order networks, and numerous examples of higher order network systems arising in applications.…”

Chaos and Chaos Control in Network Dynamical Systems

“…Note that the requirement that the functions are invariant under permutations of the inputs y imposes conditions on the coefficients a (m+1) n (z): For example, we have to have a A particular choice of interaction function still leaves some ambiguity in terms of the network; see also [9] for a more detailed discussion. First, the interaction function G (m) (z; y 1 , .…”

“…These have famously been introduced and algebraically studied in [24,25,26,27] for coupled cell networks. More recently, first advances have been made to generalise balanced partitions to network dynamical systems with higher-order interactions [9,7,28,10]. For instance in order for vertices 1 and 2 to synchronize, the partition {1, 2}, {3} needs to be balanced, meaning that any vertex in an element of the partition needs to receive the same 'kind' of inputs from any element of the partition as any other vertex in that same element.…”

“…Aguiar and coworkers [9] developed a coupled cell network framework for network dynamics with higher order interactions. They consider dynamics on directed hypergraphs H [9] = (V [9] , H [9] ) where each edge e ∈ H [9] carries weight w e ∈ R; while they allow the tail T (e) to be a multiset, we restrict to the subclass where T (e) is a set. The dynamics of node k is determined by (6.3) ẋk = F (x k ) + e∈H [9] :k∈H(e)…”

“…have appeared in the literature, this approach allows to link the associated network structure (a hypergraph or simplicial complex) to dynamical features (such as synchronization behavior). Indeed, local dynamical features such as the existence and stability of (cluster) synchronization can be phrased in terms of the higher-order interaction structure [5,6,7,8,9,10].…”