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Synchronization of hypernetworks of coupled dynamical systems

We consider synchronization of coupled dynamical systems when different types of interactions are simultaneously present. We assume that a set of dynamical systems are coupled through the connections of two or more distinct networks (each of which corresponds to a distinct type of interaction), and we refer to such a system as a hypernetwork. Applications include neural networks formed of both electrical gap junctions and chemical synapses, the coordinated motion of shoals of fishes communicating through both vision and flow sensing, and hypernetworks of coupled chaotic oscillators. We first analyze the case of a hypernetwork formed of $m=2$ networks. We look for necessary and sufficient conditions for synchronization. We attempt at reducing the linear stability problem in a master stability function form, i.e., at decoupling the effects of the coupling functions from the structure of the networks. Unfortunately, we are unable to obtain a reduction in a master stability function form for the general case. However, we show that such a reduction is possible in three cases of interest: (i) the Laplacian matrices associated with the two networks commute; (ii) one of the two networks is unweighted and fully connected; (iii) one of the two networks is such that the coupling strength from node $i$ to node $j$ is a function of $j$ but not of $i$. Furthermore, we define a class of networks such that if either one of the two coupling networks belongs to this class, the reduction can be obtained independently of the other network. As an example of interest, we study synchronization of a neural hypernetwork for which the connections can be either chemical synapses or electrical gap junctions. We propose a generalization of our stability results to the case of hypernetworks formed of $m\geq 2$ networks.