On the propagation of instability in interconnected networks

We consider how instability, when due to local interactions between agents in one part of a network, affects other parts of the network. In this initial work, we consider a stable bipartite system with homogeneous linear dynamics in each partition. The initially stable system is driven to the onset of instability by a local gain perturbation and we define a measure which indicates how the size of the resulting oscillations decays with nodal distance. For interconnections defined on d =1;2 dimensional lattices, we determine the asymptotic value of this measure, as the size of the network increases, using a Markov chain framework. In addition, approximate results are given for interconnection topologies described by a classical random graph model.