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The need to build a link between the structure of a complex network and thedynamical properties of the corresponding complex system (comprised of multiplelow dimensional systems) has recently become apparent. Several attempts totackle this problem have been made and all focus on either the controllabilityor synchronisability of the network --- usually analyzed by way of the masterstability function, or the graph Laplacian. We take a different approach. Usingthe basic tools from dynamical systems theory we show that the dynamicalstability of a network can easily be defined in terms of the eigenvalues of anhomologue of the network adjacency matrix. This allows us to compute thestability of a network (a quantity derived from the eigenspectrum of theadjacency matrix). Numerical experiments show that this quantity is veryclosely related too, and can even be predicted from, the standard structuralnetwork properties. Following from this we show that the stability of largenetwork systems can be understood via an analytic study of the eigenvalues oftheir fixed points --- even for a very large number of fixed points.

The stability of networks --- towards a structural dynamical systems theory