Spectral theory for the stability of dynamical systems on large oriented locally tree-like graphs
We develop a mathematical theory for the linear stability of stationary states in large dynamical systems modelled by a set of randomly coupled differential equations on a locally tree-like network. Our approach provides analytical expressions for the leading eigenvalue of random matrices that describe the interactions between the degrees of freedom; the sign of the leading eigenvalue characterizes the system stability. We illustrate this approach on oriented random graphs with a prescribed degree distribution and find that the leading eigenvalue is universal in the sense that it only depends on a few ensemble parameters, including the mean degree and a degree correlation coefficient. In addition, we also characterize the unstable mode of the system of interest by deriving analytical expressions for the statistics of the components of the right and left eigenvectors associated with the leading eigenvalue. Finally, we briefly discuss how this approach can be extended to models with diagonal disorder and non-oriented couplings.