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 .
We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops .
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develop a mathematical theory for the linear stability
present a linear stability analysis
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(or fixed points)
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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
defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase
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only depends
depends only
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ensemble parameters, including
parameters, such as,
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we also characterize the unstable mode of the system of interest by deriving analytical expressions for the statistics of
in the unstable regime we characterize
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components of the
nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the
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Finally, we briefly discuss how this approach
We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory
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models
graphs
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non-oriented couplings
to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops