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Emergent stability in complex network dynamics
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DOI：10.1038/s41567-023-02020-8.png)
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The stable functionality of networked systems is a hallmark of their natural ability to coordinate between their multiple interacting components. Yet, real-world networks often appear random and highly irregular, raising the question of what are the naturally emerging organizing principles of complex system stability. The answer is encoded within the system's stability matrix-the Jacobian-but is hard to retrieve, due to the scale and diversity of the relevant systems, their broad parameter space and their nonlinear interaction dynamics. Here we introduce the dynamic Jacobian ensemble, which allows us to systematically investigate the fixed-point dynamics of a range of relevant network-based models. Within this ensemble, we find that complex systems exhibit discrete stability classes. These range from asymptotically unstable (where stability is unattainable) to sensitive (where stability abides within a bounded range of system parameters). Alongside these two classes, we uncover a third asymptotically stable class in which a sufficiently large and heterogeneous network acquires a guaranteed stability, independent of its microscopic parameters and robust against external perturbation. Hence, in this ensemble, two of the most ubiquitous characteristics of real-world networks-scale and heterogeneity-emerge as natural organizing principles to ensure fixed-point stability in the face of changing environmental conditions. Despite looking highly irregular, most real-world networks exhibit natural stability to external perturbations. A study of the properties of the stability matrix of networks now sheds light on the principles underlying this emerging stability.
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