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.
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This work is dedicated in memory of Robert May. We wish to thank I. Conforti for designing inspiring artwork to accompany our scientific research. C.M. thanks the Planning and Budgeting Committee (PBC) of the Council for Higher Education, Israel, for support. C.M. is also supported by the INSPIRE-Faculty grant (code IFA19-PH248) of the Department of Science and Technology, India. C.H. is supported by the INSPIRE-Faculty grant (code IFA17-PH193) of the Department of Science and Technology, India. S.H. has contributed to this work while visiting the mathematics department of Rutgers University, New Brunswick. S.B. acknowledges funding from the project EXPLICS granted by the Italian Ministry of Foreign Affairs and International Cooperation. This research was also supported by the Israel Science Foundation (grant no. 499/19), the Israel-China ISF-NSFC joint research program (grant no. 3552/21), the US National Science Foundation CRISP award no. 1735505, and by the Bar-Ilan University Data Science Institute grant for data science research. In memory of Robert May.
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