  The stable functionality of networked systems is a hallmark of their natural
ability to coordinate between their multiple interacting components. Yet,
strikingly, real-world networks seem random and highly irregular, apparently
lacking any design for stability. What then are the naturally emerging
organizing principles of complex-system stability? Encoded within the system's
stability matrix, the Jacobian, the answer is obscured by the scale and
diversity of the relevant systems, their broad parameter space, and their
nonlinear interaction mechanisms. To make advances, here we uncover emergent
patterns in the structure of the Jacobian, rooted in the interplay between the
network topology and the system's intrinsic nonlinear dynamics. These patterns
help us analytically identify the few relevant control parameters that
determine a system's dynamic stability. Complex systems, we find, exhibit
discrete stability classes, from asymptotically unstable, where stability is
unattainable, to sensitive, in which stability abides within a bounded range of
the system's parameters. Most crucially, alongside these two classes, we
uncover a third class, asymptotically stable, in which a sufficiently large and
heterogeneous network acquires a guaranteed stability, independent of
parameters, and therefore insensitive to external perturbation. Hence, two of
the most ubiquitous characteristics of real-world networks - scale and
heterogeneity - emerge as natural organizing principles to ensure stability in
the face of changing environmental conditions.
