Emergent stability in complex network dynamics

TL;DR

This paper uncovers how the interplay of network topology and nonlinear dynamics leads to emergent stability patterns in complex systems, revealing classes of stability and the role of heterogeneity and scale.

Contribution

It identifies emergent stability classes in complex networks and shows how scale and heterogeneity naturally promote robustness independent of parameters.

Findings

Identification of three stability classes: unstable, sensitive, and stable.

Discovery that large, heterogeneous networks can be inherently stable.

Most real-world networks fall into the stable class due to their structure.

Abstract

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.