Abstract
The relation between network structure and dynamics is determinant for the behavior of complex systems in numerous domains. An important long-standing problem concerns the properties of the networks that optimize the dynamics with respect to a given performance measure. Here, we show that such optimization can lead to sensitive dependence of the dynamics on the structure of the network. Specifically, using diffusively coupled systems as examples, we demonstrate that the stability of a dynamical state can exhibit sensitivity to unweighted structural perturbations (i.e., link removals and node additions) for undirected optimal networks and to weighted perturbations (i.e., small changes in link weights) for directed optimal networks. As mechanisms underlying this sensitivity, we identify discontinuous transitions occurring in the complement of undirected optimal networks and the prevalence of eigenvector degeneracy in directed optimal networks. These findings establish a unified characterization of networks optimized for dynamical stability, which we illustrate using Turing instability in activator-inhibitor systems, synchronization in power-grid networks, network diffusion, and several other network processes. Our results suggest that the network structure of a complex system operating near an optimum can potentially be fine-tuned for a significantly enhanced stability compared to what one might expect from simple extrapolation. On the other hand, they also suggest constraints on how close to the optimum the system can be in practice. Finally, the results have potential implications for biophysical networks, which have evolved under the competing pressures of optimizing fitness while remaining robust against perturbations.
2 More- Received 3 November 2016
DOI:https://doi.org/10.1103/PhysRevX.7.041044
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Complex systems with many interacting components can be represented as a network, where the components and their interactions are described graphically with nodes and links, respectively. Such systems include chemical-reaction pathways, social hierarchies, and electrical circuits. Networks are often tailored to optimize the performance of the system as a whole. Examples include neuronal networks, which may have evolved to maximize information-transfer efficiency or to minimize synaptic-wiring cost, and power grids, where both maximizing stability and minimizing cost are desirable. An important long-standing problem in the study of complex systems is thus to understand the properties of such optimized networks. We show that network optimization can lead to a sensitive dependence of the dynamics on the structure of the network.
Specifically, we demonstrate that the stability of dynamical states in optimal networks can be highly sensitive to the removal or addition of links and nodes, or to small changes in link weights. In undirected networks (where all interactions are symmetric), this sensitivity arises from discontinuous transitions caused by optimization. In directed networks, it arises from the prevalence of eigenvector degeneracy. We illustrate these findings with analyses for several different types of networks, such as networks of activator-inhibitor systems and networks of synchronizing power generators.
Our results suggest that the network structure of a complex system operating near an optimum can potentially be fine-tuned or controlled to significantly enhance stability while also setting limits on how close to the optimum the system can be in practice. These findings have potential implications for man-made networks, where optimization may inadvertently lead to fragility, and for biophysical networks, which appear to have evolved under the competing pressures of optimizing fitness while remaining robust against perturbations.