Sensitive Dependence of Optimal Network Dynamics on Network Structure

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.

Matches the published version. A video explaining the main results is available at http://youtu.be/M8s2oXgEC1w

References in corpus (18)

• Synchronization in complex networks
• Diffusion dynamics on multiplex networks
• Reaction-diffusion processes and metapopulation models in heterogeneous networks
• Turing patterns in network-organized activator-inhibitor systems
• Pinning-controllability of complex networks
• Remote synchronization reveals network symmetries and functional modules
• Synchronization is optimal in non-diagonalizable networks
• Self-organized adaptation of a simple neural circuit enables complex robot behaviour
• Impact of Single Links in Competitive Percolation -- How complex networks grow under competition
• Master Stability Functions for Coupled Near-Identical Dynamical Systems
• Explosive Percolation: Novel critical and supercritical phenomena
• Maximum Performance at Minimum Cost in Network Synchronization
• Robustness of Optimal Synchronization in Real Networks
• Micro-transition cascades to percolation
• Enhancing the spectral gap of networks by node removal
• Network synchronizability analysis: the theory of subgraphs and complementary graphs
• Optimal synchronizability of bearings
• Comparing the Locking Threshold for Rings and Chains of Oscillators

Cited by in corpus (8)

• Higher-order interactions improve optimal collective dynamics on networks
• Functional control of network dynamics using designed Laplacian spectra
• Finite-size scaling of geometric renormalization flows in complex networks
• From Spectra to Localized Networks: A Reverse Engineering Approach
• Inferring synchronizability of networked heterogenous oscillators with machine learning
• Scaling properties of scale-free networks in degree-thresholding renormalization flows
• Disorder-promoted stability
• Predicting attractors from spectral properties of stylized gene regulatory networks