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.
Highlights
Building on the classical fields of graph theory, statistical physics, and nonlinear dynamics, as well as on the increasing availability of large-scale network data, the field of network dynamics has flourished over the past 15 years [1,2]
A fundamental question at the core of the structuredynamics relation is that of optimization: which network structures optimize the dynamics of the system for a given function and what are the properties of such networks? The significance of addressing this question is twofold
The sensitive dependence of collective dynamics on the network structure, characterized here by a derivative that diverges at an optimal point, has several implications
Summary
Building on the classical fields of graph theory, statistical physics, and nonlinear dynamics, as well as on the increasing availability of large-scale network data, the field of network dynamics has flourished over the past 15 years [1,2]. The identification of the network structures that guarantee the best fitness of natural complex systems can provide insights into the mechanisms underlying their evolution Examples of such systems include neuronal networks, whose (synaptic) connectivity structure is believed to have been optimized through evolution or learning for categorization tasks [39], synchronization efficiency [40], dynamical complexity [41,42], information transfer efficiency [42,43], and/or wiring cost [40]. While specific examples can be found in the literature [44,45,46,47], no systematic study exists on general mechanisms and conditions for such sensitivity We provide such conditions in terms of the spectral degeneracy of the network by establishing the scaling relation between the stability and the perturbation size.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
