Universal transient behavior in large dynamical systems on networks

Abstract

We analyze how the transient dynamics of large dynamical systems in the vicinity of a stationary point, modeled by a set of randomly coupled linear differential equations, depends on the network topology. We characterize the transient response of a system through the evolution in time of the squared norm of the state vector, which is averaged over different realizations of the initial perturbation. We develop a mathematical formalism that computes this quantity for graphs that are locally tree-like. We show that for unidirectional networks the theory simplifies and general analytical results can be derived. For example, we derive analytical expressions for the average squared norm for random directed graphs with a prescribed degree distribution. These analytical results reveal that unidirectional systems exhibit a high degree of universality in the sense that the average squared norm only depends on a single parameter encoding the average interaction strength between the individual constituents. In addition, we derive analytical expressions for the average squared norm for unidirectional systems with fixed diagonal disorder and with bimodal diagonal disorder. We illustrate these results with numerical experiments on large random graphs and on real-world networks.

19 pages, 7 figures. We corrected a sign typo in the new version of the paper

References in corpus (16)

• Power-law distributions in empirical data
• Critical phenomena in complex networks
• Scale-free networks are rare
• Scale-free Networks Well Done
• Network models of financial systemic risk: A review
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• May's Instability in Large Economies
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Topological resilience in non-normal networked systems
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• What drives transient behaviour in complex systems?
• Universal hypotrochoidic law for random matrices with cyclic correlations
• From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
• Power law decay for systems of randomly coupled differential equations

Cited by in corpus (14)

• Emergent stability in complex network dynamics
• Network structural origin of instabilities in large complex systems
• Mitigating long transient time in deterministic systems by resetting
• Linear stability analysis for large dynamical systems on directed random graphs
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations
• Nonlinearity-generated Resilience in Large Complex Systems
• The spectral density of dense random networks and the breakdown of the Wigner semicircle law
• Eigenvalue spectra and stability of directed complex networks
• Stability of large complex systems with heterogeneous relaxation dynamics
• Non-normality and non-monotonic dynamics in complex reaction networks
• Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems
• Local sign stability and its implications for spectra of sparse random graphs and stability of ecosystems
• Multimodal distribution of transient time of predator extinction in a three-species food chain