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.
Highlights
Networks of interacting constituents appear in the study of systems as diverse as ecosystems [1,2,3], neural networks [4,5,6,7,8], financial markets [9,10,11,12], and signaling networks [13,14,15]; for more examples, see Refs. [16,17]
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 treelike
Summary
Networks of interacting constituents appear in the study of systems as diverse as ecosystems [1,2,3], neural networks [4,5,6,7,8], financial markets [9,10,11,12], and signaling networks [13,14,15]; for more examples, see Refs. [16,17]. We characterize the transient response to a random perturbation for dynamical systems on random graphs with a prescribed degree distribution by deriving analytical expressions for S(t ). This analytical progress is made possible because (i) the general theory developed in this paper simplifies for locally oriented (or unidirectional) networks (see Sec. III C for a precise statement), and (ii) directed random graphs with a prescribed degree distribution are, with high probability, locally oriented [37,38]. In Appendix C, we make a study of the spectra of random graphs with a prescribed degree distribution and a bimodal distribution of diagonal matrix entries, and in Appendix D, we present numerical results for random graphs with power-law degree distributions
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