# Universal transient behavior in large dynamical systems on networks

## TL;DR

This paper studies how the transient response of large, randomly coupled linear dynamical systems depends on network topology, revealing universal behaviors and deriving analytical expressions for different network types.

## Contribution

It develops a formalism for analyzing transient dynamics on locally tree-like graphs and derives universal analytical results for unidirectional networks with various disorders.

## Key findings

- Unidirectional networks show universal transient behavior depending on a single parameter.
- Analytical expressions for average transient norms are derived for different disorder types.
- Numerical experiments confirm the theoretical predictions on real-world 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.