# Dynamics Concentration of Large-Scale Tightly-Connected Networks

## TL;DR

This paper demonstrates that in large, tightly-connected linear networks with random transfer functions, nodes exhibit nearly identical responses to inputs, a phenomenon termed dynamics concentration, with implications for model reduction and robustness.

## Contribution

It introduces the concept of dynamics concentration in large networks and provides a probabilistic analysis showing uniform response convergence regardless of input source.

## Key findings

- Nodes in large networks follow the same response despite heterogeneity.
- Network transfer matrix converges to a unique dynamic response.
- Similar phenomena observed in small deterministic networks within certain frequency bands.

## Abstract

The ability to achieve coordinated behavior --engineered or emergent-- on networked systems has attracted widespread interest over several fields. This has led to remarkable advances on the development of a theoretical understanding of the conditions under which agents within a network can reach agreement (consensus) or develop coordinated behaviors such as synchronization. However, fewer advances have been made toward explaining another commonly observed phenomena in tightly-connected networks systems: output responses of nodes in the networks are almost identical to each other despite heterogeneity in their individual dynamics. In this paper, we leverage tools from high-dimensional probability to provide an initial answer to this phenomena. More precisely, we show that for linear networks of nodal random transfer functions, as the network size and connectivity grows, every node in the network follows the same response to an input or disturbance --irrespectively of the source of this input. We term this behavior as dynamics concentration since it stems from the fact that the network transfer matrix uniformly converges in probability, i.e., it concentrates, to a unique dynamic response determined by the distribution of the random transfer function of each node. We further discuss the implications of our analysis in the context of model reduction and robustness, and provide numerical evidence that similar phenomena occur in small deterministic networks over a properly defined frequency band.