# Centrality in Time-Delay Consensus Networks with Structured Uncertainties

## TL;DR

This paper explores how network centrality in time-delay consensus networks is affected by uncertainties and communication delays, providing explicit formulas and ranking agents based on their centrality.

## Contribution

It introduces novel centrality measures for time-delay consensus networks considering structured uncertainties, with explicit formulas and analysis of their dependence on delays and noise.

## Key findings

- Centrality measures are highly sensitive to time-delay.
- Explicit formulas for centrality under various uncertainties are derived.
- Agents and links can be ranked based on their centrality indices.

## Abstract

We investigate notions of network centrality in terms of the underlying coupling graph of the network, structure of exogenous uncertainties, and communication time-delay. Our focus is on time-delay linear consensus networks, where uncertainty is modeled by structured additive noise on the dynamics of agents. The centrality measures are defined using the $\mathcal H_2$-norm of the network. We quantify the centrality measures as functions of time-delay, the graph Laplacian, and the covariance matrix of the input noise. Several practically relevant uncertainty structures are considered, where we discuss two notions of centrality: one w.r.t intensity of the noise and the other one w.r.t coupling strength between the agents. Furthermore, explicit formulas for the centrality measures are obtained for all types of uncertainty structures. Lastly, we rank agents and communication links based on their centrality indices and highlight the role of time-delay and uncertainty structure in each scenario. Our counter intuitive grasp is that some of centrality measures are highly volatile with respect to time-delay.