Systemic Measures for Performance and Robustness of Large-Scale Interconnected Dynamical Networks

TL;DR

This paper introduces a unified framework for analyzing performance and robustness in large-scale linear interconnected networks using systemic measures, supported by convex optimization strategies and fundamental limits on network improvements.

Contribution

It develops a comprehensive set of systemic measures for network analysis, classifies their properties, and proposes optimization methods to enhance network performance and robustness.

Findings

Existing measures are classified as convex or Schur-convex systemic measures.

New systemic measures based on Riemann zeta function and system norms are introduced.

Optimal strategies for improving network measures via edge additions are characterized.

Abstract

In this paper, we develop a novel unified methodology for performance and robustness analysis of linear dynamical networks. We introduce the notion of systemic measures for the class of first--order linear consensus networks. We classify two important types of performance and robustness measures according to their functional properties: convex systemic measures and Schur--convex systemic measures. It is shown that a viable systemic measure should satisfy several fundamental properties such as homogeneity, monotonicity, convexity, and orthogonal invariance. In order to support our proposed unified framework, we verify functional properties of several existing performance and robustness measures from the literature and show that they all belong to the class of systemic measures. Moreover, we introduce new classes of systemic measures based on (a version of) the well--known Riemann zeta function, input--output system norms, and etc. Then, it is shown that for a given linear dynamical network one can take several different strategies to optimize a given performance and robustness systemic measure via convex optimization. Finally, we characterized an interesting fundamental limit on the best achievable value of a given systemic measure after adding some certain number of new weighted edges to the underlying graph of the network.