Time-Delay Origins of Fundamental Tradeoffs Between Risk of Large Fluctuations and Network Connectivity

TL;DR

This paper derives explicit formulas for large fluctuation risks in noisy time-delay consensus networks, revealing a fundamental tradeoff where increased connectivity raises fluctuation risk, with implications for network design.

Contribution

It provides explicit risk formulas based on Laplacian spectra and eigenvectors, and establishes a fundamental tradeoff between risk and network connectivity.

Findings

Increasing network connectivity raises the risk of large fluctuations.

Explicit formulas relate risk to Laplacian spectrum and eigenvectors.

Risk scales with network size for specific topologies.

Abstract

For the class of noisy time-delay linear consensus networks, we obtain explicit formulas for risk of large fluctuations of a scalar observable as a function of Laplacian spectrum and its eigenvectors. It is shown that there is an intrinsic tradeoff between risk and effective resistance of the underlying coupling graph of the network. The main implication is that increasing network connectivity, increases the risk of large fluctuations. For vector-valued observables, we obtain computationally tractable lower and upper bounds for joint risk measures. Then, we study behavior of risk measures for networks with specific graph topologies and show how risk scales with network size.