Time-Delay Origins of Fundamental Tradeoffs Between Risk of Large Fluctuations and Network Connectivity
For researchers studying networked control systems, this work identifies a fundamental tradeoff between connectivity and fluctuation risk, though the analysis is limited to linear consensus networks with time delays.
This paper derives explicit formulas for the risk of large fluctuations in noisy time-delay linear consensus networks, revealing an intrinsic tradeoff where increasing network connectivity raises the risk of large fluctuations.
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.