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

For the class of noisy time-delay linear consensus networks, we obtain explicit formulae 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 of large fluctuations and effective resistance of the underlying coupling graph of the network. The main implication is that increasing network connectivity makes network riskier to 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 of large fluctuations scales with network size.