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For the class of noisy time-delay linear consensus networks, we obtainexplicit formulas for risk of large fluctuations of a scalar observable as afunction of Laplacian spectrum and its eigenvectors. It is shown that there isan intrinsic tradeoff between risk and effective resistance of the underlyingcoupling graph of the network. The main implication is that increasing networkconnectivity, increases the risk of large fluctuations. For vector-valuedobservables, we obtain computationally tractable lower and upper bounds forjoint risk measures. Then, we study behavior of risk measures for networks withspecific graph topologies and show how risk scales with network size.

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