Aggregate fluctuations in time-delay linear consensus networks: A systemic risk perspective

We discuss the notion of systemic risk in noisy consensus systems with delay as a measure of robustness and resilience. We propose a risk measure to characterize systemic events of the dynamic network, based on quantile functions theory. We provide explicit calculations or estimates for our consensus dynamic both in transient and steady-state dynamics. Our analysis highlights the dependence of risk on the spectral properties of the communication graph and the delay of the network. Unlike consensus networks with no delay, the risk measure exhibits a non-monotonic behavior in presence of time-delay. This non-monotonicity is the origin of potential aggregate fluctuation which is an undesirable systemic event of our network.