Centrality in Time-Delay Consensus Networks with Structured Uncertainties

We investigate notions of network centrality in terms of the underlying coupling graph of the network, structure of exogenous uncertainties, and communication time-delay. Our focus is on time-delay linear consensus networks, where uncertainty is modeled by structured additive noise on the dynamics of agents. The centrality measures are defined using the $\mathcal H_2$-norm of the network. We quantify the centrality measures as functions of time-delay, the graph Laplacian, and the covariance matrix of the input noise. Several practically relevant uncertainty structures are considered, where we discuss two notions of centrality: one w.r.t intensity of the noise and the other one w.r.t coupling strength between the agents. Furthermore, explicit formulas for the centrality measures are obtained for all types of uncertainty structures. Lastly, we rank agents and communication links based on their centrality indices and highlight the role of time-delay and uncertainty structure in each scenario. Our counter intuitive grasp is that some of centrality measures are highly volatile with respect to time-delay

Dynamics Concentration of Large-Scale Tightly-Connected Networks

The ability to achieve coordinated behavior --engineered or emergent-- on networked systems has attracted widespread interest over several fields. This has led to remarkable advances on the development of a theoretical understanding of the conditions under which agents within a network can reach agreement (consensus) or develop coordinated behaviors such as synchronization. However, fewer advances have been made toward explaining another commonly observed phenomena in tightly-connected networks systems: output responses of nodes in the networks are almost identical to each other despite heterogeneity in their individual dynamics. In this paper, we leverage tools from high-dimensional probability to provide an initial answer to this phenomena. More precisely, we show that for linear networks of nodal random transfer functions, as the network size and connectivity grows, every node in the network follows the same response to an input or disturbance --irrespectively of the source of this input. We term this behavior as dynamics concentration since it stems from the fact that the network transfer matrix uniformly converges in probability, i.e., it concentrates, to a unique dynamic response determined by the distribution of the random transfer function of each node. We further discuss the implications of our analysis in the context of model reduction and robustness, and provide numerical evidence that similar phenomena occur in small deterministic networks over a properly defined frequency band

Centrality-Based Traffic Restriction in Delayed Epidemic Networks

During an epidemic, infectious individuals might not be detectable until some time after becoming infected. The studies show that carriers with mild or no symptoms are the main contributors to the transmission of a virus within the population. The average time it takes to develop the symptoms causes a delay in the spread dynamics of the disease. When considering the influence of delay on the disease propagation in epidemic networks, depending on the value of the time-delay and the network topology, the peak of epidemic could be considerably different in time, duration, and intensity. Motivated by the recent worldwide outbreak of the COVID-19 virus and the topological extent in which this virus has spread over the course of a few months, this study aims to highlight the effect of time-delay in the progress of such infectious diseases in the meta-population networks rather than individuals or a single population. In this regard, the notions of epidemic network centrality in terms of the underlying interaction graph of the network, structure of the uncertainties, and symptom development duration are investigated to establish a centrality-based analysis of the disease evolution. A convex traffic volume optimization method is then developed to control the outbreak. The control process is done by identifying the sub-populations with the highest centrality and then isolating them while maintaining the same overall traffic volume (motivated by economic considerations) in the meta-population level. The numerical results, along with the theoretical expectations, highlight the impact of time-delay as well as the importance of considering the worst-case scenarios in investigating the most effective methods of epidemic containment