Eminence in Noisy Bilinear Networks

When measuring nodes’ importance in a network, the interconnections and dynamics are often supposed to be perfectly known. In this paper, we consider networks of agents with both uncertain couplings and dynamics. The network uncertainty is modeled by structured additive stochastic disturbances on each agent’s update dynamics and coupling weights. We then study how these uncertainties change the network’s centralities. Disturbances on the couplings between agents result in bilinear dynamics, and classical centrality indices from linear network theory need to be redefined. To do that, we first show that, similarly to its linear counterpart, the squared ${\mathcal{H}_2}$ norm of bilinear systems measures the trace of the steady-state error covariance matrix subject to stochastic disturbances. This makes the ${\mathcal{H}_2}$ norm a natural candidate for a performance metric of the system. We propose a centrality index for the agents based on the ${\mathcal{H}_2}$ norm, and show how it depends on the network topology and the noise structure. Finally, we simulate a few graphs to illustrate how uncertainties on different couplings affect the agents’ centrality rankings compared to a linearized model of the same system.