Achieving fast consensus by edge elimination in a class of consensus dynamics with large delays

A class of linear time-invariant consensus network dynamics with homogeneous delays is studied in terms of agents' convergence speed depending on the network graph. For this system, it is known that consensus is guaranteed only if the inter-agent delay τ is less than a certain margin τ* known as the delay margin. Since an analytical calculation of settling time for time delay systems is extremely difficult, if not impossible, and since rightmost roots only provide a rough estimation of state decay rate, here we take a statistical approach and analyze numerous case studies through time-domain simulations of the consensus system impulse response. On benchmark problems, we show that the graph underlying the consensus network together with the ratio τ/τ* and agents' initial conditions determine reaching consensus faster/slower. Specifically, we report that (i) for relatively large delays with τ < τ*, there is a set of graphs obtained by removing some edges between the agents with which fast consensus can be achieved, by about an order of magnitude reduction in settling time, and (ii) removing from a fully connected graph the edge corresponding to the largest difference of initial state values will reduce the settling time in the presence of large time delay.