Distributed Tracking of Network Size, Diameter, Radius, and Node Eccentricities in Open Multi-Agent Systems

We present a distributed algorithm enabling the dynamic tracking of diameter and radius (DR) of time-varying open networks (ON) together with nodes' eccentricities and network size (ES), called $\rm {DR-ON-ES}$ , which does not require the disclosure of nodes' identity nor any a priori information on the network. The convergence properties of $\rm {DR-ON-ES}$ are discussed within the framework of open multi-agent systems (OMASs): sufficient conditions are provided to ensure that the TPI of an OMAS executing $\rm {DR-ON-ES}$ is globally asymptotically open stable, meaning that all trajectories converge to within a neighborhood of the sought parameters. The parameters' estimations at equilibrium points are characterized in terms of mean expected value and mean squared error. As an ancillary result, we remark that $\rm {DR-ON-ES}$ exploits the OSTDMC Protocol – a novel distributed protocol formalized and analyzed in this manuscript – to achieve consensus on the time-varying maximum value of a set of signals fed locally to the agents of the network.