Linear analysis of the vectorial network model in the presence of leaders

Abstract

a b s t r a c tIn this work, we study the collective behavior of a multi-agent system of stochastically interacting leaders and followers. In this time-varying network, the nodes update their states through a noisy interaction with a randomly selected subset of neighbors, including leaders whose average orientation is not updated in time. By linearizing the system dynamics in the vicinity of the leaders' common trajectory, we establish a toolbox of closed-form expressions that aid in the understanding of the influence of noise, number of connected neighbors, network size, and proportion of leaders on the group coordination.

Group coordination in a biologically-inspired vectorial network model

“…The analytical solution provides good predictions of the polarization for small noise intensities, where the polarization monotonically decreases with the square of the noise scaling factor. Similar to variants of the VNM studied in [30,31,34,35], when the noise intensity increases, a phase transition is observed for large group size while a plateau in the polarization is found for small groups. Compared to our previous results in [30] for the VNM−ω, group coordination in the VNM with turn rate noise is more affected by the noise, whereby we observe a complex interplay between the parameters shaping the turn rate noise and the group coordination.…”

“…In the case of uniform additive noise, group coordination has been investigated for various group size N , number of connected neighbors K, and noise intensity η [3,34]. A linear approximation of the problem has also been proposed in [35] and in [31] by considering a fraction of individuals as group leaders.…”

“…As such, the VNM should be considered as a feasible approximation of the Vicsek model under the assumption that individuals are so fast that they completely mix at each time step. This model differs from the case study considered in [30] in the mode of selection of the neighbors, which is completely random in the vein of the traditional VNM [31,35]. Specifically, the approach considered in [30] is based on the notion of numerosity-constrained networks [2], where each individual consistently uses its prior heading for deciding its next heading.…”

“…For any time step k, we define the expected value of the disagreement norm as δ(k) = E ξ(k) 2 . Similar to [35,31], we investigate the mean square behavior of the system in Eq. (10) through the second moment matrix Ξ(k) = E ξ(k)ξ(k) T whose trace corresponds to the disagreement norm δ(k).…”

Memory Matters in Synchronization of Stochastically Coupled Maps

“…Experiments on analog circuits to demonstrate the possibility of fast switching synchronization for periodically coupled units were presented in [57,59], building on the theoretical insight in [60] on almost sure stability of synchronization and in [58] on the control of synchronization. In the discrete-time setting, our work has contributed to an improved understanding of mean square stability of consensus [1,2,46,55] and synchronization of nonlinear maps [53,54], and it has put forward criteria for the control of synchronization [3,47,48]. We have also investigated the effect of inherent noise in the maps, which could destroy synchronization [1] and potentially lead to a phase transition [56].…”