On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies

In this article, we investigate the convergence behavior of two classes of gathering protocols with fixed circulant topologies using tools from dynamical systems. Given a fixed number of mobile entities moving in the Euclidean plane, we model a gathering protocol as a system of (linear) ordinary differential equations whose equilibria are exactly all possible gathering points. Then, for a circulant topology we derive a decomposition of the state space into stable invariant subspaces with different convergence rates by utilizing tools from dynamical systems theory. It turns out, that this decomposition is identical for every linear circulant gathering protocol, whereas only the convergence rates depend on the weights in interaction graph itself. In the second part, we consider a normalized nonlinear version of the equation of motion that is obtained by scaling the speed of each entity. Again, we find a similar decomposition of the state space that is based on our findings in the linear case. Finally, we also consider visibility preservation properties of the two classes of system.