On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies

Abstract

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. For a circulant topology, we derive a decomposition of the state space into stable invariant subspaces with different convergence rates. This decomposition is identical for every linear circulant gathering protocol. Only the convergence rates depend on the weights in the interaction graph. 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. In both situations, we also consider visibility preservation properties.

Publication
Schmid, U., Kuznets, R. (eds) Structural Information and Communication Complexity. SIROCCO 2025. Lecture Notes in Computer Science, vol 15671. Springer, Cham.
Sören von der Gracht
Sören von der Gracht
PostDoc in Dynamical Systems

Research in network dynamical systems and its applications.