Abstract / Description
In realistic swarm robotic deployments, inter-robot communication links are inherently dynamic, appearing and disappearing as robots move. Consequently, static connectivity assumptions become invalid. This work extends Collatz graph dynamics (CGD), a deterministic synchronization framework inspired by the Collatz conjecture, from static connected graphs to mobile robotic networks characterized by switching communication topologies. A novel framework, termed Switching Collatz Graph Dynamics (S-CGD), is introduced. In this model, each robot updates its integer-valued state according to the Collatz rule while interacting exclusively with its currently reachable neighbors. It is proven that S-CGD achieves deterministic synchronization under the jointly connected condition, in which the union of communication graphs over any sliding time window $W$ remains connected. Upper bounds on synchronization time are established as $O(\log ^{2} M \cdot W \cdot \mathrm {diam}(\mathcal {G}))$ , shown to be tight (matched by an explicit lower bound) for path, cycle, and random geometric graph topologies, and conjectured tight in general. This bound explicitly decomposes the contributions of arithmetic dynamics, temporal switching, and network topology. Robustness properties are further analyzed, including resilience to simultaneous link failures and transient disconnections. An asynchronous convergence theorem is established for heterogeneous update rates across robots. In addition, exact complexity bounds are derived for canonical graph classes, including paths, cycles, stars, trees, complete graphs, expander graphs, and random geometric graphs. A fully distributed algorithm is presented, and its message, space, and energy complexities are rigorously analyzed. A theoretical comparison with linear consensus methods and random-walk-based rendezvous schemes demonstrates that S-CGD achieves deterministic synchronization over integer-valued states with complexity competitive with state-of-the-art linear approaches, while requiring strictly weaker assumptions on the underlying state structure.