Cyclic Cellular Automata on Networks and Cohomological Waves

A dynamic coverage problem for sensor networks that are sufficiently dense but not localized is considered. By maintaining only a small fraction of sensors on at any time, we are aimed to find a decentralized protocol for establishing dynamic, sweeping barriers of awake-state sensors. Network cyclic cellular automata is used to generate waves. By rigorously analyzing network-based cyclic cellular automata in the context of a system of narrow hallways, it shows that waves of awake-state nodes turn corners and automatically solve pusuit/evasion-type problems without centralized coordination. As a corollary of this work, we unearth some interesting topological interpretations of features previously observed in cyclic cellular automata (CCA). By considering CCA over networks and completing to simplicial complexes, we induce dynamics on the higher-dimensional complex. In this setting, waves are seen to be generated by topological defects with a nontrivial degree (or winding number). The simplicial complex has the topological type of the underlying map of the workspace (a subset of the plane), and the resulting waves can be classified cohomologically. This allows one to "program" pulses in the sensor network according to cohomology class. We give a realization theorem for such pulse waves.