Cyclic Network Automata and Cohomological Waves

Abstract

This paper considers a dynamic coverage problem for sensor networks that are sufficiently dense but not localized. Only a small fraction of sensors may be in an awake state at any given time. The goal is to find a decentralized protocol for establishing dynamic, sweeping barriers of awake-state sensors. Following Baryshnikov-Coffman-Kwak, we use network cyclic cellular automata to generate waves. This paper gives a rigorous analysis of network-based cyclic cellular automata in the context of a system of narrow hallways and 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.