On excitable beta-skeletons

A beta-skeleton is a planar proximity undirected graph of an Euclidean point set where nodes are connected by an edge if their lune-based neighborhood contains no other points of the given set. Parameter $\beta$ determines size and shape of the nodes’ neighborhoods. In an excitable beta-skeleton every node takes three states — resting, excited and refractory, and updates its state in discrete time depending on states of its neighbors. We design families of beta-skeletons with absolute and relative thresholds of excitability and demonstrate that several distinct classes of space-time excitation dynamics can be selected using beta. The classes include spiral and target waves of excitation, branching domains of excitation and oscillating localizations.