On excitable beta-skeletons

TL;DR

This paper explores how varying the parameter beta in excitable beta-skeletons influences complex space-time excitation patterns, including waves, domains, and oscillations, in a planar proximity graph model.

Contribution

It introduces families of beta-skeletons with adjustable excitability thresholds and demonstrates diverse excitation dynamics controlled by beta.

Findings

Different classes of excitation patterns can be achieved by tuning beta.

Beta-skeletons exhibit spiral, target, branching, and oscillating excitation behaviors.

The study provides a framework for controlling excitation dynamics in proximity graphs.

Abstract

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.