Never-Ending Lonely Distancing Dance Graphs

Abstract

Lonely distancing is the idea that people can crave company at the same time they fear too much of it. Taking inspiration from this, we construct a scenario in which k players on an n × n grid jockey to occupy space next to a single companion while maintaining at least a minimum distance from everyone else. The tension behind these conflicting needs—companionship versus isolation—induces a dynamical system on the grid we model with directed graphs. Depending on n and k, players are likely to redistribute themselves into one or more equilibrium states. In very special cases, no such resolution is possible, and the players enter a cycle of perpetual redistribution in what we call a never-ending lonely distancing dance.