Sharing tea on a graph

TL;DR

This paper studies a process of sharing and equalizing tea on a graph, providing bounds on the amount of tea at each vertex based on its distance from a starting point, and analyzing the set of possible weight distributions.

Contribution

It establishes an optimal bound on tea distribution relative to distance and characterizes the set of reachable weight distributions on any finite graph.

Findings

Maximum tea at a vertex is at most 1/(distance+1) from the source.

The bound is proven to be optimal.

The set of reachable weight distributions is compact.

Abstract

Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$.