Random Walks and the Best Meeting Time for Trees

TL;DR

This work analyzes the best meeting time $T_{\mathrm{bestmeet}}(G)=\min_{v}\sum_{u}\pi_u H(u,v)$ for random walks on trees with order $n$ and diameter $d$, introducing the joining time $J(v)=2|E|H(\pi,v)$ and the barycenter as a centrality notion. The authors show that among such trees the balanced lever $L_{n,d}$ uniquely minimizes $T_{\mathrm{bestmeet}}$, while the balanced double broom $D_{n,d}$ uniquely maximizes it, deriving explicit parity-dependent formulas for these extremal values. They also provide global bounds for all trees of order $n$, showing $T_{\mathrm{bestmeet}}(S_n)=\tfrac{1}{2}$ as the minimum, and giving maximizing structures as the path $P_n$ when $n$ is even and the broom $B_{n,n-2}$ (rather than the path) when $n$ is odd and large enough. The results identify how diameter-constrained tree shapes influence centrality-like measures of random walks and offer precise, exploitable formulas for extremal meeting times.

Abstract

We consider random walks on a tree $G=(V,E)$ with stationary distribution $π_v = \mathrm{deg}(v)/2|E|$ for $v \in V$. Let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at vertex $v$ to reach vertex $w$. We characterize the extremal tree structures for the best meeting time $T_{\mathrm{bestmeet}}(G) = \min_{w \in V} \sum_{v \in V} π_v H(v,w)$ for trees of order $n$ with diameter $d$. The best meeting time is maximized by the balanced double broom graph, and it is minimized by the balanced lever graph.

Figures (3)

• Figure 1.1: The balanced lever $L_{11,5}$, the broom $B_{11,5}$ and the balanced double broom $D_{11,5}$.
• Figure 2.1: A tree $G$ and its $v$-split into $\deg(v)=3$ subgraphs.
• Figure 3.1: The three-phase process that minimizes $J_{\min}(G)$. We start with a tree $G \in \mathcal{T}_{37,12}$ whose barycenter is not on a geodesic path. Phase One: repeatedly move all leaves besides $v_0,v_{12}$ to be adjacent to $c$. Phase Two: Move all non-path vertices to become leaves adjacent to $v_5$. Phase Three: move the fulcrum of the lever from $v_5$ to $v_6$.

Theorems & Definitions (39)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Theorem 2.2: Theorem 1.2 of BW2013