We analyze the discrete incremental voting process (DIV) introduced by Cooper, Radzik, and Shiraga [OPODIS '23]. In this process, we consider a set $V$ of $n$ nodes connected in an undirected graph $G = (V, E)$ where each node has an integer opinion. In one step a randomly selected node interacts with its randomly selected neighbor and changes its opinion by $1$ in the direction of the neighbour's opinion. The process converges to a unique opinion that, in expectation, is the degree-weighted average of the initial opinions. We show that if the graph has conductance $Φ(G)$, the ratio of the average to smallest degree is $γ(G)$, and the maximal difference between initial opinions is $K$, then the expected convergence time is ${O}\left({n\left(K\log (Kn)+γ(G) n \right)}/{Φ(G)^2}\right)$. This bound is essentially optimal for a large class of graphs of bounded expansion. We also show that for regular graphs, if the second largest eigenvalue is $o(1/\log^2 n)$ and $K$ is $o\left({n}/{\log^2 n}\right)$, then w.h.p.\ DIV converges to the initial average opinion (rounded up or down).

翻译：我们分析由 Cooper、Radzik 和 Shiraga [OPODIS '23] 提出的离散增量投票过程（DIV）。在该过程中，考虑一个由 $n$ 个节点组成的集合 $V$，这些节点连接在一个无向图 $G = (V, E)$ 中，每个节点拥有一个整数意见。在某一步中，随机选择一个节点与其随机选择的邻居交互，并将其意见向邻居意见方向改变 $1$。该过程收敛至唯一意见，该意见期望上等于初始意见的度加权平均值。我们证明，若图的电导率为 $Φ(G)$，平均度与最小度之比为 $γ(G)$，且初始意见的最大差值为 $K$，则期望收敛时间为 ${O}\left({n\left(K\log (Kn)+γ(G) n \right)}/{Φ(G)^2}\right)$。该界对于一大类有界扩展图本质上是紧的。我们还证明，对于正则图，若第二大特征值为 $o(1/\log^2 n)$ 且 $K$ 为 $o\left({n}/{\log^2 n}\right)$，则 D IV 以高概率收敛至初始平均意见（向上或向下取整）。