Pull voting is a random process in which vertices of a connected graph have initial opinions chosen from a set of $k$ distinct opinions, and at each step a random vertex alters its opinion to that of a randomly chosen neighbour. If the system reaches a state where each vertex holds the same opinion, then this opinion will persist forthwith. In general the opinions are regarded as incommensurate, whereas in this paper we consider a type of pull voting suitable for integer opinions such as $\{1,2,\ldots,k\}$ which can be compared on a linear scale; for example, 1 ('disagree strongly'), 2 ('disagree'), $\ldots,$ 5 ('agree strongly'). On observing the opinion of a random neighbour, a vertex updates its opinion by a discrete change towards the value of the neighbour's opinion, if different. Discrete incremental voting is a pull voting process which mimics this behaviour. At each step a random vertex alters its opinion towards that of a randomly chosen neighbour; increasing its opinion by $+1$ if the opinion of the chosen neighbour is larger, or decreasing its opinion by $-1$, if the opinion of the neighbour is smaller. If initially there are only two adjacent integer opinions, for example $\{0,1\}$, incremental voting coincides with pull voting, but if initially there are more than two opinions this is not the case. For an $n$-vertex graph $G=(V,E)$, let $\lambda$ be the absolute second eigenvalue of the transition matrix $P$ of a simple random walk on $G$. Let the initial opinions of the vertices be chosen from $\{1,2,\ldots,k\}$. Let $c=\sum_{v \in V} \pi_v X_v$, where $X_v$ is the initial opinion of vertex $v$, and $\pi_v$ is the stationary distribution of the vertex. Then provided $\lambda k=o(1)$ and $k=o(n/\log n)$, with high probability the final opinion is the initial weighted average $c$ suitably rounded to $\lfloor c \rfloor$ or $\lceil c\rceil$.

翻译：拉取投票是一种随机过程，其中连通图的顶点从 $k$ 个不同意见的集合中选取初始意见，每一步中随机选择一个顶点，将其意见更改为随机选择的邻居的意见。如果系统达到每个顶点持有相同意见的状态，则该意见将立即持续。通常，这些意见被视为不可比较的，而在本文中，我们考虑一种适用于整数意见（如 $\{1,2,\ldots,k\}$）的拉取投票类型，这些意见可以在线性尺度上进行比较；例如，1（“强烈反对”）、2（“反对”）、$\ldots$、5（“强烈同意”）。在观察到随机邻居的意见后，如果意见不同，顶点会通过离散变化更新其意见，使其更接近邻居的意见值。离散增量投票是一种模拟这种行为的拉取投票过程。每一步中，随机选择一个顶点，将其意见更改为随机选择的邻居的意见；如果所选邻居的意见较大，则将其意见增加 $+1$，如果邻居的意见较小，则将其意见减少 $-1$。如果初始时只有两个相邻的整数意见，例如 $\{0,1\}$，增量投票与拉取投票一致，但如果初始时有两个以上的意见，则情况并非如此。对于一个 $n$ 个顶点的图 $G=(V,E)$，令 $\lambda$ 为简单随机游走在 $G$ 上的转移矩阵 $P$ 的绝对第二特征值。令顶点的初始意见从 $\{1,2,\ldots,k\}$ 中选择。令 $c=\sum_{v \in V} \pi_v X_v$，其中 $X_v$ 是顶点 $v$ 的初始意见，$\pi_v$ 是顶点的平稳分布。那么，在 $\lambda k=o(1)$ 且 $k=o(n/\log n)$ 的条件下，最终意见以高概率为初始加权平均值 $c$ 适当舍入到 $\lfloor c \rfloor$ 或 $\lceil c\rceil$。