We consider two simple asynchronous opinion dynamics on arbitrary graphs where each node $u$ of the graph has an initial value $\xi_u(0)$. In the first process, the $NodeModel$, at each time step $t\ge 0$, a random node $u$ and a random sample of $k$ of its neighbours $v_1,v_2,\cdots,v_k$ are selected. Then $u$ updates its current value $\xi_u(t)$ to $\xi_u(t+1)=\alpha\xi_u(t)+\frac{(1-\alpha)}{k}\sum_{i=1}^k\xi_{v_i}(t)$, where $\alpha\in(0,1)$ and $k\ge1$ are parameters of the process. In the second process, the $EdgeModel$, at each step a random edge $(u,v)$ is selected. Node $u$ updates its value equivalently to the $NodeModel$ with $k=1$ and $v$ as the selected neighbour. For both processes the values of all nodes converge to the same value $F$, which is a random variable depending on the random choices made in each step. For the $NodeModel$ and regular graphs, and for the $EdgeModel$ and arbitrary graphs, the expectation of $F$ is the average of the initial values $\frac{1}{n}\sum_{u\in V}\xi_u(0)$. For the $NodeModel$ and non-regular graphs, the expectation of $F$ is the degree-weighted average of the initial values. Our results are two-fold. We consider the concentration of $F$ and show tight bounds on the variance of $F$ for regular graphs. We show that when the initial load does not depend on the number of nodes, the variance is negligible and the nodes are able to estimate the initial average of the node values. Interestingly, this variance does not depend on the graph structure. For the proof we introduce a duality between our processes and a process of two correlated random walks. We also analyse the convergence time for both models and for arbitrary graphs, showing bounds on the time $T_\varepsilon$ needed to make all node values `$\varepsilon$-close' to each other. Our bounds are asymptotically tight under some assumptions on the distribution of the starting values.

翻译：我们研究任意图上的两种简单异步意见动态，其中图的每个节点$u$具有初始值$\xi_u(0)$。在第一个过程$NodeModel$中，在每个时间步$t\ge 0$，随机选择一个节点$u$及其$k$个邻居的随机样本$v_1,v_2,\cdots,v_k$。然后$u$将其当前值$\xi_u(t)$更新为$\xi_u(t+1)=\alpha\xi_u(t)+\frac{(1-\alpha)}{k}\sum_{i=1}^k\xi_{v_i}(t)$，其中$\alpha\in(0,1)$和$k\ge1$是过程参数。在第二个过程$EdgeModel$中，在每个时间步随机选择一条边$(u,v)$。节点$u$以与$NodeModel$中$k=1$且$v$为选定邻居相同的方式更新其值。对于这两个过程，所有节点的值收敛到同一值$F$，这是一个依赖于每步随机选择的随机变量。对于$NodeModel$和正则图，以及对于$EdgeModel$和任意图，$F$的期望是初始值的平均值$\frac{1}{n}\sum_{u\in V}\xi_u(0)$。对于$NodeModel$和非正则图，$F$的期望是初始值的度数加权平均值。我们的结果有两个方面。我们考虑$F$的集中性，并给出正则图上$F$方差的紧界。我们证明当初始负载不依赖于节点数量时，方差可忽略不计，且节点能够估计节点值的初始平均值。有趣的是，该方差不依赖于图结构。在证明中，我们引入过程与两个相关随机游走过程之间的对偶性。我们还分析了两种模型在任意图上的收敛时间，给出了使所有节点值“$\varepsilon$-接近”所需时间$T_\varepsilon$的界。在起始值分布的某些假设下，我们的界是渐近紧的。