We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node $u$ 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 $k\ge 1$ are parameters of the process. In the second process, the EdgeModel, at each step a random pair of adjacent nodes $(u,v)$ is selected, and then 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 $F$, 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 values do not depend on the number of nodes, then the variance is negligible, hence 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 $T_\varepsilon$ required to make all node values `$\varepsilon$-close' to each other. Our bounds are asymptotically tight under assumptions on the distribution of the initial values.

翻译：我们考虑任意图上的两种简单异步意见动态过程，其中每个节点$u$具有初始值$\xi_u(0)$。在第一个过程（节点模型）中，每个时间步$t\ge 0$随机选择一个节点$u$及其$k$个邻居$v_1,v_2,\cdots,v_k$的随机样本。随后，$u$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$k\ge 1$为过程参数。在第二个过程（边模型）中，每个时间步随机选择一对相邻节点$(u,v)$，然后节点$u$按照$k=1$且$v$作为选定邻居的节点模型等价方式进行值更新。对于两种过程，所有节点的值收敛于$F$，该随机变量取决于每一步的随机选择。对于正则图上的节点模型和任意图上的边模型，$F$的期望值为初始值的平均值$\frac{1}{n}\sum_{u\in V} \xi_u(0)$。对于非正则图上的节点模型，$F$的期望值为初始值的度数加权平均值。我们的研究具有双重性：首先考虑$F$的集中性，并给出正则图上$F$方差的紧界。研究表明，当初始值与节点数无关时，方差可忽略不计，因此节点能够估计节点值的初始平均值。有趣的是，该方差与图结构无关。在证明中，我们引入了过程与两个相关随机游走过程的对偶性。此外，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达到'$\varepsilon$-接近'所需时间$T_\varepsilon$的界。在初始值分布假设下，我们的界具有渐近紧致性。