We consider a simple one-way averaging protocol on graphs. Initially, every node of the graph has a value. A node $u$ is chosen uniformly at random and $u$ samples $k$ neighbours $v_1,v_2,\cdots, v_k \in N(u)$ uniformly at random. Then, $u$ averages its value with $v$ as follows: $\xi_u(t+1) = \alpha \xi_u(t) + \frac{(1-\alpha)}{k} \sum_{i=1}^k \xi_{v_i}(t)$ for some $\alpha \in (0,1)$, where $\xi_u(t)$ is the value of node $u$ at time $t$. Note that, in contrast to neighbourhood value balancing, only $u$ changes its value. Hence, the sum (and the average) of the values of all nodes changes over time. Our results are two-fold. First, we show a bound on the convergence time (the time it takes until all values are roughly the same) that is asymptotically tight for some initial assignments of values to the nodes. Our second set of results concerns the ability of this protocol to approximate well the initial average of all values: we bound the probability that the final outcome is significantly away from the initial average. Interestingly, the variance of the outcome does not depend on the graph structure. The proof introduces an interesting generalisation of the duality between coalescing random walks and the voter model.

翻译：我们考虑的是图表上简单的单向平均协议。 最初, 图表的每个节点都有值。 随机选择的节点美元是统一的, 随机选择的节点美元是统一的, 随机选择的是美元样本是美元 $v_ 1,v_2,\cdots, v_k\inN( u) 美元, 任意选择的是美元。 然后, 美元是它的平均值, 以美元计算如下: $xi_ u( t+1) = ALpha\xx_ u( t) +\ frac{( 1- ALpha)\ ( 1 -\ ALpha)\ k}\ sum ⁇ i=1\ k\ rock\ rool_ c_ i} (t), 以美元标点是 $\ kpalpha 。 。 美元的平均值与 相对而言, 美元值的平衡值只有 $u$u 值。 因此, 所有不规则值的数值的数值的数值的数值总平均值值是两重。 首先值, 我们的初值的精度值的精度值是直值的精度值。 。 。 。 在最初结果的精度上, 我们的精度的精度值的精度的精度的精度的精度的精度的精度, 直度的精度, 。