In the context of the Cobb-Douglas productivity model we consider the $N \times N$ input-output linkage matrix $W$ for a network of $N$ firms $f_1, f_2, \cdots, f_N$. The associated influence vector $v_w$ of $W$ is defined in terms of the Leontief inverse $L_W$ of $W$ as $v_W = \frac{\alpha}{N} L_W \vec{\mathbf{1}}$ where $L_W = (I - (1-\alpha) W')^{-1}$, $W'$ denotes the transpose of $W$ and $I$ is the identity matrix. Here $\vec{\mathbf{1}}$ is the $N \times 1$ vector whose entries are all one. The influence vector is a metric of the importance for the firms in the production network. Under the realistic assumption that the data to compute the influence vector is incomplete, we prove bounds on the worst-case error for the influence vector that are sharp up to a constant factor. We also consider the situation where the missing data is binomially distributed and contextualize the bound on the influence vector accordingly. We also investigate how far off the influence vector can be when we only have data on nodes and connections that are within distance $k$ of some source node. A comparison of our results is juxtaposed against PageRank analogues. We close with a discussion on a possible extension beyond Cobb-Douglas to the Constant Elasticity of Substitution model, as well as the possibility of considering other probability distributions for missing data.

翻译：在柯布-道格拉斯生产率模型框架下，我们考虑由N个企业$f_1, f_2, \cdots, f_N$构成的网络，其投入产出关联矩阵$W$为$N \times N$阶矩阵。对应的影响力向量$v_w$基于Leontief逆矩阵$L_W$定义为$v_W = \frac{\alpha}{N} L_W \vec{\mathbf{1}}$，其中$L_W = (I - (1-\alpha) W')^{-1}$，$W'$表示$W$的转置矩阵，$I$为单位矩阵。此处$\vec{\mathbf{1}}$为所有元素均为1的$N \times 1$向量。该影响力向量是衡量生产网络中企业重要性的指标。在计算影响力向量的数据不完整这一现实假设下，我们证明了影响力向量最坏情况误差的渐近紧界（紧至常数因子）。同时考虑缺失数据服从二项分布的情形，并据此对影响力向量的边界进行情境化分析。我们还研究了当仅能获取距某源节点距离$k$以内的节点与连接数据时，影响力向量的最大偏移量。研究结果与PageRank相关结论进行了对比。最后探讨了将模型从柯布-道格拉斯扩展至常替代弹性模型的可能性，以及考虑缺失数据服从其他概率分布的可能性。