Measurement errors are omnipresent in network data. Most studies observe an erroneous network instead of the desired error-free network. It is well known that such errors can have a severe impact on network metrics, especially on centrality measures: a central node in the observed network might be less central in the underlying, error-free network. The robustness is a common concept to measure these effects. Studies have shown that the robustness primarily depends on the centrality measure, the type of error (e.g., missing edges or missing nodes), and the network topology (e.g., tree-like, core-periphery). Previous findings regarding the influence of network size on the robustness are, however, inconclusive. We present empirical evidence and analytical arguments indicating that there exist arbitrary large robust and non-robust networks and that the average degree is well suited to explain the robustness. We demonstrate that networks with a higher average degree are often more robust. For the degree centrality and Erdős–Rényi (ER) graphs, we present explicit formulas for the computation of the robustness, mainly based on the joint distribution of node degrees and degree changes which allow us to analyze the robustness for ER graphs with a constant average degree or increasing average degree.

中文翻译：

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网络规模和平均度对中心性度量鲁棒性的影响

测量误差在网络数据中无处不在。大多数研究观察的是错误的网络，而不是期望的无错误网络。众所周知，此类错误会对网络指标产生严重影响，尤其是对中心性度量：观察到的网络中的中心节点在底层的无错误网络中可能不那么中心。稳健性是衡量这些影响的常用概念。研究表明，鲁棒性主要取决于中心性度量、错误类型（例如，缺失边或缺失节点）和网络拓扑（例如，树状、核心-外围）。然而，先前关于网络大小对鲁棒性影响的研究结果尚无定论。我们提出了经验证据和分析论据，表明存在任意大的鲁棒和非鲁棒网络，并且平均程度非常适合解释鲁棒性。我们证明了具有更高平均程度的网络通常更健壮。对于度中心性和 Erdős-Rényi (ER) 图，我们提出了计算鲁棒性的明确公式，主要基于节点度和度变化的联合分布，这使我们能够分析具有恒定平均值的 ER 图的鲁棒性学位或增加平均学位。