The Role of Network Size for the Robustness of Centrality Measures

Measurement errors are omnipresent in network data. Studies have shown that these errors have a severe impact on the robustness of centrality measures. It has been observed that the robustness mainly depends on the network structure, the centrality measure, and the type of error. Previous findings regarding the influence of network size on robustness are, however, inconclusive. Based on twenty-four empirical networks, we investigate the relationship between global network measures, especially network size and average degree, and the robustness of the degree, eigenvector centrality, and PageRank. We demonstrate that, in the vast majority of cases, networks with a higher average degree are more robust. For random graphs, we observe that the robustness of Erdős-Rényi (ER) networks decreases with an increasing average degree, whereas with Barabàsi-Albert networks, the opposite effect occurs: with an increasing average degree, the robustness also increases. As a first step into an analytical discussion, we prove that for ER networks of different size but with the same average degree, the robustness of the degree centrality remains stable.