Correlations among centrality indices and a class of uniquely ranked graphs

- Correlations among centralities do not necessarily stem from formal similarities.
- Measures of centrality are perfectly correlated on threshold graphs.
- Correlation decreases on graphs with increasing distance from threshold graphs.
- Indices yield competing explanations for observed effects on core-periphery networks.

Various centrality indices have been proposed to capture different aspects of structural importance but relations among them are largely unexplained. The most common strategy appears to be the pairwise comparison of centrality indices via correlation. While correlation between centralities is often read as an inherent property of the indices, we argue that it is confounded by network structure in a systematic way. In fact, correlations may be even more indicative of network structure than of relationships between indices. This has substantial implications for the interpretation of centrality effects as it implies that competing explanations embodied in different indices cannot be separated from each other if the network structure is close to a certain generalization of star graphs.