This paper investigates the effects of contagion in interbank-lending networks, with a special focus on the theoretical grounding of centrality measures. I introduce a new systemic importance measure based on the harmonic distance of Acemoglu, Ozdaglar and Tahbaz-Salehi from 2015. Then, motivated by their theoretical results for concentration centrality in a later work of the same year, I compare this measure with well-known centrality measures that have already been applied in the systemic risk literature, and that do not take into account the structure of a contagion mechanism. I derive an explicit formula of size-adjusted harmonic distances and extend it by using liquid assets for a heterogeneous banking system. The simulation results on extended Barabasi–Albert-type networks and complete networks do not confirm that these new measures would perform better than “off-the-shelf” centralities, but a linear combination of these measures forecasts losses better than any of the individual measures. The time variations of centrality measures are presented in an interbank network. I also test for the scale-free property of the Hungarian interbank-lending network and suggest that network measures be used as systemic risk indicators in systemic stress indexes. Finally, this work points out that, in some cases, the application of centrality measures might be misleading, and specific clearing algorithms provide more reliable results.