Sensitivity Analysis of Centralities on Unweighted Networks

Revealing important vertices is a fundamental task in network analysis. As such, many indicators have been proposed for doing so, which are collectively called centralities. However, the abundance of studies on centralities blurs their differences. In this work, we compare centralities based on their sensivitity to modifications in the graph. Specifically, we introduce a quantitative measure called (average-case) edge sensitivity, which measures how much the centrality value of a uniformly chosen vertex (or an edge) changes when we remove a uniformly chosen edge. Edge sensitivity is applicable to unweighted graphs, regarding which, to our knowledge, there has been no theoretical analysis of the centralities. We conducted a theoretical analysis of the edge sensitivities of six major centralities: the closeness centrality, harmonic centrality, betweenness centrality, endpoint betweenness centrality, PageRank, and spanning tree centrality. Our experimental results on synthetic and real graphs confirm the tendency predicted by the theoretical analysis. We also discuss an extension of edge sensitivity to the setting that we remove a uniformly chosen set of edges of size k for an integer k = 1.