How to choose the root: centrality measures over tree structures

Centrality measures are commonly used to analyze graph-structured data; however, the principles of this diverse world of measures are not well understood. Even in the case of tree structures, there is no consensus on which node should be selected as the most central. In this work, we embark on studying centrality measures over trees, specifically, on understanding what is the structure of various centrality rankings in trees. We introduce a property of \emph{rooting trees} that states a measure selects one or two adjacent nodes as the most important and the importance decreases from them in all directions. This property is satisfied by closeness centrality, but violated by PageRank. We show that for all standard centralities that root trees the comparison of adjacent nodes can be inferred by \emph{potential functions} that assess the quality of trees. We use these functions to give fundamental insights on rooting and derive a characterization that explains why some measures root trees. Moreover, we provide a polynomial-time algorithm to compute the root of a graph by using potential functions. Finally, using a family of potential functions, we show that there exist infinitely many ways of rooting trees with desirable properties.