How do centrality measures choose the root of trees?

TL;DR

This paper investigates how different centrality measures select roots in trees, introducing potential functions to analyze and compute root nodes efficiently, revealing multiple rooting strategies with desirable properties.

Contribution

It introduces a novel approach focusing on classes of graphs, especially trees, to understand centrality measures, and develops an almost linear-time algorithm for tree rooting based on potential functions.

Findings

Closeness centrality satisfies the tree rooting property.

PageRank violates the tree rooting property.

Multiple tree rooting methods with desirable properties are identified.

Abstract

Centrality measures are widely used to assign importance to graph-structured data. Recently, understanding the principles of such measures has attracted a lot of attention. Given that measures are diverse, this research has usually focused on classes of centrality measures. In this work, we provide a different approach by focusing on classes of graphs instead of classes of measures to understand the underlying principles among various measures. More precisely, we study the class of trees. We observe that even in \fix{the} case of trees, there is no consensus on which node should be selected as the most central. To analyze the behavior of centrality measures on trees, we introduce a property of \emph{tree rooting} 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 several centrality measures 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 explaining why some measure root trees. Moreover, we provide an almost liner-time algorithm to compute the root of a graph by using potential functions. Finally, using a family of potential functions, we show that many ways of tree rooting exist with desirable properties.