Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg