Working paper

Issue number:

2019.09

Series:

Documents de Travail du Centre d’Economie de la Sorbonne

Publisher:

Centre d’Economie de la Sorbonne

Year:

2019

Working Paper [1]

One of the most famous ranking methods for digraphs is the ranking by Copeland score.
The Copeland score of a node in a digraph is the difference between its outdegree (i.e. its number of
outgoing arcs) and its indegree (i.e. its number of ingoing arcs). In the ranking by Copeland score,
a node is ranked higher, the higher is its Copeland score. In this paper, we deal with an alternative
to rank nodes according to their out- and indegree, namely ranking the nodes according to their
degree ratio, i.e. the outdegree divided by the indegree. To avoid dividing by a zero indegree, we
implicitly take the out- and indegree of the reflexive digraph. We provide an axiomatization of the
ranking by degree ratio using a sibling neutrality axiom, which says that the entrance of a sibling
(i.e. a node that is in some sense similar to the original node) does not change the ranking among
the original nodes. We also provide a new axiomatization of the ranking by Copeland score using
the same axioms except that this method satisfies a different sibling neutrality. Finally, we modify
the ranking by degree ratio by not considering the reflexive digraph, but by definition assume
nodes with indegree zero to be ranked higher than nodes with a positive indegree. We provide an
axiomatization of this ranking by modified degree ratio using yet another sibling neutrality and
a maximal property. In this way, we can compare the three ranking methods by their respective
sibling neutrality.