Copeland centrality
Copeland centrality
[2] is a centrality measure inspired by the Copeland voting rule from social choice theory, which aggregates the preferences of voters over a given set of alternatives based on the majority relation [3, 4]. In networks, the preferences of the nodes can be defined over a set of other nodes based on their shortest-path distances. Specifically, for a node \(i \in \mathcal{N}\), the
distance-based preference
relation is defined as
\[
j \succ_i k \quad \text{if and only if} \quad d_{ij} < d_{ik},
\]
that is, node \(j\) is preferred to node \(k\) by node \(i\) if \(j\) is strictly closer to \(i\) than \(k\) is.
The
majority relation
\(μ\) between two nodes \(j\) and \(k\) is then defined as
\[
j μ k \quad \text{if and only if} \quad
\big|\{ i \in \mathcal{N} \setminus \{j,k\} : j \succ_i k \}\big|
>
\big|\{ i \in \mathcal{N} \setminus \{j,k\} : k \succ_i j \}\big|.
\]
In other words, node \(j\) is said to dominate node \(k\) if a strict majority of nodes in the network prefer \(j\) to \(k\), that is, if \(j\) is closer than \(k\) to more nodes.
The
Copeland score
of a node \(i\) is then defined as
\[
c_{\mathrm{Copeland}}(i)
=
\big|\{ k \in \mathcal{N} \setminus \{i\} : i μ k \}\big|
-
\big|\{ k \in \mathcal{N} \setminus \{i\} : k μ i \}\big|.
\]
Hence, the Copeland score of node \(i\) is computed as the difference between the number of nodes it dominates and the number of nodes that dominate it. A node receives a higher Copeland score if it is preferred by more nodes, which corresponds to being relatively close to many nodes in the network.