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. Thus, the distance-based preference relation of node \(i\) constitutes a weak order (irreflexive, transitive and negatively transitive binary relation) over the set \(\mathcal{N} \setminus \{i\}\), where each layer \(k\) corresponds to the indifference class of nodes located at distance \(k \in \{1, \ldots, \max_j d_{ij}\}\) from node \(i\).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\[c_{\mathrm{Copeland}}(i) = \big|\{ k \in \mathcal{N} \setminus \{i\} : i μ k \}\big|- \big|\{ k \in \mathcal{N} \setminus \{i\} : k μ i \}\big|.\]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.\vfill