Hierarchical closeness

Hierarchical closeness (HC) is a structural centrality measure used in network theory or graph theory. It is extended from closeness centrality to rank how centrally located a node is in a directed network. While the original closeness centrality of a directed network considers the most important node to be that with the least total distance from all other nodes, hierarchical closeness evaluates the most important node as the one which reaches the most nodes by the shortest paths. The hierarchical closeness explicitly includes information about the range of other nodes that can be affected by the given node. In a directed network formula_1 where formula_2 is the set of nodes and formula_3 is the set of interactions, hierarchical closeness of a node formula_4 ∈ formula_2 called formula_6 was proposed by Tran and Kwon as follows:
where:
In the formula, formula_19 represents the number of nodes in formula_2 that can be reachable from formula_4. It can also represent the hierarchical position of a node in a directed network. It notes that if formula_22, then formula_23 because formula_24 is formula_25. In cases where formula_26, the reachability is a dominant factor because formula_27 but formula_28. In other words, the first term indicates the level of the global hierarchy and the second term presents the level of the local centrality.
Application.
Hierarchical closeness can be used in biological networks to rank the risk of genes to carry diseases.