Centrality scores

The simplest of centrality scores is the degree of a vertex. In a directed network such as a citation network, there are two degrees, the in-degree and the out-degree. It is reasonable, for instance, to imagine that important or influential vertices in a citation network will receive many citations and therefore have high in-degree. A more sophisticated version of the same idea is eigenvector centrality, in which, rather than merely counting the number of citations a vertex gets, we award a higher score when the citing vertices are themselves influential.

The simplest way to do this is to define the centrality to be proportional to the sum of the centralities of the citing vertices, which makes the centralities proportional to the elements of the leading eigenvector of the adjacency matrix. Unfortunately, this method does not work for acyclic directed networks, such as citation networks, for which all such centralities turn out to be zero.