See Measures and metrics for networks
One can distinguish two types of important nodes in directed networks. We describe them for the case of an information network, like WWW first:
This idea was implemented by Kleinberg into the hyperlink-induced topic search or HITS algorithm. The mathematical definitions that tries to capture the above intuition are:
Mathematically,
where and are the authority and hub centralities, respectively. These equations combine to show that these centralities are in fact the eigenvectors of and , respectively, with the same eigenvalue (which must be the leading one, suing similar arguments as cases above, and which is equal to . (or , but not both) is a free parameter that can be chosen to be as we don't care about relative centralities.
This connection means that these centralities are similar to the eigenvector centralities for the cocitation and bibliographic coupling network, respectively (see Mathematics of networks).