Relationships between centrality measures of networks with diameter 2

From an intuitive analysis of the betweeness of networks in which we detail its components (not only the basic definition of the betweeness as the number of short paths through a node), we propose a relationship between the betweeness, the degree and the clustering coefficient of a node in networks with diameter 2. We verify the validity of this relationship on seven diameter 2 networks: the global migration network, the dual networks of the underground systems of Barcelona, Tokyo, London and three model networks. We show also that the dependence of the closeness with the degree is an exact linear relation with slope equal to −1.