Analytical relationships between metric and centrality measures of a network and its dual

The centrality and efficiency measures of a network GG are strongly related to the respective measures on the dual G⋆G⋆ and the bipartite B(G)B(G) associated networks. We show some relationships between the Bonacich centralities c(G)c(G), c(G⋆)c(G⋆) and c(B(G))c(B(G)) and between the efficiencies E(G)E(G) and E(G⋆)E(G⋆) and we compute the behavior of these parameters in some examples.