  Complex network theory (CNT) is gaining a lot of attention in the scientific
community, due to its capability to model and interpret an impressive number of
natural and anthropic phenomena. One of the most active CNT field concerns the
evaluation of the centrality of vertices and edges in the network. Several
metrics have been proposed, but all of them share a topological point of view,
namely centrality descends from the local or global connectivity structure of
the network. However, vertices can exhibit their own intrinsic relevance
independent from topology; e.g., vertices representing strategic locations
(e.g., hospitals, water and energy sources, etc.) or institutional roles (e.g.,
presidents, agencies, etc.). In these cases, the connectivity network structure
and vertex intrinsic relevance mutually concur to define the centrality of
vertices and edges. The purpose of this work is to embed the information about
the intrinsic relevance of vertices into CNT tools to enhance the network
analysis. We focus on the degree, closeness and betweenness metrics, being
among the most used. Two examples, concerning a social (the historical Florence
family marriage network) and an infrastructure (a water supply system) network,
demonstrate the effectiveness of the proposed relevance-embedding extension of
the centrality metrics.
