  Typing Yesterday into the search-bar of your browser provides a long list of
websites with, in top places, a link to a video by The Beatles. The order your
browser shows its search results is a notable example of the use of network
centrality. Centrality is a measure of the importance of the nodes in a network
and it plays a crucial role in a huge number of fields, ranging from sociology
to engineering, and from biology to economics. Many metrics are available to
evaluate centrality. However, centrality measures are generally based on ad hoc
assumptions, and there is no commonly accepted way to compare the effectiveness
and reliability of different metrics. Here we propose a new perspective where
centrality definition arises naturally from the most basic feature of a
network, its adjacency matrix. Following this perspective, different centrality
measures naturally emerge, including the degree, eigenvector, and hub-authority
centrality. Within this theoretical framework, the accuracy of different
metrics can be compared. Tests on a large set of networks show that the
standard centrality metrics perform unsatisfactorily, highlighting intrinsic
limitations of these metrics for describing the centrality of nodes in complex
networks. More informative multi-component centrality metrics are proposed as
the natural extension of standard metrics.
