  Evaluating the importance of a network node is a crucial task in network
science and graph data mining. H-index is a popular centrality measure for this
task, however, there is still a lack of its interpretation from a rigorous
statistical aspect. Here we show the statistical nature of h-index from the
perspective of order statistics, and we obtain a new family of centrality
indices by generalizing the h-index along this direction. The theoretical and
empirical evidences show that such a statistical interpretation enables us to
obtain a general and versatile framework for quantifying the importance of a
network node. Under this framework, many new centrality indices can be derived
and some of which can be more accurate and robust than h-index. We believe that
this research opens up new avenues for developing more effective indices for
node importance quantification from a viewpoint that still remains unexplored.
