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.

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