p-means centrality

The evaluation of the importance or “status” of the nodes in social networks is of great interest to the researchers of several areas of knowledge. The importance of nodes depends on the positioning in the network structure, being obtained through centrality measures. There are several centrality measures in the literature, with different criteria that define which nodes are more central. Geometric metrics, for example, consider the geodesic paths among pairs of nodes. The closeness centrality and its alternative version harmonic centrality are both geometric metrics and present the characteristic of being reciprocal, respectively, of the arithmetic and harmonic means of geodesic distances. We propose in this article a generalized measure, which we call p-means centrality. Depending on the value given to the parameter p, we obtain several measures of centrality, based on the averages of the geodesic distances. The degree, closeness, harmonic and eccentricity centralities are special cases of the p-means centrality. We have shown that the most central nodes, defined by negative values of p, are closer to the nodes with the greatest spreading capacity, defined by the SIR model. We also verify for which values of p, the p-means centrality satisfies the centrality axioms.