Group centralization of network indices

In social network analysis various centrality indices are introduced to quantify importance of nodes in networks. Group centrality indices, introduced in 1999 by Borgatti and Everett, measure the importance of groups of nodes in networks. While centrality measures compare the importance of different nodes within a graph, the associated notion of centralization, as introduced in 1979 by Freeman allows us to compare the relative importance of nodes within their respective graphs. In this paper, we study the notion of group centralization with respect to eccentricity, degree and betweenness centrality measures. For groups of size 2, we determine the maximum achieved value of group eccentricity and group betweenness centralization and describe the corresponding extremal graphs. For group degree centralization we do the same with arbitrary size of group. We conclude with posing few open problems.