Unnormalized and normalized forms of gefura measures in directed and undirected networks

In some networks nodes belong to predefined groups （e.g., authors belong to institutions）. Common network cen- trality measures do not take this structure into account. Gefura measures are designed as indicators of a node＇s brokerage role between such groups. They are defined as variants of betweeuness centrality and consider to what extent a node belongs to shortest paths between nodes from different groups. In this article we make the following new contributions to their study： （1） We systematically study unnormalized gefura measures and show that, next to the ＇structural＇ normalization that has hitherto been applied, a ＇basic＇ normalization procedure is possible. While the former normalizes at the level of groups, the latter normalizes at the level of nodes. （2） Treating undirected networks as equivalent to symmetric directed networks, we expand the definition of gefura measures to the directed case. （3） It is shown how Brandes＇ algorithm for betweenness centrality can be adjusted to cover these cases.

In some networks nodes belong to predefined groups(e.g.,authors belong to institutions).Common network centrality measures do not take this structure into account.Gefura measures are designed as indicators of a node's brokerage role between such groups.They are defined as variants of betweenness centrality and consider to what extent a node belongs to shortest paths between nodes from different groups.In this article we make the following new contributions to their study:(1) We systematically study unnormalized gefura measures and show that,next to the ‘structural' normalization that has hitherto been applied,a ‘basic' normalization procedure is possible.While the former normalizes at the level of groups,the latter normalizes at the level of nodes.(2) Treating undirected networks as equivalent to symmetric directed networks,we expand the definition of gefura measures to the directed case.(3) It is shown how Brandes' algorithm for betweenness centrality can be adjusted to cover these cases.