In graph theory and network analysis, group centrality generalizes the concept of centrality to sets of nodes in a network. Introduced by Everett and Borgatti in 1999, group centrality measures are intended to lift the restriction of usual centrality measures only applying to individual nodes. For instance, the analysis of the centrality or marginalization of certain groups in the network of an organization can be enabled by group centrality measures. One complication when searching for groups of a given size k {\textstyle k} with maximum group centrality arises from the fact that for many measures, this problem is NP-hard. Therefore, heuristics and approximation algorithms are typically used in practical applications.
Group degree centrality In analogy to the conceptually simple degree centrality, which measures the degree of a node, the group degree centrality of a group S {\textstyle S} of nodes is simply defined as the number of non-group nodes adjacent to nodes in S {\textstyle S} . Formally, the group degree centrality C D ( S ) {\textstyle C_{D}(S)} of a group S ⊆ V {\textstyle S\subseteq V} of a given graph G = ( V , E ) {\textstyle G=(V,E)} is defined as
C D ( S ) = | ⋃ v ∈ S N ( v ) ∖ S | {\displaystyle C_{D}(S)=\left|\bigcup _{v\in S}N(v)\setminus S\right|}
When the size k {\textstyle k} is part of the input, the problem of determining a set S {\textstyle S} of size k {\textstyle k} with maximum group degree centrality is NP-hard.
Group closeness centrality In a connected graph, the normalized group closeness centrality (or group closeness) of a group is the reciprocal of the average length of the shortest path between the group and all other nodes in the graph. More precisely, the group closeness centrality C ( S ) {\textstyle C(S)} of a group S ⊆ V {\textstyle S\subseteq V} in a connected graph G = ( V , E ) {\textstyle G=(V,E)} is defined as
C ( S ) = | V | − | S | ∑ v ∈ V dist ( v , S ) {\displaystyle C(S)={\frac {|V|-|S|}{\sum _{v\in V}{\text{dist}}(v,S)}}} , where
dist ( v , S ) := min u ∈ S dist ( v , u ) {\displaystyle {\text{dist}}(v,S):=\min _{u\in S}{\text{dist}}(v,u)}
denotes the distance of v {\textstyle v} to S {\textstyle S} , that is, the distance to the closest node in S {\textstyle S} . Hence, just like Closeness centrality, this measure can be regarded as the normalized variant of the reciprocal of a farness measure, which for the group case is given as
f ( S ) = ∑ v ∈ V dist ( v , S ) {\textstyle f(S)=\sum _{v\in V}{\text{dist}}(v,S)} . The problem of finding a group S {\textstyle S} of given size k {\textstyle k} with maximum group closeness is NP-hard.
Group betweenness centrality Group betweenness centrality is the generalization of betweenness centrality and similarly quantifies the number of times group nodes act as a bridge along the shortest path between two other nodes. Formally, for a graph G = ( V , E ) {\textstyle G=(V,E)} it is defined as
C B ( S ) = ∑ s ≠ v ≠ t ∈ V σ s t ( S ) σ s t {\displaystyle C_{B}(S)=\sum _{s\neq v\neq t\in V}{\frac {\sigma _{st}(S)}{\sigma _{st}}}}
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