# Group Betweenness Centrality

#### Definition

Group betweenness centrality (GBC) shows the proportion of geodesics connecting pairs of non-group members that pass through a group of nodes.

GBC of a group of nodes stands for the fraction of shortest paths between pairs of nodes in a network that passes through at least one of the nodes in the group. Thus, GBC can be used for estimating the ability of a group of collaborating nodes to monitor the network traffic. [EVERETT, M. G. 1999, PUZIS, R., 2009]

Defined the group betweenness of u and v as:### B(u, v) = C

where:
for i

[KOLACZYK, E. D., 2009]

GBC of a group of nodes stands for the fraction of shortest paths between pairs of nodes in a network that passes through at least one of the nodes in the group. Thus, GBC can be used for estimating the ability of a group of collaborating nodes to monitor the network traffic. [EVERETT, M. G. 1999, PUZIS, R., 2009]

Defined the group betweenness of u and v as:

### B(u, v) = C_{{u,v}}(u, u) + C_{{u,v}}(v, v) − C_{{u,v}}(u, v),

_{1}, i_{2}∈ {u, v}, and σ_{st}(i_{1}, i_{2}) is the number of geodesic paths between vertices s and t that pass through both i_{1}and i_{2}.[KOLACZYK, E. D., 2009]

**Faster computation of successive bounds on the group betweenness centrality**

Dinler, D. and Tural, M.K., Faster computation of successive bounds on the group betweenness centrality. Networks - Wiley Online Library.

**K-step Group Betweenness Centrality**

Akgün, M.K. and Tural, M.K., 2020. k-step betweenness centrality. Computational and Mathematical Organization Theory, 26(1), pp.55-87.

#### Software

#### References

- EVERETT, M. G. & BORGATTI, S. P. 1999. The centrality of groups and classes. Journal of Mathematical Sociology, 23, 181-201. DOI: 10.1080/0022250X.1999.9990219
- KOLACZYK, E. D., CHUA, D. B. & BARTHÉLEMY, M. 2009. Group betweenness and co-betweenness: Inter-related notions of coalition centrality. Social Networks, 31, 190-203. DOI: 10.1016/j.socnet.2009.02.003
- PUZIS, R., YAGIL, D., ELOVICI, Y. & BRAHA, D. 2009. Collaborative attack on Internet users' anonymity. Internet Research, 19, 60-77. DOI: 10.1108/10662240910927821