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# κ-Betweenness Centrality (Bounded-Distance Betweenness)

#### Definition

κ-betweenness centrality is given by:
For an arbitrary graph G(V,E), let d(s, t) denote
the length of the shortest path between vertices s and t. Define σ

This definition of κ-betweenness centrality subsumes Freeman’s definition of betweenness centrality for κ = 0.

_{stk}to be the number of paths between s and t whose length is less than or equal to d(s, t) + κ. Likewise, σ_{st}(v) is the count of the subset of paths that pass through vertex v.This definition of κ-betweenness centrality subsumes Freeman’s definition of betweenness centrality for κ = 0.

#### Software

#### References

- Borgatti, S.P. and Everett, M.G., 2006. A graph-theoretic perspective on centrality. Social networks, 28(4), pp.466-484.
- JIANG, K., EDIGER, D. & BADER, D. A. Generalizing k-Betweenness centrality using short paths and a parallel multithreaded implementation. Parallel Processing, 2009. ICPP'09. International Conference on, 2009. IEEE, 542-549.