κ-Betweenness Centrality (Bounded-Distance Betweenness)

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_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.

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