# NC - Normalized α-Centrality

#### Definition

Bonacich defined α-centrality C

_{i,j}(α,β,n) as the total number of attenuated paths between nodes i and j, with β and α giving the attenuation factors along direct edges (from i) and indirect edges (from interme- diate nodes) in the path from i to j, respectively, and n is the length of the longest path. Given the adjacency matrix of the network A, α-centrality matrix is defined as follows: Normalized α-centrality defined as matrix as: Just like the original α-centrality, normalized α-centrality contains a tunable parameter α that sets the length scale of interactions. For α = 0, (normalized) α-centrality takes into account direct edges only. As α increases, NC(α,β,n → ∞) becomes a more global measure, taking into account ever larger network components.#### Software

#### References

- GHOSH, R. & LERMAN, K. 2011. Parameterized centrality metric for network analysis. Physical Review E, 83, 066118.