X-Degree Centrality

X-degree (Torres et al. 2021, equation 3.15) counts the oriented nonbacktracking walks of four edges whose middle node is $i$. It depends only on the degrees $d_j$ of the neighbors of $i$: $$Xdeg(i) = \Big(\sum_{j \in N(i)} (d_j - 1)\Big)^2 - \sum_{j \in N(i)} (d_j - 1)^2.$$

Usage

centrality_x_degree(x, ...)

Arguments

x: Network input accepted by centrality.
...: Further arguments to centrality, such as normalized (divide by the maximum, default FALSE).

Value

A named numeric vector with one score per node, in input node order.

Details

The measure is computed on the simple undirected skeleton of the network, so direction, weights, loops and parallel edges are ignored. Isolated nodes and leaves score zero, and every node of a star scores zero. Components are scored independently. The function computes the score on the supplied network. The iterative node-removal immunization procedure of the paper is a separate algorithm.

References

Torres, L., Chan, K. S., Tong, H., & Eliassi-Rad, T. (2021). Nonbacktracking Eigenvalues under Node Removal: X-Centrality and Targeted Immunization. SIAM Journal on Mathematics of Data Science, 3(2), 656-675. doi:10.1137/20M1352132 .

Examples

centrality_x_degree(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>        386        558        768        516        348        422        298 
#>   Evaluate     Create      Share 
#>        498        604        498