ControlRank Centrality

ControlRank (Zhou, Yu and Lu 2019) is the smallest eigenvalue of the symmetric part of the graph Laplacian $L = D - A$ after the row and column of the node are deleted: $$CR_i = \lambda_{\min}\left(\left(\frac{L + L^T}{2}\right)_{-i,-i} \right).$$ Larger values rank higher.

Usage

centrality_controlrank(x, ...)

Arguments

x: Network input accepted by centrality.
...: Further arguments to centrality. The measure uses weighted (use edge weights, default TRUE) and normalized (divide by the maximum, default FALSE).

Value

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

Details

Edge weights must be finite and nonnegative, weighted = FALSE gives every edge weight one, and loops are removed. In a directed network $A_{ij}$ is the arc from $i$ to $j$, $D$ holds the out-strengths and scores can be negative. On a connected undirected network with at least two nodes all scores are positive, and on a disconnected undirected network every score is zero. A single node scores zero. With normalized = TRUE the scores are divided by their maximum when it is positive, so negative directed scores stay negative. For matrix input with very small weights, set directed = TRUE explicitly, because symmetry detection is approximate. A weight range beyond double precision or an unresolved spectrum raises an error.

References

Zhou, J., Yu, X. and Lu, J.-A. (2019). Node Importance in Controlled Complex Networks. IEEE Transactions on Circuits and Systems II: Express Briefs, 66(3), 437-441. doi:10.1109/TCSII.2018.2845940 .

Examples

centrality_controlrank(regulation_net)
#>     Explore        Plan     Monitor       Adapt     Reflect     Discuss 
#> -0.02646079 -0.08739707 -0.04912661 -0.06026944  0.03788145 -0.04541166 
#>  Synthesize    Evaluate      Create       Share 
#> -0.07026868 -0.07521652 -0.07620639 -0.07268062