Degree and Importance of Lines

The degree and importance of lines (DIL; Liu et al. 2016) adds to the degree of a node a share of the importance of every line incident to it. A line $e_{ij}$ that lies on $p$ triangles has importance $I_{ij} = (k_i - p - 1)(k_j - p - 1) / \lambda$ with $\lambda = p/2 + 1$, and this importance is split between the endpoints in proportion to their degrees minus one: $$L_i = k_i + \sum_{j \in \Gamma_i} I_{ij} \frac{k_i - 1}{k_i + k_j - 2}.$$

Usage

centrality_dil(x, ...)

Arguments

x: Network input accepted by centrality.
...: Further arguments to centrality, such as normalized.

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. Every score is at least the degree of the node, and isolated nodes score 0. The share is undefined only on an isolated edge, whose importance is zero. That share is set to zero, so both endpoints score 1. The text layer of the published article reads $\lambda$ as $2p + 1$. The printed page and the worked example of the article give $p/2 + 1$, which is the value used here. normalized = TRUE divides the scores by their maximum.

References

Liu, J., Xiong, Q., Shi, W., Shi, X. and Wang, K. (2016). Evaluating the importance of nodes in complex networks. Physica A: Statistical Mechanics and its Applications, 452, 209-219. doi:10.1016/j.physa.2016.02.049 .

Examples

centrality_dil(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  11.866667  17.580808  13.101818  22.646465  17.885714  16.644444   9.714286 
#>   Evaluate     Create      Share 
#>  14.222222  12.702020   9.502222