Immediate Effects Centrality

Immediate effects centrality (Friedkin 1991) is the reciprocal of the mean length of the influence sequences that end at a node. The influence matrix $W$ is the adjacency matrix with a unit diagonal, divided by its row sums. With $c$ its stationary vector, $Z = (I - W + \mathbf{1}c')^{-1}$ and the mean first passage times $M = (I - Z + E Z_{dg})\,\mathrm{diag}(1/c)$: $$IEC_j = \frac{n - 1}{\sum_{i \ne j} m_{ij}}$$

Usage

centrality_iec(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, or NA at every node when the influence chain is reducible or the network has one node.

Details

Direction is used, and weights, loops and parallel edges are ignored, so the unit diagonal is calibrated against unit edges. mode has no effect. The measure needs an irreducible chain, that is a connected undirected or a strongly connected directed network. Otherwise, and on a single node, every score is NA with a cograph_undefined_measure warning. The measure differs from centrality_markov, which omits the unit diagonal and divides by $n$, and the two can rank nodes differently. The measure is costly and is computed only when requested by name or through include.

References

Friedkin, N. E. (1991). Theoretical foundations for centrality measures. American Journal of Sociology, 96(6), 1478-1504. doi:10.1086/229694 .

Examples

centrality_iec(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#> 0.12812742 0.04122807 0.16167772 0.13181762 0.10049616 0.04665442 0.03461014 
#>   Evaluate     Create      Share 
#> 0.03095393 0.09086872 0.09704518