Length-Scaled, Delta and Ego Betweenness, and Delta Closeness

Length-scaled betweenness (Borgatti and Everett 2006; Brandes 2008) weights each pair $s,t$ in the betweenness sum by $1/d(s,t)$. Delta betweenness (Agneessens et al. 2017) uses the pair weight $(h(s,t)-1)^{-\delta}$, where $h(s,t)$ is the number of edges on a shortest path, so $\delta = 0$ gives ordinary betweenness. Ego betweenness (Everett and Borgatti 2005) is the betweenness of a node inside its own ego network. Delta closeness (Agneessens et al. 2017, eq. 2) is $$C_\delta(i) = \frac{1}{n-1} \sum_{j \ne i} d_{ij}^{-\delta}.$$

Usage

centrality_length_scaled_betweenness(x, ...)

centrality_delta_betweenness(x, betweenness_delta = 1, ...)

centrality_ego_betweenness(x, ...)

centrality_delta_closeness(x, mode = "all", closeness_delta = 1, ...)

Arguments

x: Network input accepted by centrality.
...: Further arguments to centrality. The weighted measures use weighted (default TRUE), invert_weights (default NULL, which inverts for tna input only) and alpha (inversion exponent, default 1). Delta closeness also uses cutoff (default -1, no limit).
betweenness_delta: Decay exponent $\delta$ of delta betweenness. Default 1.
mode: Direction for delta closeness: "all" (default), "out" or "in".
closeness_delta: Distance exponent $\delta$ of delta closeness. Default 1.

Value

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

Details

On a directed network the three betweenness measures count directed paths. Length-scaled and delta betweenness and delta closeness read edge weights as distances, and invert_weights = TRUE converts weights to distances $1/w^\alpha$. In delta betweenness the weighted distances decide which paths are shortest, and the pair weight $(h - 1)^{-\delta}$ uses the number of edges $h$ on a shortest path (the fewest when several tie), so $h - 1$ is the number of intermediaries. On a binary network $h = d(s,t)$. A pair joined by a one-edge shortest path contributes nothing. weighted = FALSE uses hop counts. Ego betweenness ignores weights, and a node with fewer than two neighbors scores 0. Delta closeness follows mode and cutoff. On hop distances $\delta = 1$ gives harmonic closeness divided by $n-1$ and $\delta = 0$ gives the share of nodes reached. Bounded-distance betweenness, listed in the Centrality Zoo as k-betweenness, is centrality_betweenness(x, cutoff = k).

References

Borgatti, S. P., & Everett, M. G. (2006). A graph-theoretic perspective on centrality. Social Networks, 28(4), 466-484. doi:10.1016/j.socnet.2005.11.005 .

Agneessens, F., Borgatti, S. P., & Everett, M. G. (2017). Geodesic based centrality: Unifying the local and the global. Social Networks, 49, 12-26.

Brandes, U. (2008). On variants of shortest-path betweenness centrality and their generic computation. Social Networks, 30(2), 136-145.

Everett, M., & Borgatti, S. P. (2005). Ego network betweenness. Social Networks, 27(1), 31-38.

Examples

centrality_length_scaled_betweenness(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  11.060744  28.746387  38.946145  27.957440  28.003107   1.041667  15.379439 
#>   Evaluate     Create      Share 
#>   5.258329  27.863965  14.631523 
centrality_delta_betweenness(regulation_net, weighted = FALSE)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   4.883333   6.991667   8.458333  12.475000   3.000000   2.883333   3.533333 
#>   Evaluate     Create      Share 
#>   2.491667   7.116667   8.166667 
centrality_ego_betweenness(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>    8.00000    9.00000   11.83333   17.00000    4.00000    3.00000    7.00000 
#>   Evaluate     Create      Share 
#>    4.00000   13.16667   10.00000 
centrality_delta_closeness(regulation_net, closeness_delta = 2)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   67.51029   26.77763   49.56442   15.65708   82.11082   13.46055   44.62772 
#>   Evaluate     Create      Share 
#>   38.75376   27.73156   11.54804