gilschmidt computes the Gil-Schmidt Power Index for all nodes in dat, with or without normalization.
gilschmidt(dat, g = 1, nodes = NULL, gmode = "digraph", diag = FALSE,
tmaxdev = FALSE, normalize = TRUE)A vector of centrality scores.
one or more input graphs (for best performance, sna edgelists or network objects are suggested).
integer indicating the index of the graph for which centralities are to be calculated (or a vector thereof). By default, g=1.
list indicating which nodes are to be included in the calculation. By default, all nodes are included.
string indicating the type of graph being evaluated. "digraph" indicates that edges should be interpreted as directed; "graph" indicates that edges are undirected. gmode is set to "digraph" by default.
boolean indicating whether or not the diagonal should be treated as valid data. (This has no effect on this index, but is included for compatibility with centralization.
boolean indicating whether or not the theoretical maximum absolute deviation from the maximum nodal centrality should be returned. By default, tmaxdev==FALSE.
logical; should the index scores be normalized?
Carter T. Butts, buttsc@uci.edu
For graph \(G=(V,E)\), let \(R(v,G)\) be the set of vertices reachable by \(v\) in \(V\setminus v\). Then the Gil-Schmidt power index is defined as $$C_{GS}(v) = \frac{\sum_{i \in R(v,G)} \frac{1}{d(v,i)}}{|R(v,G)|}.$$ where \(d(v,i)\) is the geodesic distance from \(v\) to \(i\) in \(G\); the index is taken to be 0 for isolates. The measure takes a value of 1 when \(v\) is adjacent to all reachable vertices, and approaches 0 as the distance from \(v\) to each vertex approaches infinity. (For finite \(N=|V|\), the minimum value is 0 if \(v\) is an isolate, and otherwise \(1/(N-1)\).)
If normalize=FALSE is selected, then normalization by \(|R(v,G)|\) is not performed. This measure has been proposed as a better-behaved alternative to closeness (to which it is closely related).
The closeness function in the sna library can also be used to compute this index.
Gil, J. and Schmidt, S. (1996). “The Origin of the Mexican Network of Power”. Proceedings of the International Social Network Conference, Charleston, SC, 22-25.
Sinclair, P.A. (2009). “Network Centralization with the Gil Schmidt Power Centrality Index” Social Networks, 29, 81-92.
closeness, centralization
data(coleman) #Load Coleman friendship network
gs<-gilschmidt(coleman,g=1:2) #Compute the Gil-Schmidt index
#Plot G-S values in the fall, versus spring
plot(gs,xlab="Fall",ylab="Spring",main="G-S Index")
abline(0,1)
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