Compute the Gil-Schmidt Power Index
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)dat |
one or more input graphs (for best performance, sna edgelists or network objects are suggested). |
g |
integer indicating the index of the graph for which centralities are to be calculated (or a vector thereof). By default, |
nodes |
list indicating which nodes are to be included in the calculation. By default, all nodes are included. |
gmode |
string indicating the type of graph being evaluated. |
diag |
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 |
tmaxdev |
boolean indicating whether or not the theoretical maximum absolute deviation from the maximum nodal centrality should be returned. By default, |
normalize |
logical; should the index scores be normalized? |
For graph G=(V,E), let R(v,G) be the set of vertices reachable by v in V \ v. Then the Gil-Schmidt power index is defined as
C_GS(v) = sum( 1/d(v,i), i in R(v,G) )/|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.
A vector of centrality scores.
Carter T. Butts, buttsc@uci.edu
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.
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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