R: Compute the Gil-Schmidt Power Index

gilschmidt {sna} | R Documentation

## Compute the Gil-Schmidt Power Index

### Description

`gilschmidt` computes the Gil-Schmidt Power Index for all nodes in `dat`, with or without normalization.

### Usage

    gilschmidt(dat, g = 1, nodes = NULL, gmode = "digraph", diag = FALSE,
        tmaxdev = FALSE, normalize = TRUE)

### Arguments

`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, `g`=1.
`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. `"digraph"` indicates that edges should be interpreted as directed; `"graph"` indicates that edges are undirected. `gmode` is set to `"digraph"` by default.
`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 `centralization`.
`tmaxdev` | boolean indicating whether or not the theoretical maximum absolute deviation from the maximum nodal centrality should be returned. By default, `tmaxdev==FALSE`.
`normalize` | logical; should the index scores be normalized?

### Details

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.

### Value

A vector of centrality scores.

### Author(s)

Carter T. Butts, buttsc@uci.edu

### References

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.

### See Also

`closeness, centralization`

### Examples

    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)

[Package sna version 2.6 Index]