R: Find the Centralization of a Given Network, for Some Measure...

centralization {sna} | R Documentation

## Find the Centralization of a Given Network, for Some Measure of Centrality

### Description

`Centralization` returns the centralization GLI (graph-level index) for a given graph in `dat`, given a (node) centrality measure `FUN`. `Centralization` follows Freeman's (1979) generalized definition of network centralization, and can be used with any properly defined centrality measure. This measure must be implemented separately; see the references below for examples.

### Usage

    centralization(dat, FUN, g=NULL, mode="digraph", diag=FALSE,
        normalize=TRUE, ...)

### Arguments

`dat` | one or more input graphs.
`FUN` | Function to return nodal centrality scores.
`g` | Integer indicating the index of the graph for which centralization should be computed. By default, all graphs are employed.
`mode` | String indicating the type of graph being evaluated. "digraph" indicates that edges should be interpreted as directed; "graph" indicates that edges are undirected. `mode` is set to "digraph" by default.
`diag` | Boolean indicating whether or not the diagonal should be treated as valid data. Set this true if and only if the data can contain loops. `diag` is `FALSE` by default.
`normalize` | Boolean indicating whether or not the centralization score should be normalized to the theoretical maximum. (Note that this function relies on `FUN` to return this value when called with `tmaxdev==TRUE`.) By default, `tmaxdev==TRUE`.
`...` | Additional arguments to `FUN`.

### Details

The centralization of a graph G for centrality measure C(v) is defined (as per Freeman (1979)) to be:

C^*(G) = sum( |max(C(v))-C(i)|, i in V(G) )

Or, equivalently, the absolute deviation from the maximum of C on G. Generally, this value is normalized by the theoretical maximum centralization score, conditional on |V(G)|. (Here, this functionality is activated by `normalize`.) `Centralization` depends on the function specified by `FUN` to return the vector of nodal centralities when called with `dat` and `g`, and to return the theoretical maximum value when called with the above and `tmaxdev==TRUE`. For an example of such a centrality routine, see `degree`.

### Value

The centralization of the specified graph.

### Note

See `cugtest` for null hypothesis tests involving centralization scores.

### Author(s)

Carter T. Butts buttsc@uci.edu

### References

Freeman, L.C. (1979). “Centrality in Social Networks I: Conceptual Clarification.” Social Networks, 1, 215-239.

Wasserman, S., and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.

### See Also

`cugtest`

### Examples

    #Generate some random graphs
    dat<-rgraph(5,10)
    #How centralized is the third one on indegree?
    centralization(dat,g=3,degree,cmode="indegree")
    #How about on total (Freeman) degree?
    centralization(dat,g=3,degree)

[Package sna version 2.6 Index]