`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.

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

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`

.

The centralization of the specified graph.

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

$$C^*(G) = \sum_{i \in V(G)} \left|\max_{v \in V(G)}(C(v))-C(i)\right|$$

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`

.

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.

```
# NOT RUN {
#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)
# }
```

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