infocent {sna} | R Documentation

## Find Information Centrality Scores of Network Positions

### Description

`infocent` takes one or more graphs (`dat`) and returns the information centralities of positions (selected by `nodes`) within the graphs indicated by `g`. This function is compatible with `centralization`, and will return the theoretical maximum absolute deviation (from maximum) conditional on size (which is used by `centralization` to normalize the observed centralization score).

### Usage

    infocent(dat, g=1, nodes=NULL, gmode="digraph", diag=FALSE,
        cmode="weak", tmaxdev=FALSE, rescale=FALSE,tol=1e-20)

### Arguments

`dat` | one or more input graphs.
`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. This is currently ignored.
`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.
`cmode` | the rule to be used by `symmetrize` when symmetrizing dichotomous data; must be one of `"weak"` (for an `OR` rule), `"strong"` for an `AND` rule), `"upper"` (for a `max` rule), or `"lower"` (for a `min` rule). Set to `"weak"` by default, this parameter obviously has no effect on symmetric data.
`tmaxdev` | boolean indicating whether or not the theoretical maximum absolute deviation from the maximum nodal centrality should be returned. By default, `tmaxdev==FALSE`.
`rescale` | if true, centrality scores are rescaled such that they sum to 1.
`tol` | tolerance for near-singularities during matrix inversion (see `solve`).

### Details

Actor information centrality is a hybrid measure which relates to both path-length indices (e.g., closeness, graph centrality) and to walk-based eigenmeasures (e.g., eigenvector centrality, Bonacich power). In particular, the information centrality of a given actor can be understood to be the harmonic average of the ``bandwidth'' for all paths originating with said individual (where the bandwidth is taken to be inversely related to path length). Formally, the index is constructed as follows. First, we take G to be an undirected (but possibly valued) graph – symmetrizing if necessary – with (possibly valued) adjacency matrix A. From this, we remove all isolates (whose information centralities are zero in any event) and proceed to create the weighted connection matrix

C = B^-1

where B is a pseudo-adjacency matrix formed by replacing the diagonal of 1-A with one plus each actor's degree. Given the above, let T be the trace of C with sum S_T, and let S_R be an arbitrary row sum (all rows of C have the same sum). The information centrality scores are then equal to

C_I = ( T + (S_T-2S_R)/|V(G)| )^-1

(recalling that the scores for any omitted vertices are 0).

In general, actors with higher information centrality are predicted to have greater control over the flow of information within a network; highly information-central individuals tend to have a large number of short paths to many others within the social structure. Because the raw centrality values can be difficult to interpret directly, rescaled values are sometimes preferred (see the `rescale` option). Though the use of path weights suggest information centrality as a possible replacement for closeness, the problem of inverting the B matrix poses problems of its own; as with all such measures, caution is advised on disconnected or degenerate structures.

### Value

A vector, matrix, or list containing the centrality scores (depending on the number and size of the input graphs).

### Note

The theoretical maximum deviation used here is not obtained with the star network; rather, the maximum occurs for an empty graph with one complete dyad, which is the model used here.

### Author(s)

David Barron david.barron@jesus.ox.ac.uk

Carter T. Butts buttsc@uci.edu

### References

Stephenson, K., and Zelen, M. (1989). ``Rethinking Centrality: Methods and Applications.'' Social Networks, 11, 1-37.

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

### See Also

`evcent`, `bonpow`, `closeness`, `graphcent`, `centralization`

### Examples

    #Generate some test data
    dat<-rgraph(10,mode="graph")
    #Compute information centrality scores
    infocent(dat)

[Package sna version 1.1 Index]