Perform a GouldFernandez Brokerage Analysis
Description
Performs the brokerage analysis of Gould and Fernandez on one or more input graphs, given a class membership vector.
Usage
1  brokerage(g, cl)

Arguments
g 
one or more input graphs. 
cl 
a vector of class memberships. 
Details
Gould and Fernandez (following Marsden and others) describe brokerage as the role played by a social actor who mediates contact between two alters. More formally, vertex v is a broker for distinct vertices a and b iff a > v > b and a !> b. Where actors belong to a priori distinct groups, group membership may be used to segment brokerage roles into particular types. Let A > B > C denote the twopath associated with a brokerage structure, such that some vertex from group B brokers the connection from some vertex from group A to a vertex in group C. The types of brokerage roles defined by Gould and Fernandez (and their accompanying twopath structures) are then defined in terms of group membership as follows:

w_I: Coordinator role; the broker mediates contact between two individuals from his or her own group. Twopath structure: A > A > A

w_O: Itinerant broker role; the broker mediates contact between two individuals from a single group to which he or she does not belong. Twopath structure: A > B > A

b_{IO}: Representative role; the broker mediates an incoming contact from an outgroup member to an ingroup member. Twopath structure: A > B > B

b_{OI}: Gatekeeper role; the broker mediates an outgoing contact from an ingroup member to an outgroup member. Twopath structure: A > A > B

b_O: Liaison role; the broker mediates contact between two individuals from different groups, neither of which is the group to which he or she belongs. Twopath structure: A > B > C

t: Total (cumulative) brokerage role occupancy. (Any of the above twopaths.)
The brokerage score for a given vertex with respect to a given role is the number of ordered pairs having the appropriate group membership(s) brokered by said vertex. brokerage
computes the brokerage scores for each vertex, given an input graph and vector of class memberships. Aggregate scores are also computed at the graph level, which correspond to the total frequency of each role type within the network structure. Expectations and variances of the brokerage scores conditional on size and density are computed, along with approximate ztests for incidence of brokerage. (Note that the accuracy of the normality assumption is not known in the general case; see Gould and Fernandez (1989) for details. Simulationbased tests may be desirable as an alternative.)
Value
An object of class brokerage
, containing the following elements:
raw.nli 
The matrix of observed brokerage scores, by vertex 
exp.nli 
The matrix of expected brokerage scores, by vertex 
sd.nli 
The matrix of predicted brokerage score standard deviations, by vertex 
z.nli 
The matrix of standardized brokerage scores, by vertex 
raw.gli 
The vector of observed aggregate brokerage scores 
exp.gli 
The vector of expected aggregate brokerage scores 
sd.gli 
The vector of predicted aggregate brokerage score standard deviations 
z.gli 
The vector of standardized aggregate brokerage scores 
exp.grp 
The matrix of expected brokerage scores, by group 
sd.grp 
The matrix of predicted brokerage score standard deviations, by group 
cl 
The vector of class memberships 
clid 
The original class names 
n 
The input class sizes 
N 
The order of the input network 
Author(s)
Carter T. Butts buttsc@uci.edu
References
Gould, R.V. and Fernandez, R.M. 1989. “Structures of Mediation: A Formal Approach to Brokerage in Transaction Networks.” Sociological Methodology, 19: 89126.
See Also
triad.census
, gtrans
Examples
1 2 3 4 5 6 7 