# brokerage: Perform a Gould-Fernandez Brokerage Analysis In sna: Tools for Social Network 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 two-path 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 two-path 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. Two-path 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. Two-path structure: A -> B -> A

• b_{IO}: Representative role; the broker mediates an incoming contact from an out-group member to an in-group member. Two-path structure: A -> B -> B

• b_{OI}: Gatekeeper role; the broker mediates an outgoing contact from an in-group member to an out-group member. Two-path 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. Two-path structure: A -> B -> C

• t: Total (cumulative) brokerage role occupancy. (Any of the above two-paths.)

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 z-tests 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. Simulation-based 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 [email protected]

## References

Gould, R.V. and Fernandez, R.M. 1989. “Structures of Mediation: A Formal Approach to Brokerage in Transaction Networks.” Sociological Methodology, 19: 89-126.

`triad.census`, `gtrans`

## Examples

 ```1 2 3 4 5 6 7``` ```#Draw a random network with 3 groups g<-rgraph(15) cl<-rep(1:3,5) #Compute a brokerage object b<-brokerage(g,cl) summary(b) ```

### Example output

```Loading required package: statnet.common
network: Classes for Relational Data
Version 1.13.0 created on 2015-08-31.
copyright (c) 2005, Carter T. Butts, University of California-Irvine
Mark S. Handcock, University of California -- Los Angeles
David R. Hunter, Penn State University
Martina Morris, University of Washington
Skye Bender-deMoll, University of Washington
For citation information, type citation("network").
Type help("network-package") to get started.

sna: Tools for Social Network Analysis
Version 2.4 created on 2016-07-23.
copyright (c) 2005, Carter T. Butts, University of California-Irvine
For citation information, type citation("sna").
Type help(package="sna") to get started.

Gould-Fernandez Brokerage Analysis

Global Brokerage Properties
t     E(t)    Sd(t)       z Pr(>|z|)
w_I   22.0000  22.7122   5.5569 -0.1282   0.8980
w_O   93.0000  75.7074  16.8818  1.0243   0.3057
b_IO  81.0000  75.7074  14.4164  0.3671   0.7135
b_OI  80.0000  75.7074  14.4164  0.2978   0.7659
b_O   95.0000  94.6343  11.2836  0.0324   0.9741
t    371.0000 344.4688  29.6401  0.8951   0.3707

Individual Properties (by Group)

Group ID: 1
w_I w_O b_IO b_OI b_O  t        w_I        w_O       b_IO       b_OI
[1,]   2   6    5    7   6 26  0.3374547  0.3335747 -0.0136666  0.5659030
[2,]   2   6    5    4   5 22  0.3374547  0.3335747 -0.0136666 -0.3034514
[3,]   0   7    0    9   5 21 -1.0516718  0.6836598 -1.4625905  1.1454725
[4,]   2   4    7    7   8 28  0.3374547 -0.3665956  0.5659030  0.5659030
[5,]   1   6   10    2   5 24 -0.3571086  0.3335747  1.4352573 -0.8830210
b_O          t
[1,] -0.09008752  0.3230293
[2,] -0.38167864 -0.1026511
[3,] -0.38167864 -0.2090712
[4,]  0.49309473  0.5358695
[5,] -0.38167864  0.1101891

Group ID: 2
w_I w_O b_IO b_OI b_O  t        w_I        w_O       b_IO       b_OI
[1,]   2   7   10    6  10 35  0.3374547  0.6836598  1.4352573  0.2761182
[2,]   4   2    8    3   3 20  1.7265813 -1.0667659  0.8556878 -0.5932362
[3,]   1   7    3    9   6 26 -0.3571086  0.6836598 -0.5932362  1.1454725
[4,]   1   8   10    3   9 31 -0.3571086  1.0337450  1.4352573 -0.5932362
[5,]   3   9    8   14   7 41  1.0320180  1.3838301  0.8556878  2.5943965
b_O          t
[1,]  1.07627698  1.2808102
[2,] -0.96486089 -0.3154913
[3,] -0.09008752  0.3230293
[4,]  0.78468585  0.8551298
[5,]  0.20150360  1.9193309

Group ID: 3
w_I w_O b_IO b_OI b_O  t        w_I         w_O       b_IO       b_OI
[1,]   1   3    4    4   3 15 -0.3571086 -0.71668073 -0.3034514 -0.3034514
[2,]   1  11    3    7  11 33 -0.3571086  2.08400043 -0.5932362  0.5659030
[3,]   1   9    3    3   8 24 -0.3571086  1.38383014 -0.5932362 -0.5932362
[4,]   0   5    0    1   3  9 -1.0516718 -0.01651044 -1.4625905 -1.1728057
[5,]   1   3    5    1   6 16 -0.3571086 -0.71668073 -0.0136666 -1.1728057
b_O          t
[1,] -0.96486089 -0.8475918
[2,]  1.36786810  1.0679700
[3,]  0.49309473  0.1101891
[4,] -0.96486089 -1.4861124
[5,] -0.09008752 -0.7411717
```

sna documentation built on May 30, 2017, 12:18 a.m.