Introduction

This lab will examine how to measure individuals’ perceived structure of social networks (cognitive social structures or CSS) and how to analyze these perceptions, drawing on notions of structural equivalence and quadratic assignment procedure discussed in class.

We will be using the Krackhardt’s Advice and Friendship data sets. 11 This is the data set reported in Krackhardt, D. (1987) “Cognitive Social Structures,” Social Networks, 9: 109—34. Reading that paper before carrying out the analysis is strongly recommended. The 21 respondents are managers in a company.

There are four sections to this lab below: CSS analysis, visualization, structural equivalence, and differences and correlation. We are not looking for an essay response to every question, but you should succinctly convey that you understand how to interpret and make inferences based on the outputs from these analyses.

This assignment is designed to use the sna package in the R statistical programming language.22 See Butts, Carter T., sna: Tools for Social Network Analysis, R package version 2.4.; see also ?? sna for documentation and Butts, Carter T. (2008). “Social Network Analysis with sna.” Journal of Statistical Software, 24(6). You are provided the RData file. krackhardt_css_data.RData

Our visualization for this exercise will be done using ggnet2,33 Moritz Marbach and Francois Briatte, with help from Heike Hoffmann, Pedro Jordano and Ming-Yu Liu; see ?? ggnet2. a visualization package which applies the visualization framework developed in ggplot2, an up-and-coming visualization framework created by RStudio that is well on its way to being recognized as the professional standard in R visualization.44 See ?? ggplot2, and the tidyverse website.

I. CSS Analysis and Extraction (20 pts)

We’ll begin by viewing an example response matrix. Notice that this is a binary sociomatrix.

Advice Matrix Table

The advice_nets and friendship_nets objects are R lists that each contain 21 networks, one for each respondent’s perception about what the advice and friendship networks look like. Let’s view the characteristics of a sample friendship network. We’ll visualize the ties within that network in the next Part.

Individual Respondent Network

Next, we’ll aggregate the individual observations of each actor within the network into a single network. There are multiple ways to do so. Each presents a different manner of combining the 21 responses into a single aggregated network. These include four locally aggregated structures (LAS) and one consensus aggregated structure. First, we calculate the four LAS: row, column, intersection, and union.

Friendship, Column.

Friendship, Column Matrix

Friendship, Row.

Friendship, Row

Friendship, Intersection.

Friendship Intersection

Friendship, Union.

Friendship Union

Friendship, Median.

Friendship Median

ad_column <- consensus(advice_nets, mode="digraph", diag=FALSE, method="OR.col")
ad_row    <- consensus(advice_nets, mode="digraph", diag=FALSE, method="OR.row")
ad_intersection <- consensus(advice_nets, mode="digraph", diag=FALSE, method="LAS.intersection")
ad_union  <- consensus(advice_nets, mode="digraph", diag=FALSE, method="LAS.union")
ad_median <- consensus(advice_nets, mode="digraph", diag=FALSE, method="central.graph")

II. Visualization (25 pts)

Define Network Structure

First, we’re going to define the position of the nodes on the network so that it is easier to compare edges across graphs.

Base Graph Structure

Plot Two Aggregated Friendship Networks

Using our initial node placement as a template, we will now visualize the ties for aggregated networks.

# If you pass ggnet2 the mode value of a matrix, it will use the first two vectors to position the nodes on their x and y axes. Thus, if we call baseLayout throughout the rest of the visualizations, the nodes will remain in place but the edges drawn between the visualizations will change. 
ggnet2(fr_union_net, mode = baseLayout, label = TRUE,  arrow.size = 8, edge.color="red", arrow.gap = 0.03) + theme(panel.background = element_rect(fill = "#fffff8"))

Friendship, Union.

Analysis

Describe what this network shows in your own words.
ANSWER: This graph shows that not only the number of relationships vary a lot across nodes, but some relationships are not mutual. Nodes 2, 14 and 21 have a high degree of mutual connectivity. Node 11 thinks that it has a lot of relationships, but they are not mutual. Some outliers: 18 has no friends and 10 only considers one other node as a friend, but that is not reciprocal.

Friendship, Row.

Plot Two Aggregated Advice Networks

Choose two of the aggregated advice networks calculated above to visualize.

# Feel free to edit this portion of the code if you would like to plot different aggregated measures.
# Plot the ad_intersection network.
ad_union_net <-  igraph::graph.adjacency(ad_union) # make an igraph network object from the advice union adjacency matrix

ggnet2(ad_union_net, mode = baseLayout, label = TRUE,  arrow.size = 8, edge.color="pink", arrow.gap = 0.03) + theme(panel.background = element_rect(fill = "#fffff8")) # plots the advice intersection network

Advice, Union

Analysis

Describe what this network shows in your own words.
ANSWER: This network shows that some node have an connectivity when the perspective of all observers are accounted for when it comes to advice-seeking. High recall, but is it low precision? Node 18 is a particularly striking example, since he is seeked by many other nodes, despite having no friendships according to the other data. It seems that there is a strong percepction that nodes feel that they are requesting and providing advice.

Advice, Median.

Plot Two Individual Self-Report Networks

Next, we’ll plot two of the individual self-report networks. Choose two respondents (by number, 1—21) from either advice_nets or friendship_nets, or visualize both of a single respondent’s self-reports.

Respondent Networks

Respondent Network 1: node 18.

Respondent Network 2: node 20

Plot the Intersection of the FR_Union and Ad_Union Networks

Finally, we’re going to plot the intersection of two networks.

Intersection of Friendship, Union, and Advice, Union.

III. Structural Equivalence (25 pts)

In this section, we will compute the structural equivalence among the actors using the locally aggregated structure (LAS). Based upon the exploratory visualizations you created in Part II, choose one LAS structure for both the AD and FR relation type to compute structural equivalence (e.g. ad_union and fr_union).

Structural Equivalence Matrices

Now, we’ll generate the matrices used to evaluate structural equivalence. Note that we’ll be using the Euclidean distance method.55 Hint: the way you interpret results using the Pearson correlation method and the Euclidean distance method are inverse. You should review the readings or slides from class to make sure you understand how to interpret results.

Advice SEM

Advice SEM

Friendship SEM

Take a moment to compare the structural equivalence matrices (SEM) for the advice and friendship networks that you analyzed. You might want to refer to previous visualizations. Notice that it’s challenging to decode this information visually in matrix form, even for a relatively small network. We’ll search the matrix programmatically to understand more about it.

Next, we will identify the two nodes with the highest and lowest SEM Euclidean distance in each matrix as well as the mean value of distance across both networks.

Friendship SEM Summary Statitistics

## [1] "Friendship SEM"

##    row col
## 3    3   1
## 8    8   1
## 9    9   1
## 14  14   1
## 17  17   1
## 18  18   1
## 1    1   3
## 6    6   3
## 7    7   3
## 10  10   3
## 11  11   3
## 13  13   3
## 15  15   3
## 16  16   3
## 20  20   3
## 12  12   4
## 19  19   5
## 3    3   6
## 8    8   6
## 9    9   6
## 14  14   6
## 17  17   6
## 18  18   6
## 3    3   7
## 8    8   7
## 9    9   7
## 14  14   7
## 17  17   7
## 18  18   7
## 1    1   8
## 6    6   8
## 7    7   8
## 10  10   8
## 11  11   8
## 12  12   8
## 13  13   8
## 15  15   8
## 16  16   8
## 20  20   8
## 1    1   9
## 6    6   9
## 7    7   9
## 10  10   9
## 11  11   9
## 13  13   9
## 15  15   9
## 16  16   9
## 19  19   9
## 20  20   9
## 3    3  10
## 8    8  10
## 9    9  10
## 14  14  10
## 17  17  10
## 18  18  10
## 3    3  11
## 8    8  11
## 9    9  11
## 14  14  11
## 17  17  11
## 18  18  11
## 4    4  12
## 8    8  12
## 3    3  13
## 8    8  13
## 9    9  13
## 14  14  13
## 17  17  13
## 18  18  13
## 1    1  14
## 6    6  14
## 7    7  14
## 10  10  14
## 11  11  14
## 13  13  14
## 15  15  14
## 16  16  14
## 20  20  14
## 3    3  15
## 8    8  15
## 9    9  15
## 14  14  15
## 17  17  15
## 18  18  15
## 3    3  16
## 8    8  16
## 9    9  16
## 14  14  16
## 17  17  16
## 18  18  16
## 1    1  17
## 6    6  17
## 7    7  17
## 10  10  17
## 11  11  17
## 13  13  17
## 15  15  17
## 16  16  17
## 20  20  17
## 1    1  18
## 6    6  18
## 7    7  18
## 10  10  18
## 11  11  18
## 13  13  18
## 15  15  18
## 16  16  18
## 20  20  18
## 5    5  19
## 9    9  19
## 3    3  20
## 8    8  20
## 9    9  20
## 14  14  20
## 17  17  20
## 18  18  20

## [1] "Min. Value:"

## [1] 1

##    row col
## 4    4   2
## 5    5   2
## 2    2   4
## 5    5   4
## 21  21   4
## 2    2   5
## 4    4   5
## 21  21   5
## 4    4  21
## 5    5  21

## [1] "Max Value:"

## [1] 2.44949

## [1] "Mean Value:"

## [1] 1.482616

Advice SEM Summary Statistics

## [1] "Advice SEM"

##    row col
## 16  16   1
## 17  17   6
## 13  13   9
## 9    9  13
## 1    1  16
## 6    6  17

## [1] "Min. Value:"

## [1] 1.414214

##    row col
## 18  18  12
## 12  12  18

## [1] "Max. Value:"

## [1] 5

## [1] "Mean Value:"

## [1] 3.246036

IV. Differences and Correlation (30 pts)

Advice Networks

Now we will perform the QAP analysis on the advice networks by looping over every network in the list of networks and compare it against the median network we created in Part I. Let’s take a look at one of those values.

## 
## QAP Test Results
## 
## Estimated p-values:
##  p(f(perm) >= f(d)): 0 
##  p(f(perm) <= f(d)): 1 
## 
## Test Diagnostics:
##  Test Value (f(d)): 0.280373 
##  Replications: 1000 
##  Distribution Summary:
##      Min:     -0.1519819 
##      1stQ:    -0.03488578 
##      Med:     0.004146261 
##      Mean:    0.002020516 
##      3rdQ:    0.04017584 
##      Max:     0.2083139

The summary of the QAP test includes a number of values:

Estimated p-values: These estimate the probability of observing the test statistic (graph correlation in this instance) value. Qaptest will show both the probability of observing a value higher than or lower than the value observed. If the correlation is substantially higher than zero, these values will often be 1 and 0. This means that, during the QAP process no value was observed that was higher (or potentially lower) than the observed value. To confirm this, look at the Min/Max values in the distribution summary (see below).

Test value: This is the observed correlation between the two graphs.

Distribution summary: This summarizes the distribution of values calculated during the QAP process.

Look over the results that R printed to the console. Each result should begin with the respondent’s index number. Below, we’ll summarize the results of the correlation between the consensus network and each of our 21 respondents.

QAP Plot, Advice

Let’s plot the QAP distribution for our advice networks.

Advice QAP

Based on the results of the QAP test, is the most accurate observer’s correlation with the consensus network significant or spurious? How does the graph above help you make that determination? Where would you draw a vertical line?
ANSWER: I believe it’s significant, since the QAP replications are more tense around 0, and the maximum value is around 0.3. The vertical line representing respondent 16’s perception would not even be inside the present chart.

Friendship Networks

We will repeat this process for friendship networks. Take a look at your console output to answer the following question.

Considering Centrality

Next, we will investigate the correlation between various centrality measures and the union consensus network.

## Correlation with Degree Centrality: 0.214675

## Correlation with Betweennesss Centrality: 0.1479718

## Correlation with Closeness Centrality: 0.3286186

## Correlation with Eigenvector Centrality: 0.08114582

Identifying Most and Least Similar Respondent Viewpoints

Next, we’ll run the QAP test on the individual advice/friendship networks. Take a look at your console output to answer this question.

## Proportion of draws which were >= observed value: 0

## Proportion of draws which were <= observed value: 1

Submitting the Lab (5 pts)

After knitting your file to RPubs, copy the URL and paste it into the comment field of the Lab 2 Assignment on Canvas. Save this .Rmd file and submit it in the file portion of your Canvas assignment. Make sure to review your file and its formatting. Run spell check (built into RStudio) and proofread your answers before submitting.