Identifying cliques and clique comembers in R using the SNA package

The R package SNA provides a number of tools for analyzing social network data. This post reviews the function clique.census from the SNA package, and shows how it can be used to better understand the group structure among a list of network members.

Let’s start by creating a toy network. Say Henry, Sarah, Rick, Joe and Annie are all colleagues in a criminology department. Now imagine they’ve been asked to identify the most important people they collaborate with. The results of this fictitious effort are shown in the following edgelist.

edgelist <- matrix(c("Henry", "Sarah", "Henry", "Joe", "Sarah", "Joe", "Henry", "Sarah", "Henry", "Rick", "Sarah", "Rick", "Sarah", "Henry", "Sarah", "Rick", "Henry", "Rick", "Sarah", "Henry", "Sarah", "Joe", "Henry", "Joe", "Sarah", "Rick", "Sarah", "Annie", "Rick", "Annie"), ncol = 2)

Recall, a clique is a group where everyone in it “likes” everyone else. To identify cliques among our network of criminology researchers, we first transform it into a network object and then apply the SNA function clique.census.

library("network")

net tabulate.by.vertex=FALSE, enumerate=TRUE, clique.comembership="bysize") # Identify cliques
net_cc$clique.count

 

sna clique comembership 1

After applying the function clique.census, we see there were two cliques among our respondents, each involving three researchers.

To identify the comembers of these cliques, we inspect the contents of the variable net_cc$clique.comemb[3, , ].

To paraphrase the SNA documentation, the variable net_cc$clique.comemb is a three dimensional matrix of size max clique size x n x n. In this example we observed cliques involving only three network members each. As such, the 3-clique comembership information is stored in the variable net_cc$clique.comemb[3, , ]. (Note: if we observed one or more cliques with more than three members, say, a 4-clique, we could examine their comembership using the variable net_cc$clique.comemb[4, , ]).

Notably, the format by which net_cc$clique.comemb[3, , ] organizes clique comembership takes some getting used to. In fact, the main point of this post is to explain this organizational scheme in a more everyday kind of way.

Given our criminologist cliques, here’s how we find out who was in them. Recall, the largest clique we observed contained three individuals. Further recall that our network only contained five respondents. As such, the matrix net_cc$clique.comemb[3, , ] is of size 5 x 5. This matrix mimics the network structure itself. That is, network members are listed along the rows and, in the exact same order, listed again across the columns. The values within the matrix then identify researchers who were in cliques together.

sna clique comembership 2

Let’s go through a couple columns together to better understand what this matrix is telling us exactly.

The values in the column Joe show who Joe was in a clique with. Zero values indicate who Joe was not in a clique with, while values greater than zero indicate who he was in a clique with. More than just showing who Joe was in a clique with, these values identify the different cliques he was a part of. Reading the column from top to bottom, Joe and Annie were in zero cliques together. Next we see that Joe was in a clique with Henry. Notably, Joe was in one clique with himself (i.e., there was only one clique that involved Joe). The rest of the column values show that Joe was in zero cliques with Rick and one clique with Sarah. In total, this column tell us there was one 3-clique that involved Joe (see Joe’s value for Joe) and that this clique involved the researchers Henry and Sarah.

Let’s look at a trickier column. The values in the column for Henry ranged from 0 to 2. As with Joe, the values show who Henry was in a clique with and how many times they were in the same clique. Going down the column, we see a zero value for Henry and Annie. That is, Henry and Annie were not part of the same clique. Notably, Henry is in two cliques with himself. That is, there were two cliques, each of which involved Henry. The rest of the values show that Henry was in one clique with Joe, another clique with Rick and two cliques with Sarah.

Combined, these column values tell us there were two 3-cliques: one clique involving Henry, Sarah and Joe and another involving Henry, Sarah and Rick.

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Plot Network Data in R with iGraph

I recently had a conversation on Twitter about a plot I made a while back. Recall, the plot showed my Twitter network, my friends and my friend’s friends.

Here’s the Twitter thread:

And here’s the R code:

#### Load R libraries
library("iGraph")

#### Load edgelist
r <- read.csv(file="edgelist_friends.csv-03-25.csv",header=TRUE,stringsAsFactors=FALSE)[,-1]

#### Convert to graph object
gr <- graph.data.frame(r,directed=TRUE)

#### gr
# Describe graph
summary(gr)
ecount(gr) # Edge count
vcount(gr) # Node count
diameter(gr) # Network diameter
farthest.nodes(gr) # Nodes furthest apart
V(gr)$indegree = degree(gr,mode="in") # Calculate indegree

#### Plot graph
E(gr)$color = "gray"
E(gr)$width = .5
E(gr)$arrow.width = .25
V(gr)$label.color = "black"
V(gr)$color = "dodgerblue"
V(gr)$size = 4

set.seed(40134541)
l <- layout.fruchterman.reingold(gr)

pdf("network_friends_plot.pdf")
plot(gr,layout=l,rescale=TRUE,axes=FALSE,ylim=c(-1,1),asp=0,vertex.label=NA)
dev.off()

Convert Qualitative Codes into a Binary Response Matrix in R

Content analysis is a qualitative method for identifying themes among a collection of documents. The themes themselves are either derived from the content reviewed or specified a priori according to some established theoretical perspective or set or research questions. Documents are read, content is considered and themes (represented as letter “codes”) are applied. It’s not uncommon for documents to exhibit multiple themes. In this way, results from a content analysis are not unlike responses to the “select all that apply” type questions found in survey research. Once a set of documents is coded, it’s often of interest to know the proportion of times the codes were observed.

The following R code transforms codes on a set of documents, stored as a list of lists, into a binary matrix.

#### Load R packages
library("XLConnect")
library("stringr")
library("vcd")


#### Working directory
getwd()
setwd("C:/Users/chernoff/Desktop")


#### Read data
theData <- readWorksheet(loadWorkbook("dummyData.xlsx"),sheet="Sheet1",header=TRUE)


#### Parse codes
theData2 <- str_extract_all(theData$codes,"[a-zA-Z]")

codeList <- unique(unlist(theData2))

theData2 <- lapply(X=theData2,function(X) as.data.frame(matrix(as.numeric(codeList%in%X),ncol=length(codeList),dimnames=list(c(),codeList))))
theData3 <- do.call(rbind,theData2)

theData4 <- cbind(theData,theData3)

If we print the data to the screen, we see themes are represented as binary variables, where 1 indicates a theme was observed and 0 indicates it was not.

binaryMatrix

Once the data are organized as a binary matrix, we can calculate column totals colSums(theData4[,codeList]) to see which themes were more popular and which ones were least popular.

binaryMatrixColSums

And lastly, if we want to get fancy, we can represent the data using a mosaic plot.

totals <- table(theData4$a,theData4$b,theData4$c,dnn=c("A","B","C"))

png("mosaicPlot.png")
mosaicplot(totals,sort=3:1,col=hcl(c(120,10)),"Mosaic Plot")
dev.off()

mosaicPlot

A mosaic plot shows the relative proportion of each theme compared to one or more of the other themes. The main two rows show the levels of theme B. The main two columns represent theme C’s levels. And the two columns within the two main columns represent the levels of A. By default, the label for theme A is not shown. The cell in the upper left-hand corner, i.e. cell (1,1), shows there were some but not many documents without any themes. Cells (1,3) and (1,4) show there were an equal number of documents with theme C as there were that involved themes A and C combined. The remaining cell in this first row (1,2) shows there were more documents pertaining solely to theme A than all other document types not containing theme B. Interpretations of the remaining rectangles follow similarly.

How to extract a network subgraph using R

In a previous post I wrote about highlighting a subgraph of a larger network graph. In response to this post, I was asked how extract a subgraph from a larger graph while retaining all essential characteristics among the extracted nodes.

Vinay wrote:

Dear Will,
The code is well written and only highlights the members of a subgraph. I need to fetch them out from the main graph as a separate subgraph (including nodes and edges). Any suggestions please.

Thanks.

Extract subgraph
For a given list of subgraph members, we can extract their essential characteristics (i.e., tie structure and attributes) from a larger graph using the iGraph function induced.subgraph(). For instance,

library(igraph)                   # Load R packages

set.seed(654654)                  # Set seed value, for reproducibility
g <- graph.ring(10)               # Generate random graph object
E(g)$label <- runif(10,0,1)       # Add an edge attribute

# Plot graph
png('graph.png')
par(mar=c(0,0,0,0))
plot.igraph(g)
dev.off()

g2 <- induced.subgraph(g, 1:7)    # Extract subgraph

# Plot subgraph
png('subgraph.png')
par(mar=c(0,0,0,0))
plot.igraph(g2)
dev.off()

Graph
graph

Subgraph
subgraph