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 and Highlight All Clique Triads in VISONE

snaCliques

Description

This post describes how to identify group structures among a network of respondents in VISONE. For a network of selections we identify any cliques involving three or more members. A clique is defined as a group containing three or more members where everyone has chosen everyone else.

Identify all triads

A tried is a network structure containing exactly three members. There are many types of triads. A group of three members where everyone chooses everyone else is a triad (i.e., a clique). A group of three members where two people choose each other and nobody choose the third member is another type of triad. There are 16 unique ways three people can select each other.

Identify all triads. Click the ‘analysis’ tab. Next to ‘task’, select ‘grouping’ from the drop down list of available options. Select ‘cohesiveness’ from the drop down list next to ‘class’. Select the option ‘triad census’ next to ‘measure’. Click ‘analyze’.

Highlight all cliques

Highlight all cliques. Click an empty part of the graph. Press the keys ‘Ctrl’ and ‘a’. Open the attribute manager. Click the ‘link’ button. Click the ‘filter’ button. Select ‘default value’ from the first drop down list. Select ‘triadType300’ from the second drop down list. Select ‘has individual value’ from the third drop down list. Click the radial button ‘replace’. Click ‘select’. Click ‘close’.

From the main VISONE drop down bar, select ‘links’. Click ‘properties’. Click the given color next to ‘color:’. Select ‘rgb’ tab. Set the ‘red’, ‘green’, and ‘blue’ values to 0. Set the ‘alpha’ value to 255. Set ‘opacity’ to 50%. Click the ‘close’ button. Set the ‘width:’ value to 5.0. From the ‘edge properties’ dialogue box, click the ‘apply’ button. Click ‘close.

Reduce visibility of all non-clique selections. Select all nodes and links. Click an empty part of the graph. Press the keys ‘Ctrl’ and ‘a’. Open the attribute manager. Click the ‘link’ button. Click the ‘filter’ button. Select ‘default value’ from the first drop down list. Select ‘triadType300’ from the second drop down list. Select ‘has individual value’ from the third drop down list. Click the radial button ‘remove’. Click ‘select’. Click ‘close’.

From the main VISONE drop down bar, select ‘links’. Click ‘properties’. Click the given color next to ‘color:’. Set ‘opacity’ to 20%. Set the ‘width:’ value to 2.0. From the ‘edge properties’ dialogue box, click the ‘apply’ button. Click ‘close.

How to plot a network subgraph on a network graph using R

plotSubgraphOnGraph
Here is an example of how to highlight the members of a subgraph on a plot of a network graph.

## Load R libraries
library(igraph)

# Set adjacency matrix
g <- matrix(c(0,1,1,1, 1,0,1,0, 1,1,0,1, 0,0,1,0),nrow=4,ncol=4,byrow=TRUE)

# Set adjacency matrix to graph object
g <- graph.adjacency(g,mode="directed")

# Add node attribute label and name values
V(g)$name <- c("n1","n2","n3","n4")

# Set subgraph members
c <- c("n1","n2","n3")

# Add edge attribute id values
E(g)$id <- seq(ecount(g))

# Extract supgraph
ccsg <- induced.subgraph(graph=g,vids=c)

# Extract edge attribute id values of subgraph
ccsgId <- E(ccsg)$id

# Set graph and subgraph edge and node colors and sizes
E(g)$color="grey"
E(g)$width=2
E(g)$arrow.size=1
E(g)$arrow.width=1
E(g)[ccsgId]$color <- "#DC143C" # Crimson
E(g)[ccsgId]$width <- 2
V(g)$size <- 4
V(g)$color="#00FFFF" # Cyan
V(g)$label.color="#00FFFF" # Cyan
V(g)$label.cex <-1.5
V(g)[c]$label.color <- "#DC143C" # Crimson
V(g)[c]$color <- "#DC143C" # Crimson

# Set seed value
set.seed(40041)

# Set layout options
l <- layout.fruchterman.reingold(g)

# Plot graph and subgraph
plot.igraph(x=g,layout=l)

Simple, no?