11 Problems

Q1 Consider the interaction network at week 3 among the 18 participants in the Borgatti Camp92 dataset. The data are rank-order values, with 17 indicating the least interaction and 1 indicating the most interaction. Dichotomize the network so that there is a tie if the alter is among the ego’s top 3 interactants. Now symmetrize the network using the or/union approach (i.e., symmetrizing by maximum) and visualize the resulting network. Based on the visualization, how might you characterize the subgroups among actors in the network? How many communities are there in this network? Are there nodes who are members of more than one community?

campnet = xDichotomize(Borgatti_Camp92$MostInteractionsT3,3,Type="LTE")

campnet = xSymmetrize(campnet,"Max")

sna::gplot(campnet, displaylabels = T, usearrows=F)

Q2 Run a clique analysis on the data from Problem 1 and a hierarchical cluster analysis on the clique co-membership matrix. How many subgroups would you say there are in this network? Interpret the result.

c = xCliquesMembership(campnet)

c

CL_1 CL_2 CL_3 CL_4 CL_5 CL_6 CL_7 CL_8 CL_9 CL_10

HOLLY 0 0 0 0 0 0 0 0 1 0

BRAZEY 0 0 0 1 0 0 0 0 0 0

CAROL 0 0 0 0 1 1 0 0 0 0

PAM 0 0 0 0 1 0 1 1 0 0

PAT 0 0 0 0 0 1 0 0 0 0

JENNIE 0 0 0 0 0 0 1 0 0 0

PAULINE 0 0 0 0 1 1 0 1 0 0

ANN 0 0 0 0 0 0 1 1 0 0

MICHAEL 0 0 0 0 0 0 0 0 1 1

BILL 0 0 0 0 0 0 0 0 0 1

LEE 0 0 0 1 0 0 0 0 0 0

DON 0 0 0 0 0 0 0 0 1 1

JOHN 0 1 0 0 0 0 0 0 0 0

HARRY 0 0 0 0 0 0 0 0 1 1

GERY 0 1 1 0 0 0 0 0 0 0

STEVE 1 0 1 1 0 0 0 0 0 0

BERT 1 0 0 1 0 0 0 0 0 0

RUSS 1 1 1 0 0 0 0 0 0 0

Ten cliques were found. The person-by-clique membership matrix forms a 2-mode datasets. We can draw a bipartite representation of clique membership as follows:

sna::gplot(c, displaylabels = T,usearrows=F,label.cex=.6)

Next we can compute the co-membership matrix and cluster it.

comemb = xCliquesCoMembership(campnet)

xHierarchicalClustering(comemb, Method="average", Input="Similarities")

Q3 Next, submit the data to some of the modularity optimization techniques, namely, fast greedy, walktrap, Girvan–Newman and Louvain. Compare the modularity scores and the partitions. How do these compare with the results of your clique analysis (in Problem 2)?

#get partitions from 4 methods

fg = xFastGreedy(campnet)

wt = xWalkTrap(campnet)

gn = xGirvanNewman(campnet)

lo = xLouvainMethod(campnet)

#display partitions side by side

cbind(fg,wt,gn,lo)

CL_4 CL_3 CL_4 CL_4

HOLLY 2 2 2 3

BRAZEY 4 1 3 2

CAROL 3 3 1 1

PAM 3 3 1 1

PAT 3 3 1 1

JENNIE 3 3 1 1

PAULINE 3 3 1 1

ANN 3 3 1 1

MICHAEL 2 2 2 3

BILL 2 2 2 3

LEE 4 1 3 2

DON 2 2 2 3

JOHN 1 1 4 4

HARRY 2 2 2 3

GERY 1 1 4 4

STEVE 4 1 3 2

BERT 4 1 3 2

RUSS 1 1 4 4

We can see that all of the methods except walktrap find four groups. Walktrap gives three groups, which fits our intuition based on the visualization. The three methods that give four groups all have the same solution, although they are numbered differently. We can tell that by cross-tabulating the partition vectors.

table(fg,wt)

wt

fg 1 2 3

1 3 0 0

2 0 5 0

3 0 0 6

4 4 0 0

We can see that the walktrap solution combines groups 1 and 4 from the fastgreedy partition.

> table(fg,gn)

gn

fg 1 2 3 4

1 0 0 0 3

2 0 5 0 0

3 6 0 0 0

4 0 0 4 0

The 1 to 1 correspondence between the fastgreedy and girvan-newman partitions (taking into account the different numbering) shows they have arrived at the same solution.

> table(fg,lo)

lo

fg 1 2 3 4

1 0 0 0 3

2 0 0 5 0

3 6 0 0 0

4 0 4 0 0

We can see fastgreedy and louvain also get the same answers. Here is the plot generated by the xLouvain function:

Q5 Consider the collaboration network of 960 scientists in a research institute (called Borgatti_Scientists960). Using the valued, symmetric network, find the communities in this network using the modularity optimization techniques discussed in this chapter.