Ch 9
End-of-Chapter Problems

Q1 For each of the following centrality measures discuss the appropriateness of analyzing both directed and undirected binary networks.

a. Degree centrality

For undirected networks, degree centrality is simply the number of ties a node has. This is a useful property to know.

For directed networks we can measure outdegree (the number of ties from a node to others) and indegree (the number of ties from others to the node). Both are useful and easily interpreted. If the ties mean 'seeks advice from', then outdegree is the number of different nodes a node seeks advice from, and indegree is the number of people who seek out the node for advice. Often, indegree is more interesting.

b. Betweenness centrality

Betweenness applies to both directed and undirected networks. For undirected networks, betweenness is (loosely-speaking) the number of times a node is along the shortest path between two nodes, where we consider all pairs of nodes. For directed networks, the meaning is the same: we need only take care to respect the direction of ties. For example, if A-->B-->C, then B gets a point for being between A and C. But if A-->B<--C, B does not get a point, because there is no directed path from A to C.

c. Eigenvector centrality

In undirected networks, eigenvector centrality can be thought of as a measure of popularity, where popularity means being connected to many people who are themselves popular. For directed data, we can define left eigenvector centrality, in which a node has a high score if they receive many ties from people who themselves have high scores, and right eigenvector centrality, in which a node has a high score if they send ties to many people who are themselves high scorers. In practice, left and right eigenvectors are often uninterpretable or consist of complex numbers. For directed data, it is better to use beta centrality.

d. Closeness centrality

For undirected networks, closeness centrality (specifically, Freeman's version) is the sum of distances from a node to all others. A high score can be interpreted in terms of being dependent on intermediaries, or in terms of how long it takes for something flowing from a node to reach all others (and vice-versa). For directed networks we can interpret out-closeness in terms of how long things take to get from a node to all others, and in-closeness as how long it takes things to reach a node from all others. In practice, there are problems measuring closeness in directed networks because often there is no directed path from a given node to another. This then requires implementing repair strategies such as assigning an artificial large value to the distance to an unreachable node, or calculating reciprocal distances and assigning zeros to unreachable pairs.

e. Beta centrality

Out-beta centrality can be interpreted as the total number of walks from a node to all others, weighted inversely by their length. In-beta centrality can be interpreted as the total number of walks from all nodes to a given node, again weighted inversely by their length. In undirected networks, as beta approaches 1/lambda (where lambda is the largest eigenvalue of the adjacency matrix) from below, beta centrality approaches eigenvector centrality. Thus, beta centrality is nicely interpretable for both directed and undirected networks.

Q2 Consider the Hawthorne bank wiring room games network, which is a binary, undirected graph of who plays games with whom among a set of workers. Calculate each of the following centrality measures.

a. Degree centrality

b. Closeness centrality

c. Eigenvector centrality

d. Beta centrality, with b parameters = 0, 0.1 and 0.18.

e. Betweenness centrality

#get the data

games = Hawthorne_BankWiring$Games

#calc measures

degree = xDegreeCentrality(games)[,1]

closeness = xClosenessCentrality(games)[,1]

eigenvec = xEigenvectorCentrality(games)[,1]

beta = xBetaCentrality(games,Beta=c(0,0.1,0.18))

between = xBetweennessCentrality(games)[,1]

#display side-by-side

meas = cbind(degree,closeness,eigenvec,beta,between)

meas

degree closeness eigenvec Beta.0 Beta.0.1 Beta.0.18 between

I1 4 37 0.30665931 4 8.879110 87.42515 0.0000000

I3 0 65 0.00000000 0 0.000000 0.00000 0.0000000

W1 6 30 0.41691182 6 12.669063 119.70150 7.5000000

W2 5 36 0.36535214 5 10.783908 104.36858 0.5000000

W3 6 30 0.41691182 6 12.669063 119.70150 7.5000000

W4 6 30 0.41691182 6 12.669063 119.70150 7.5000000

W5 5 27 0.32327806 5 10.736704 94.96027 60.0000000

W6 3 37 0.02878007 3 5.145091 16.46479 0.0000000

W7 5 29 0.08492990 5 8.407072 35.15675 56.6666667

W8 4 36 0.03337021 4 6.521917 19.82383 0.6666667

W9 4 36 0.03337021 4 6.521917 19.82383 0.6666667

S1 5 31 0.36800266 5 10.952780 105.51800 3.0000000

S2 0 65 0.00000000 0 0.000000 0.00000 0.0000000

S4 3 37 0.02878007 3 5.145091 16.46479 0.0000000

cor(meas)

degree closeness eigenvec Beta.0 Beta.0.1 Beta.0.18 between

degree 1.0000000 -0.9408099 0.7726720 1.0000000 0.9753798 0.8167369 0.3147072

closeness -0.9408099 1.0000000 -0.5894508 -0.9408099 -0.8781057 -0.6454806 -0.4044009

eigenvec 0.7726720 -0.5894508 1.0000000 0.7726720 0.8922781 0.9973232 0.1207582

Beta.0 1.0000000 -0.9408099 0.7726720 1.0000000 0.9753798 0.8167369 0.3147072

Beta.0.1 0.9753798 -0.8781057 0.8922781 0.9753798 1.0000000 0.9228719 0.2700887

Beta.0.18 0.8167369 -0.6454806 0.9973232 0.8167369 0.9228719 1.0000000 0.1473599

between 0.3147072 -0.4044009 0.1207582 0.3147072 0.2700887 0.1473599 1.0000000

Notes:

• beta.0 = degree

• beta.0.18 = eigenvector centrality

• betweenness centrality is the measure least similar to all the others.

Q3 Create four visualizations of the Hawthorne bank wiring room games network. In each of the visualizations make the size of the nodes proportional to the value of each of the four centrality measures (degree, closeness, eigenvector, betweenness). Compare and contrast the differences and similarities of the measures across the four visualizations. From a social and behavioral perspective, how might you interpret these comparisons?

#a function for rescaling attributes when sizing nodes

rescale<-function(vec, dmin=0, dmax=1)

{

amax <- max(vec,na.rm=T)

amin <- min(vec,na.rm=T)

rng <- amax - amin

normed <- dmin + (dmax-dmin)*(vec-amin)/rng

return(normed)

}

#a function to package up the drawing commands

draw<-function(amat, sizeby=NULL, gm="graph", crd=NULL)

{

sizeby<-rescale(sizeby,.5,3)

par(mar=c(0,0,0,0))

crd <- sna::gplot(amat, jitter = F, mode="kamadakawai", displaylabels = T,

label.cex = .9, gmode=gm,

vertex.cex=sizeby, coord=crd)

return(crd)

}

#draw the plots in a 4-by-4 square

par(mfrow=c(2,2))

#draw the networks!

draw(games,degree)

draw(games,-closeness)

draw(games,eigenvec)

draw(games,between)

Q4 Using the advice network in the Krackhardt high-tech managers dataset, provide an analysis of node centrality for each of the measures below. The network is directed and dichotomous.

1. Indegree and outdegree centrality. Create visualizations of the advice network, with one having node size proportional to indegree centrality and one with node size proportional to outdegree centrality. Compare the results.

2. Incoming and outgoing k-reach centrality, with k = 1, 2, 3

3. Beta reach centrality for outgoing ties, with b parameters = 0, 0.2, 0.4 and 0.6. Interpret the results.

4. Beta centrality – both ‘in’ and ‘out’

#get data

advice = Krackhardt_HighTech$Advice

#get degree measures

outdegree = rowSums(advice)

indegree = colSums(advice)

#format plots

par(mfrow=c(2,1))

#draw sized by indegree, and save coordinates

crd = draw(advice,indegree,"digraph")

#draw sized by outdegree, using saved coordinates

draw(advice,outdegree,"diagraph",crd)

#calc outgoing and incoming k-reach centrality

outreach = xReachCentrality(advice,kReach = c(1,2,3))

inreach = xReachCentrality(t(advice),kReach = c(1,2,3))

#First three are outgoing, last three are incoming

cbind(outreach,inreach)

kReach.1 kReach.2 kReach.3 kReach.1 kReach.2 kReach.3

A01 6 20 20 13 17 20

A02 3 12 20 18 20 20

A03 15 20 20 5 19 20

A04 12 20 20 8 20 20

A05 15 20 20 5 15 20

A06 1 11 20 10 20 20

A07 8 20 20 13 20 20

A08 8 20 20 10 20 20

A09 13 20 20 4 15 20

A10 14 20 20 9 16 20

A11 3 12 20 11 19 20

A12 2 12 20 7 19 20

A13 6 20 20 4 15 20

A14 4 20 20 10 20 20

A15 20 20 20 4 15 20

A16 4 19 20 8 17 20

A17 5 15 20 9 20 20

A18 17 20 20 15 20 20

A19 11 20 20 4 15 20

A20 12 20 20 8 19 20

A21 11 20 20 15 20 20

Every node can reach every other in three steps, and every node can be reached by every other in three steps. A15 can reach everyone in 1 step, which means A15 seeks advice from everyone.

xBetaReachCentrality(advice,Beta=c(0,.2,.4,.6,1))

Beta.0 Beta.0.2 Beta.0.4 Beta.0.6

A01 6 8.80 11.60 14.40

A02 3 5.12 7.88 11.28

A03 15 16.00 17.00 18.00

A04 12 13.60 15.20 16.80

A05 15 16.00 17.00 18.00

A06 1 3.36 6.44 10.24

A07 8 10.40 12.80 15.20

A08 8 10.40 12.80 15.20

A09 13 14.40 15.80 17.20

A10 14 15.20 16.40 17.60

A11 3 5.12 7.88 11.28

A12 2 4.32 7.28 10.88

A13 6 8.80 11.60 14.40

A14 4 7.20 10.40 13.60

A15 20 20.00 20.00 20.00

A16 4 7.04 10.16 13.36

A17 5 7.20 9.80 12.80

A18 17 17.60 18.20 18.80

A19 11 12.80 14.60 16.40

A20 12 13.60 15.20 16.80

A21 11 12.80 14.60 16.40

When beta = 0, beta reach centrality gives you (out)degree. Nodes reached with paths longer than 1 link are not counted at all. When beta = 0.2, the measure counts all of the nodes a given node can reach, but only gives full weight to those it is immediately connected to. The others are severely discounted. As beta gets larger, the measure is more liberal and gives even distant nodes some weight.

Q5 Sampson (1969) collected the top three choices of liking and disliking among a set of monks in a monastery. The liking network was collected (retrospectively) for three points in time. We will use T3

• Ignoring the values of choices for the liking network (i.e., dichotomizing at greater than zero), provide the k reach centrality for outgoing ties, with k = 1, 2, 3 and 4. Interpret the results.

• Ignoring the values of choices for the disliking network, provide the k reach centrality for outgoing ties, with k = 1, 2, 3 and 4. Interpret the results and compare to the results for the liking network.

• Finally, ignoring the values for both networks, apply the PN centrality measure and interpret the results.

#get the data

like = xDichotomize(Sampson_Monastery$LikeT3)

dislike = xDichotomize(Sampson_Monastery$Dislike)

#calc outgoing k-reach centrality for each network

krlike = xReachCentrality(like,kReach=c(1,2,3,4))

krdislike = xReachCentrality(dislike,kReach=c(1,2,3,4))

meas = cbind(krlike,krdislike)

meas

kReach.1 kReach.2 kReach.3 kReach.4 kReach.1 kReach.2 kReach.3 kReach.4

ROMUALD 4 10 16 17 0 0 0 0

BONAVENTURE 3 7 11 13 0 0 0 0

AMBROSE 3 9 12 16 3 9 13 14

BERTHOLD 3 6 10 13 4 8 11 12

PETER 3 6 9 12 3 8 11 12

LOUIS 3 9 12 16 3 10 13 13

VICTOR 3 6 10 13 3 8 11 12

WINFRID 3 6 10 11 0 0 0 0

JOHN_BOSCO 3 9 11 14 3 7 11 12

GREGORY 3 6 10 11 3 8 12 12

HUGH 3 6 10 11 3 7 11 12

BONIFACE 3 5 7 10 3 9 13 13

MARK 3 5 7 10 3 8 11 12

ALBERT 3 5 7 10 3 8 11 12

AMAND 3 11 16 16 3 8 11 12

BASIL 4 9 14 16 4 10 12 12

ELIAS 3 7 10 14 3 8 11 12

SIMPLICIUS 3 7 10 14 3 7 12 13

Somewhat improbably, the monks tend to have more 'enemies of enemies' than 'friends of friends'. It is surprising that someone like Louis can reach 10 people in 2 steps or less, given that he only has 3 direct dislikes. The remaining 7 are people who are disliked by the people he dislikes.

#calc pn centrality, combining positive and negative ties

xPNCentrality(like,dislike)

PNcentrality

ROMUALD 1.1149650

BONAVENTURE 1.0838575

AMBROSE 0.9360470

BERTHOLD 0.8861251

PETER 0.9287938

LOUIS 0.9350133

VICTOR 0.9307870

WINFRID 1.0833276

JOHN_BOSCO 0.9227497

GREGORY 0.9516707

HUGH 0.9589060

BONIFACE 0.9467080

MARK 0.9502032

ALBERT 0.9517389

AMAND 0.9340754

BASIL 0.9196967

ELIAS 0.9412390

SIMPLICIUS 0.9322612

The results show that Romuald, Bonaventure and Winfrid have, on balance, more positive ties than negative, and that their positive ties tend not to be to those with lots of negative ties, nor are their negative ties to monks with lots of positive ties.