# Problems 10

Note that solutions for some problems rely on the solutions from previous solutions.

Q2 Calculate the density for both the marriage and business networks in the Padgett Florentine families dataset.

# get data

# calc measures

dMarriage = xDensity(mar)

# display side by side

[1,] 0.1666667 0.125

Q3 Calculate the closeness centralization for the Florentine families using the marriage network.

xMakeStar <- function(n)

{

mat = matrix(0,n,n)

mat[,1] = 1

mat[1,] = 1

mat[1,1] = 0

return(mat)

}

xClosenessCentralization <- function(mat)

{

getsumdiff = function(fmat)

{

clo = xClosenessCentrality(fmat)[,4]

maxclo = max(clo)

return(sum(maxclo-clo))

}

n = nrow(mat)

star = xMakeStar(n)

sumdiffstar = getsumdiff(star)

sumdiffnet = getsumdiff(mat)

return(sumdiffnet/sumdiffstar)

}

xClosenessCentralization(mar)

 0.4473016

The solution given uses the reciprocal closeness measure. Note that the network is not connected, which is not ideal for closeness centrality.

Q5 Calculate the connectedness and compactness for the marriage network and compare both measures with those obtained for the business network for the Padgett dataset.

results = matrix(0,2,2)

colnames(results) = c("Connectedness","Compactness")

results[1,1] = xConnectedness(mar)

results[1,2] = xCompactness(mar)

results[2,1] = xConnectedness(bus)

results[2,2] = xCompactness(bus)

results

Connectedness Compactness

Marriage 0.8750000 0.4376389

The marriage network is more connected and has shorter distances than the business network. This is largely because the business network has 5 isolates.

Q6 Freeman’s EIES friendship network was measured at two time points. Dichotomize the network at each time point, such that x(i,j) if i considers j at least a “friend”, and x(i,j) = 0 otherwise. Calculate the density, connectedness and compactness for the friendship network at both time points. Compare the results for each of the three measures.

#dichotomize the networks

t1 = xDichotomize(Freeman_EIES\$AcquaintanceT1,1,"GT")

t2 = xDichotomize(Freeman_EIES\$AcquaintanceT2,1,"GT")

#calc measures

t1meas = cbind(xDensity(t1),xConnectedness(t1),xCompactness(t1))

t2meas = cbind(xDensity(t2),xConnectedness(t2),xCompactness(t2))

#display results

results = rbind(t1meas,t2meas)

colnames(results) = c("Density","Connectedness","Compactness")

results

Density Connectedness Compactness

[1,] 0.5171371 1 0.7580645

[2,] 0.6582661 1 0.8291331

>

The network gets denser over time, and distances become shorter.

Q8 Use Freeman’s EIES friendship network at times 1 and 2 (where the network is 1 when the person considers the other a friend or a good friend) to calculate arc reciprocity. What do you conclude?

cbind(xReciprocity(t1),xReciprocity(t2))

Value Value

Mutual 220.0000000 281.0000000

Asymmetric 73.0000000 91.0000000

Null 203.0000000 124.0000000

ArcReciprocity 0.8576998 0.8606432

>

c(.8577/.5171,.8606/.6583)

 1.658673 1.307307

Reciprocity is high at both time periods -- above chance levels, as indicated by density. Because the arc reciprocity measure is about the same at the two time periods, but the network is denser at time 2, we conclude there is a slightly lower tendency to reciprocate at T2.

Q9 The Freeman EIES dataset contains information about the discipline of each node. Calculate a density table for time 2.

disc = Freeman_EIES\$Attributes\$Discipline

xd = xDensity(t2,ROWS=disc,COLS=disc)

rownames(xd) = c("Sociology","Anthropology","Mathematics","Other")

colnames(xd) = rownames(xd)

xd

Sociology Anthropology Mathematics Other

Sociology 0.7683824 0.5686275 0.6666667 0.5098039

Anthropology 0.6078431 0.9000000 0.9444444 0.6111111

Mathematics 0.7450980 0.6111111 0.6666667 0.7222222

Other 0.5784314 0.6111111 0.7222222 0.4000000

>

In general, we expect there to be more ties within group than between. Interestingly, the anthropologists have a strong tendency to regard mathematicians as friends -- stronger, in fact, than the reverse.