In this homework, we will gain experience with CUG testing, QAP testing, and exponential random graph modeling using the familiar Lord of the Rings 3 network.
You will need the LOTR3 network, the invlogit function from the lab, and the mycugtest and myqaptest utilities. These have been packaged in hwk6.zip.
Part I: Attribute setup
Create a new attribute that simplifies the nine character Types in the network into three categories using the following table:
typeTable <- data.frame(Type=c("Dwarf", "Elf", "Ent", "Hobbit", "Human", "Orc", "Undead", "Unknown", "Wizard"),
Category=c("Fellow", "Fellow", "Other", "Fellow", "Fellow", "Evil", "Evil", "Other", "Fellow"))
Create a data frame of the Type attributes from the network. Then use the dplyr inner_join function to compute the correct category for each character. Create the attribute Category for each node using the result of the join. Important: You must use as.character when setting the attribute values because the data frame contains factors and igraph attributes cannot contain factors.
For you Tolkien fans out there, yes, I am aware that Saruman and Sauron are both wizards and not part of the fellowship, but they don't appear in this network, so we are OK.
Part II: CUG and QAP
For each CUG test, display the summary (print.cug.test) and the probability plot (plot.cug.test), state the null hypothesis, and indicate whether the hypothesis is accepted or rejected.
· assortativity of the network by the original Type attribute
· remember to convert the attributes to a factor when calling the function.
· assortativity of the network by the created Category attribute
For each QAP test, display the summary (print.qaptest) and the probability plot (plot.qaptest), state the null hypothesis, and indicate whether the hypothesis is accepted or rejected.
· assortativity of the network by Type
· assortativity of the network by Category
Part III: ERGM
For each model, display the model summary, the MCMC diagnostics (where applicable), and the network goodness-of-fit with plots. For the MCMC models (all except #1), save the model fit in an Rdata file and load it when running knitr, include the call to ergm as a non-executing code block as we did in lab.
Create four models of the LOTR3 network.
· Model 1: edges and Category assortativity
· note that there are three values of this attribute (yielding six possible parameters) but there are no examples of Evil-Other or Other-Other edges, so these must be included in the "base": base=c(4,6)
· note that the fit to the degree distribution is poor, especially for degree=1
· note also that the non-matching version of the nodemix parameters were not very strong, so matching may be enough
· Model 2: edges, Category match (nodematch) and degree(1)
· note that the esp distribution is a poor match
· Model 3: Model 2 + gwesp(cutoff=7)
· the cutoff parameter controls how many esp terms are calculated
· note that the degree term now has poor fit -- it seems to be redundant with the gwesp term
· note also that degree 4 is a particularly big mismatch
· Model 4: edges, Category nodemix but with only the matching values for Evil-Evil and Fellow-Fellow base=c(2,4,5,6), gwesp(cutoff=7), and degree(2:4)
· coefficients are significant, except for degree 2 and 3. We are keeping these because they improve the fit of the degree distribution
· note that there is inadequate mixing time in this computation (autocorrelation is high)
· Model 5: Model 4 increasing burnin to 100000 and increasing interval to 5000 and samplesize to 2048. (These were the final values from lab.)
Part IV: ERGM Interpretation
· Compare the probability of an Evil-Evil edge versus a Fellow-Fellow edge vs an edge not of these types.
· Fill in the following data frame:
·
Case
edges
nodemix-EE
nodemix-FF
Log-odds
Cond. probability
Evil-Evil edge
Fellow-Fellow edge
Other edge
· Question: What does the model fit show about assortativity by category in this network? How does this add to what we learned from CUG and QAP testing?
· Compute the probability of a second edge being added to a evil character to another evil character considering all the possible degree values within the model coefficients: six cases.
· Create a data frame with the six cases, their log-odds and probabilities
· Create a side-by-side barplot comparing the conditional probabilities across conditions.
· Question: What does the model fit show about degree variation