I recently discovered Gary Weissman’s excellent post on Grey’s Anatomy Network of Sexual Relations and I felt inspired.  For those who haven’t heard of the television show before, Grey’s Anatomy is a widely popular, award-winning prime-time medical drama airing on ABC which has received no shortage of critical acclaim.  Meeting conventional medical drama expectations, the show quite regularly features members of its attractive cast “hooking up.”  Or so I am told.  In an effort to teach medical students some basic social network lessons, Weissman produced a network data set on the show’s sexual contacts between characters.  Though I’m not particularly fond of the show and both sexual and fictional networks lie outside my research interests, Weissman’s post served as a remarkable demonstration of network analysis for pedagogical purposes.

For funzies, I’d like to revisit this data on the subject of tie formation through exponential random graph modeling (ERGM).  Before I proceed, I’d first like to say that the best tutorials currently available on ERGM come directly from its creators, the statnet team.  Due to its applied nature, my favorite of their tutorials is their February 2011 Sunbelt Workshop (.pdf).  If you’re serious about learning ERGM, their Online User Guides should be your primary reference.  However, I suspect there’s an audience who are more captivated by the affairs of sexy TV doctors than, say, 16th century Florentine marriage networks.

Exponential random graph models present a way to understand the processes of network structure emergence and tie formation.  For this example, “tie formation” and “hooking up” are synonymous.  These models work by measuring a limited set of known statistics from a network and then use the distributions such statistics to generate random networks.  These random networks are then compared to the observed networks to assess the likelihood of fit.  Good ERG models produce networks isomorphically similar to the observed network using few parameters (i.e., the networks look structurally similar).  Bad ERG models produce networks that bear little semblance to the observed network using many unnecessary parameters.

Which statistics should be used?  The number of edges (i.e., “ties”, “relationships”) is the single most important statistic in ERGM.  If our question is on tie formation, it’s crucial to know the number of ties.  Also, in Weissman’s post, some of the commentators mentioned the importance of a character’s sex or perhaps their position in the hospital.  Nodes (i.e., “actors”, “vertices “) can sometimes be characterized by attributes, like an individual’s sex, age,  position in the hospital, or astrological sign.  When modeling a network, one can use nodal attributes to account for assortativity effects like general homophily (“Do members of the hospitals staff bed with others in their same position in the hospital?”), differential homophily (“Do residents bed with other residents?  Do they do so at a greater proclivity than interns among other interns?”), and assortative mixing (“Do resident-attending couples occur more frequently than we’d expect vis-à-vis nurse-attending couples?”).  Other network properties can be analyzed, however users are limited by data availability, the type of network analyzed (directed, undirected, or two-mode), and the ability of the model to converge.

The data used here begins with what Weissman made available.  I added additional nodes and edges using information cursory gleamed off of Wikipedia entries.  I only included characters who sleep with one or more characters on the show.  I also added attributes that include the character’s sex, race/ethnicity, position in the hospital, approximate age, season introduced, and the actor’s astrological sign.  I collected these attributes off of Wikipedia and IMDB.  For sake of simplicity, I coded each character’s race as either black, white, or other.  While I don’t like conceptualizing race on these terms, the number of characters and their diversity is too limited to provide meaningful nuance in this respect.   For the character’s approximate age, I used either the age of their actors or, in two cases, I made a guess.  As with Weissman, the data here is likely missing a few relationships, nodes, and simplifies the description of the characters and their relationships.

To begin, open your R terminal.  Then install and load the packages “ergm” and “RCurl” if you haven’t already done so.

install.packages("RCurl"); install.packages("ergm")
library(RCurl); library(ergm)

For the repository, just a location that’s reasonably near you.  The package “RCurl” will be used to pull the data from my Google Docs file.  The package “ergm” will include a few other packages including “network,” as ERGM must use objects produced by the network package.

Next, read in the data and produce a network object.

#First, read in the sociomatrix
ga.mat<-getURL("https://docs.google.com/spreadsheet/pub?key=0Ai--oOZQWBHSdDE3Ynp2cThMamg1b0VhbEs0al9zV0E&single=true&gid=0&output=txt",
                        ssl.verifypeer = FALSE)
ga.mat<-as.matrix(read.table(textConnection(ga.mat), sep="\t",
                  header=T, row.names=1, quote="\""))
#Second, read in the network attributes
ga.atts<-getURL("https://docs.google.com/spreadsheet/pub?key=0Ai--oOZQWBHSdDE3Ynp2cThMamg1b0VhbEs0al9zV0E&single=true&gid=1&output=txt",
                 ssl.verifypeer = FALSE)
ga.atts<-read.table(textConnection(ga.atts), sep="\t", header=T, quote="\"",
                    stringsAsFactors=F, strip.white=T, as.is=T)
#Third, create a network object using the sociomatrix and its corresponding attributes
ga.net<-network(ga.mat, vertex.attr=ga.atts, vertex.attrnames=colnames(ga.atts),
                directed=F, hyper=F, loops=F, multiple=F, bipartite=F)

Now that we’ve got the data, we’re ready to rock and roll.  Before going into the modeling, let’s take a look-see at a plot of the network, coloring the nodes pink for the women and blue for the men.

plot(ga.net, vertex.col=c("blue","pink")[1+(get.vertex.attribute(ga.net, "sex")=="F")],
          label=get.vertex.attribute(ga.net, "name"), label.cex=.75)

Layouts have some random variation, but here’s mine

We should notice a few interesting features about this network.  First, on connectivity, we have a very large component, yet the graph as a whole is disconnected into four separate components.  Should the show continue without introducing any new characters, I’d speculate that one of the smaller components would, ahem, “merge” with the larger component. Second, the majority of sexual relationships on the show are heterosexual.  Only two edges (torres-arizona and torres-hahn) are same-sex.  Likewise, there are no odd-numbered cycles: in a sexual network, cycles of three, five, seven, etc indicate at least one same-sex relationship within the cycle.  Third, reflecting the sampling strategy, the network contains no sexually lonely isolates.  The minimal degree, or number of ties incident upon a node, equals one.  Fourth, the network violates the ex’s ex’s ex taboo.  In one study on sexual networks, researchers found a prohibition against coupling with a former partner’s former partner’s former partner due to status implications.  It appears that there are at least three cycles in this network that meet the taboo’s criteria.  However, it is also well-known that awkward status implications arising from other’s sexual relationships make for compelling entertainment.

Let’s start our formal analyses with a baseline model including just the number of edges and a term, “nodematch,” that captures the network’s heterosexual tendency.

ga.base<-ergm(ga.net~edges+nodematch("sex")) #Estimate the model
summary(ga.base) #Summarize the model

Here is the output from summary(ga.base).

==========================
Summary of model fit
==========================

Formula:  ga.net ~ edges + nodematch(“sex”)

Iterations: 20

Monte Carlo MLE Results:

Estimate     Std. Error    MCMC %    p-value

edges                              -2.3003       0.1581         NA               <1e-04 ***
nodematch.sex            -3.1399        0.7260        NA               <1e-04 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Null Deviance: 1311.43 on 946 degrees of freedom
Residual Deviance: 320.47 on 944 degrees of freedom
Deviance: 990.97 on 2 degrees of freedom

AIC: 324.47 BIC: 334.17

Confirming expectations, but the number of edges and heterosexuality are significant predictors of tie formation, as the p-value is far less than the conventional significance .05 cut off.  The column for “MCMC%” is left as missing, or NA, because this model didn’t require Markov chain Monte Carlo estimation.  The coefficients, -2.30 and -3.14, are expressed as conditional log-odds, such that the log-odds of two characters hooking up equals (-2.30)*change in number of ties + (-3.14)*change in number of homophilous (same-sex) relationships.  The log-odds that a heterosexual tie will form in the network equals -2.30, equaling a probability of exp(-2.30)/(1+exp(-2.30)), or 0.09.  Likewise, the log-odds of a homosexual relationship forming equals -2.30-3.14, or -5.44, corresponding to a probability of 0.004 , or exp(-5.44)/(1+exp(-5.44)).

Let’s see what a random, simulated network from this ERGM would look like.

plot(simulate(ga.base),
          vertex.col=c("blue","pink")[ 1+(get.vertex.attribute(ga.net, "sex")=="F")])

Yours will look a bit different, but my plot looks like this

This network does accurately reflect both the number of edges and the propensity toward heterosexual ties, however there is something clearly wrong with the network: some of our nodes are isolates.  That is to say, the simulation predicts sexually abstinent individuals.  Because no sexually abstinent individuals should appear in our network due to the data collection, the model is underspecified.

While we have goodness of fit measurements produced by the likelihood function, one conventional, systematic way of determining goodness of fit is to compare unmodeled network measurements from the observed network to the same measurements from simulated networks proposed by the ERGM.

ga.base.gof<-gof(ga.base)
summary(ga.base.gof) #Summarize the goodness of fit
par(mfrow=c(3,1)); plot(ga.base.gof) #Plot three windows. It's OK to ignore the warning.

The plot reproduces the same information printed out in the summary.  The p-values in the far right columns are the probability that the simulated networks accurately capture the observed network for a given measurement.  We see that in terms of the edgewise shared partner and minimum geodesic distance measurements represent the data quite well.  However, if we look to the degree statistics we see that, confirming our visual inspection, the number of abstinent isolates (degree of 0) are overrepresented, the number of nodes with a degree of 1 (monogamists) are underrepresented, and are estimates are off for those with degrees of 3 and 9.

We’ll now add a statistic to the model that controls for the monogamy (degree of 1) effect.

ga.base.d1<-ergm(ga.net~edges+nodematch("sex")+degree(1))
summary(ga.base.d1)
plot(simulate(ga.base.d1),
     vertex.col=c("blue","pink")[1+(get.vertex.attribute(ga.net, "sex")=="F")])
ga.base.d1.gof<-gof(ga.base.d1); summary(ga.base.d1.gof)
par(mfrow=c(3,1)); plot(ga.base.d1.gof)

Using the previous model’s BIC of 334.17 as our benchmark, we do see an improvement in model fit as ga.base.d1’s BIC equals 316.56, suggesting our addition to the model increased the likelihood beyond the BIC’s penalties against additional parameters.  The monogamy effect is also statistically significant.  The plot portrays fewer isolates, confirmed by the gof() command.  Judging by the goodness of fit measurements, the model seems to be a reasonable approximation of the network, exempting the outlier character with nine sexual partners on the show.

Unlike the previous model, this model used Markov chain Monte Carlo (MCMC) estimation techniques.  When ergm() printed output like

Iteration 1 of at most 20:

the log-likelihood improved by 0.669

the program was running through probability distributions.  We can evaluate the diagnostics of the MCMC estimation.

mcmc.diagnostics(ga.base.d1)

Upon the output, we should first look at second item, which compares the sample statistics against the observed.

Are sample statistics significantly different from observed?

edges                       nodematch.sex     degree1                   Overall (Chi^2)

diff.           -0.0968000            0.0232000            0.0534000            NA
test stat.  -0.2942081             0.8328180             0.3379779              1.086016
P-val.        0.7685989             0.4049474              0.7353798             0.780451

This looks good, the estimated graphs made by the MCMC aren’t significantly different from the observed.  Let’s look at the other diagnostics.

Sample statistics auto-correlation:
Chain 1

edges                 nodematch.sex           degree1

Lag 0         1.0000000     1.00000000                1.0000000
Lag 100    0.8476292       0.47956824                 0.7334256
Lag 200    0.7332856       0.26641687                 0.6116231
Lag 300    0.6371281        0.17109432                 0.5198955
Lag 400    0.5587029       0.12489565                 0.4480439
Lag 500    0.4883954       0.08496445                0.3860807

Sample statistics burn-in diagnostic (Geweke):
Chain 1

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5

edges   nodematch.sex   degree1

-1.590  -1.068                   1.372

P-values (lower = worse):

edges              nodematch.sex    degree1

0.1117691      0.2856841            0.1701817

Overall, nothing looks alarming within the MCMC diagnostics.  The autocorrelation statistics between the lags are a bit higher than I’d prefer (they should be closer to zero and far away from one, exempting Lag 0).  Also, the Geweke statistics roughly correspond to a z-statistic and while their associated p-values aren’t significant, they’re much closer to zero than one.  Luckily, we can adjust some of the MCMC’s default settings to accomodate.

ga.base.d1<-ergm(ga.net~edges+nodematch("sex")+degree(1),
                      control=control.ergm(MCMC.burnin=50000, MCMC.interval=5000))

Increasing MCMC.burnin from 10,000 (default) to 50,000 should lower the Geweke statistics.  Increasing the MCMC.interval from 100 (default) to 5000 should lower the autocorrelation between lags.  If our sample statistics did significantly differ from observed, we would increase the sample size of the MCMC from 10,000 (default) to something like 20,000 or 50,000.  The command would be

ga.base.d1<-ergm(ga.net~edges+nodematch("sex")+degree(1),
                      control=control.ergm(MCMC.burnin=50000,
                      MCMC.interval=5000, MCMC.samplesize=50000))

We should exercise prudence in increasing these MCMC settings as they can dramatically increase the time of computation.

summary(ga.base.d1) #The figures are mostly the same
mcmc.diagnostics(ga.base.d1)

Ta-da!  The MCMC diagnostics have improved considerable, suggesting that our latest model is more robust.

Now that we understand a bit more on how to evaluate quality models, we can return to the interesting question of hook-ups!  We’ve so far modeled the network on the number of edge, the propensity toward heterosexuality, and monogamy, so what other aspects would be meaningful to analyze?  I’d suggest starting with age, because most romantic partners tend to be close in age.

ga.base.d1.age<-ergm(ga.net~edges+nodematch("sex")+degree(1)+absdiff("birthyear"),
                          control=control.ergm(MCMC.burnin=50000, MCMC.interval=5000))
summary(ga.base.d1.age)
mcmc.diagnostics(ga.base.d1.age)
summary(gof(ga.base.d1.age))

The model seems like a decent improvement!  The BIC decreased from 316.56 to 297.51, the MCMC diagnostics look good, the goodness of fit measures look generally reasonable, and the age difference is significant and negative, indicating that as the age difference between the actors increases, the chances of their characters hooking up decreases.

Lastly, let’s examine assortativity with respect to racial groupings.  Previous studies on adolescent sexual and romantic networks have uncovered assortativity according to racial groupings.  Further, the show’s creator, Shonda Rhimes, intentionally designed the cast to portray racial diversity.  Though the show does include a number of African-American characters, along with a handful of other non-white characters, it remains to be seen whether or not cast diversity resulted in racially diverse sexual relationships.   First, let’s try to fit a model of general homophily by race.

ga.base.d1.age.race<-ergm(ga.net~edges+nodematch("sex")+degree(1)
                          +absdiff("birthyear")+nodematch("race"),
                          control=control.ergm(MCMC.burnin=50000,MCMC.interval=5000))
summary(ga.base.d1.age.race)
mcmc.diagnostics(ga.base.d1.age.race)
summary(gof(ga.base.d1.age.race))

We do find a positive and significant racial assortativity effect, suggesting that tie formation is more likely between characters of the same race.  In this show, sexual relationships are more likely to be intraracial than interracial compared to chance expectations.  The MCMC diagnostics look good and the goodness of fit measures are in the ballpark.  Though the model looks good by most indicators, the BIC backslides from 297.51 in the previous model on age to 298.21 in this model.  If we’re interested in a simple explanation characterizing network structure, the previous model on age provides the better answer.

But is it possible that the white characters are homophilous and the black characters less so?  Or vice-versa?  We can refer to this process as differential homophily.  We can model it by adding the parameter “diff=T” to the racial nodematch term.

ga.base.d1.age.racediff<-ergm(ga.net ~ edges + nodematch("sex") + degree(1)
                              +absdiff("birthyear") + nodematch("race", diff=T),
                              control=control.ergm(MCMC.burnin=50000, MCMC.interval=5000))
summary(ga.base.d1.age.racediff)

While we can model differential homophily, ergm() can’t interpret the “other” category because the observed network doesn’t feature any sexual relationships between Latinas, Asian-Americans, or Latinas and Asian-Americans.  ergm() expressed this difficulty with the following statement:

Observed statistic(s) nodematch.race.Other are at their smallest attainable values. Their coefficients will be fixed at -Inf.

Likewise, the AIC, BIC, deviance, and the effects of nodematch.race.Other are all listed as NA in the summary.  Let’s try modeling this again, but excluding homophily among Latinas and Asian-Americans from the model.  To do so, we include the parameter “keep=c(1,3)” within the racial nodematch term, as “Black” and “White” were first and third categories listed in the summary(ga.base.d1.age.racediff) effects.

ga.base.d1.age.racediff<-ergm(ga.net~edges+nodematch("sex")+degree(1)
                              +absdiff("birthyear")+nodematch("race", diff=T, keep=c(1,3)),
                              control=control.ergm(MCMC.burnin=50000, MCMC.interval=5000))
summary(ga.base.d1.age.racediff)
mcmc.diagnostics(ga.base.d1.age.racediff)
summary(gof(ga.base.d1.age.racediff))

As with the previous model, we see significant assortativity by racial groups on the show.  Both black characters and white characters exhibit tendencies of racial homophily beyond random expectations.  This model also indicates that homophily effects are stronger among the African-American characters than the white characters, as indicated by the higher coefficient and lower p-value associated with the nodematch.race.Black term relative to the nodematch.race.White term.  Checking the MCMC diagnostics and goodness of fit statistics, the model is a sound representation of the original network.  That said, the BIC further backslides to 300.73, indicating that the model is less parsimonious than the general homophily model and the age model.

Let’s see a visual simulation of the model

plot(simulate(ga.base.d1.age.racediff),
          vertex.col=c("blue","pink")[1+(get.vertex.attribute(ga.net, "sex")=="F")])

Yours will look somewhat different, however this model seems to match the original data pretty well.  This simulation features a large component, few isolates (if any), a few smaller components, and while the relationships are mostly heterosexual we do see four same-sex relationships (the original featured two).

While the number of hook-ups on Grey’s Anatomy may seem surprising, the issue of dyadic tie formation represents a fairly traditional expression of sexual relationships: monogamy norms are quite prevalent and the vast majority of relationships are heterosexual between couples of similar age and racial profiles.  That said,  this exploration has omitted a number of alternative explanations that could better represent the network.  I invite the readers to play with the data and other ERG models.  Some additional examples:

#Perhaps men and women have different tendencies to form sexual partnerships?
ga.base.d1.age.sex<-ergm(ga.net~edges+nodematch("sex")+degree(1)
                         +absdiff("birthyear")+nodefactor("sex"),
                         control=control.ergm(MCMC.burnin=100000, MCMC.interval=5000))
                         #Maybe less "traditional" than previously concluded?
#Perhaps you're interested in the assortative mixture among roles in the hospital?
#Resident-resident sexual contacts (28) are the reference group,
  #and unobserved pairings between positions (3-5, 9, 12-15, 17-21, 24, 27) are omitted.
ga.base.d1.age.rolemix<-ergm(ga.net~edges+nodematch("sex")+degree(1)+absdiff("birthyear")
                             +nodemix("position", base=c(3:5, 9, 12:15, 17:21, 24, 27, 28)),
                             control=control.ergm(MCMC.burnin=50000, MCMC.interval=5000))
list.vertex.attributes(ga.net) #List the different vertex attributes in the network object
get.vertex.attribute(ga.net, "sign") #Provides each actor's astrological sign.
?ergm.terms #This command will provide a list of other terms that ergm() can model
?control.ergm #This command will present different options on the estimation methods.

Enjoy!

  • Thank you for this great exersice!

    • I’m glad you like it. If you play around with the data and ERG models, let me know what you find. Watch out that some models won’t converge, though. For example, I tried modeling the ex of an ex of an ex through a 4-cycle term and each iteration improved the likelihood by 20, a sure sign of convergence problems.

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  • martina

    Hey Benjamin — very nice tutorial! RE: the 4-cycle model — did you have all of the dyad independent terms in the model as well (nodefactor, nodematch)?

    • I’m happy to hear you liked the tutorial. Yes, I tried to fit a 4-cycle term after including nodematch(“sex”) and perhaps some other dyad independent terms. There might be ways to fit a 4-cycle here, but I’d guess it’s a secondary consideration for the show’s creators relative to the heterosexuality and age-pairing in determining the relationship limitations.

      • martina

        There are quite a few 4 cycles in that graph 8-). And several stem from the same dyad. I’m guessing it’s more than a small cast thing. The problem with the cycle terms is that they typically define a degenerate model — recall the problem with triangle terms. The 4cycle config is also a dyadwise shared partner count of 2 (or more), So you could try fitting with gwdsp. Of course, there is the temporal dimension here also…

        • Thank you, I haven’t considered those issues. 🙂 The temporal dimension is very important, I have the season of their introduction coded, but the order and direction of their relationships does complicate the matter. I’d like to do one of these with STERGM in the future, but I’d need to find the right (and fun) data.

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  • Gabriel

    So far you have created a mathematical model (and ERGM) that can produce similar networks to the observed one. How is this useful, what would you do with this model in practice? This is not a troll question, but this point really wasn’t clear from the blog post. It is not necessary to fit an ERGM to show homophily. That could be done by just looking at the observed network.

    • Gabriel

      Actually I was really hoping for an answer to this one.

    • Benjamin Lind

      The use is that ERGM can identify the mechanisms behind tie formation within a network. It’s not just about testing homophily. Homophily is just one family of mechanisms. The example here compared different types of homophily against each other, identifying which are more meaningful than others and how they operate. Further, you can see whether or not homophily holds after accounting for other network properties. In this case, the other network property examined was monogamy (degree of one). Ties here were formed not only in accordance with homophily/heterophily, but also following monogamy norms. Many other network properties could be tested alongside homophily, though. E.g., I didn’t model triadic closure because most of the ties were formed between heterosexual partners, but in other networks homophily and triadic closure could be competing effects. (In the comments here, Martina suggested including a 4-cycle term, which could be more relevant than some of the homophily terms.)

  • ERGM enthusiast

    Hi!

    I was wondering whether there is a way to increase the number of chains in the MCMC simulations. That will provide more information on the robustness of the simulations.

    • Benjamin Lind

      There probably is a way, though I don’t know it offhand within ergm. While I’m not familiar with it, the package “Bergm” has an option to set the number of chains in the function bergmS().

    • Benjamin Lind

      I made a couple edits to the below post, but they seem like they didn’t go through. I had forgotten about parallel options and I think that would be the way to do it within ergm. If you want “x” chains, set the control parameter equal to “control.ergm(…, parallel=x)”.

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  • Cody Dey

    Hi Benjamin, thanks a lot for the tutorial. I have one question: when you are using nodemix(), how can you determine the values for the pairings you want to exclude (using the base arguement)? Is it based on a mixing matrix of the “position” attribute? and if so, how can you access it so you know which values to insert after base=c(). Cheers!

    • Benjamin Lind

      Hi Cody, you’re welcome. Initially I ran the nodemix(“position”) argument without specifying the base. Doing so will issue a warning that some pairings “are at their smallest attainable values.” These pairings are unobserved. Running summary() on this model will indicate the unobserved pairings, as their coefficients are -Inf, and they are listed in the same order that they are considered in nodemix()’s “base” argument. From this model, the 3-5, 9, 12-15, 17-21, 24, and 27 pairings in the summary() returned -Inf coefficients, so that’s how I knew to exclude them and add them to the base argument. I added 28, the Resident-Resident pairing as an observed reference group.

  • Sunny Lee

    Hi Benjamin, thanks a bunch for the tutorial. I liked it very much especially because you used one of my favorite TV shows as an example. 🙂 One quick basic question I have is if you found many significant parameters (like idegree, odegree (1, 2, 3), nodefactor(gender), etc.), but mcmc diagnostics tell you they are significantly different from sample stats, what do you do? Do you keep running with different parameters until you get good diagnostics? I’m trying to see tie formation in my network data (directed with about 250 ties), and that seems to be the case.

    • Benjamin Lind

      Hi Sunny! You’re welcome; I’m glad you like it. Regarding your question, you should increase the sample size of the MCMC. The default is 10,000, so using values like 20,000 or 50,000 should correct it.

      • Sunny

        Awesome! I have been coming back to your site over and over (for referring the example), but only to find your answer to my question now. Thanks for responding, and I’ll try with your tip!

  • MOHSEN MOSLEH

    Hi Benjamin, this is an amazing example. I have a question for you. Is there any way to generate random graphs with different number of nodes from the model created by ERGM? For example, I may want to scale down the size of the network so that I can run computational extensive algorithm on the random graphs with similar statistical characteristic as the observed network.
    Thanks

    • Benjamin Lind

      Thanks. You can use a smaller empty network for the ‘basis’ parameter in ‘simulate.ergm().’ The downside is that the edges parameter will be too large. Basically, it will result in an overly dense random graph. There’s likely a way around this issue, but I don’t know of it offhand.

  • de5zcz
  • thanks

    Excellent…Thank you so much. This was really helpful. I do have one question: what precisely does it mean when you include the “degree(1)” parameter in the model? Are we saying that there are more nodes with a degree of 1 as would have been expected from the density of the graph?