Last updated 04-03-2016

This tutorial is a joint product of the Statnet Development Team:

Martina Morris (University of Washington)
Mark S. Handcock (University of California, Los Angeles)
Carter T. Butts (University of California, Irvine)
David R. Hunter (Penn State University)
Steven M. Goodreau (University of Washington)
Skye Bender de-Moll (Oakland)
Pavel N. Krivitsky (University of Wollongong)

For general questions and comments, please refer to statnet users group and mailing list
http://statnet.csde.washington.edu/statnet_users_group.shtml

1. Getting Started

Open an R session, and set your working directory to the location where you would like to save this work.

To install all of the packages in the statnet suite:

install.packages('statnet')
library(statnet)

Or, to only install the specific statnet packages needed for this tutorial:

install.packages('ergm') # will install the network package
install.packages('sna')

After the first time, to update the packages one can either repeat the commands above, or use:

update.packages('name.of.package')

For this tutorial, we will need one additional package (coda), which is recommended (but not required) by ergm:

install.packages('coda')

Make sure the packages are attached:

library(statnet)

or

library(ergm)
library(sna)
library(coda)

Check package version

# latest versions:  ergm 3.6.0 and network 1.13.0 (as of 4/03/2016)
sessionInfo()

Set seed for simulations – this is not necessary, but it ensures that we all get the same results (if we execute the same commands in the same order).

set.seed(0)

2. Statistical network modeling; the summary and ergm commands, and supporting functions

Exponential-family random graph models (ERGMs) represent a general class of models based in exponential-family theory for specifying the probability distribution for a set of random graphs or networks. Within this framework, one can—among other tasks—obtain maximum-likehood estimates for the parameters of a specified model for a given data set; test individual models for goodness-of-fit, perform various types of model comparison; and simulate additional networks with the underlying probability distribution implied by that model.

The general form for an ERGM can be written as:

\[ P(Y=y)=\frac{\exp(\theta'g(y))}{k(\theta)} \]

where Y is the random variable for the state of the network (with realization y), \(g(y)\) is a vector of model statistics for network y, \(\theta\) is the vector of coefficients for those statistics, and \(k(\theta)\) represents the quantity in the numerator summed over all possible networks (typically constrained to be all networks with the same node set as y).

This can be re-expressed in terms of the conditional log-odds of a single tie between two actors:

\[ \operatorname{logit}{(Y_{ij}=1|y^{c}_{ij})=\theta'\delta(y_{ij})} \]

where \(Y_{ij}\) is the random variable for the state of the actor pair \(i,j\) (with realization \(y_{ij}\)), and \(y^{c}_{ij}\) signifies the complement of \(y_{ij}\), i.e. all dyads in the network other than \(y_{ij}\). The vector \(\delta(y_{ij})\) contains the “change statistic” for each model term. The change statistic records how \(g(y)\) term changes if the \(y_{ij}\) tie is toggled on or off. So:

\[ \delta(y_{ij}) = g(y^{+}_{ij})-g(y^{-}_{ij}) \]

where \(y^{+}_{ij}\) is defined as \(y^{c}_{ij}\) along with \(y_{ij}\) set to 1, and \(y^{-}_{ij}\) is defined as \(y^{c}_{ij}\) along with \(y_{ij}\) set to 0. That is, \(\delta(y_{ij})\) equals the value of \(g(y)\) when \(y_{ij}=1\) minus the value of \(g(y)\) when \(y_{ij}=0\), but all other dyads are as in \(g(y)\).

This emphasizes that the coefficient \(\theta\) can be interpreted as the log-odds of an individual tie conditional on all others.

The model terms \(g(y)\) are functions of network statistics that we hypothesize may be more or less common than what would be expected in a simple random graph (where all ties have the same probability). For example, specific degree distributions, or triad configurations, or homophily on nodal attributes. We will explore some of these terms in this tutorial, and links to more information are provided in section 3.

One key distinction in model terms is worth keeping in mind: terms are either dyad independent or dyad dependent. Dyad independent terms (like nodal homophily terms) imply no dependence between dyads—the presence or absence of a tie may depend on nodal attributes, but not on the state of other ties. Dyad dependent terms (like degree terms, or triad terms), by contrast, imply dependence between dyads. Such terms have very different effects, and much of what is different about network models comes from the complex cascading effects that these terms introduce. A model with dyad dependent terms also requires a different estimation algorithm, and you will see some different components in the output.

We’ll start by running some simple models to demonstrate the use of the “summary” and “ergm” commands. The ergm package contains several network data sets that we will use for demonstration purposes here.

data(package='ergm') # tells us the datasets in our packages

Bernoulli model

We begin with the simplest possible model, the Bernoulli or Erdos-Renyi model, which contains only one term to capture the density of the network as a function of a homogenous edge probability. The ergm-term for this is edges. We’ll fit this simple model to Padgett’s Florentine marriage network. As with all data analysis, we start by looking at our data: using graphical and numerical descriptives.

data(florentine) # loads flomarriage and flobusiness data
flomarriage # Let's look at the flomarriage network properties
 Network attributes:
  vertices = 16 
  directed = FALSE 
  hyper = FALSE 
  loops = FALSE 
  multiple = FALSE 
  bipartite = FALSE 
  total edges= 20 
    missing edges= 0 
    non-missing edges= 20 

 Vertex attribute names: 
    priorates totalties vertex.names wealth 

No edge attributes
par(mfrow=c(1,2)) # Setup a 2 panel plot (for later)
plot(flomarriage, main="Florentine Marriage", cex.main=0.8) # Plot the flomarriage network
summary(flomarriage~edges) # Look at the $g(y)$ statistic for this model
edges 
   20 
flomodel.01 <- ergm(flomarriage~edges) # Estimate the model 
Evaluating log-likelihood at the estimate. 
summary(flomodel.01) # The fitted model object

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

Formula:   flomarriage ~ edges

Iterations:  5 out of 20 

Monte Carlo MLE Results:
      Estimate Std. Error MCMC % p-value    
edges  -1.6094     0.2449      0  <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 166.4  on 120  degrees of freedom
 Residual Deviance: 108.1  on 119  degrees of freedom
 
AIC: 110.1    BIC: 112.9    (Smaller is better.) 

How should we interpret the coefficient from this model? The log-odds of any tie existing is:

\[ \small{ \begin{eqnarray*} & = & -1.609\times\mbox{change in the number of ties}\\ & = & -1.609\times1 \end{eqnarray*} } \]

for all ties, since the addition of any tie to the network always changes the number of ties by 1 for a tie toggled from 0 to 1 (or by -1 for a tie toggled from 1 to 0).

The corresponding probability is:

\[ \small{ \begin{eqnarray*} & = & \exp(-1.609)/(1+\exp(-1.609))\\ & = & 0.1667 \end{eqnarray*} } \]

which corresponds to the density we observe in the flomarriage network: there are 20 ties and (16 choose 2 = 16*15/2 =) 120 dyads.

Triad formation

Let’s add a term often thought to be a measure of “clustering”: the number of completed triangles. The ergm-term for this is triangle. This is a dyad dependent term. As a result, the estimation algorithm automatically changes to MCMC, and because this is a form of stochastic estimation your results may differ slightly.

summary(flomarriage~edges+triangle) # Look at the g(y) stats for this model
   edges triangle 
      20        3 
flomodel.02 <- ergm(flomarriage~edges+triangle) 
Starting maximum likelihood estimation via MCMLE:
Iteration 1 of at most 20: 
The log-likelihood improved by 0.004492 
Step length converged once. Increasing MCMC sample size.
Iteration 2 of at most 20: 
The log-likelihood improved by 0.001591 
Step length converged twice. Stopping.
Evaluating log-likelihood at the estimate. Using 20 bridges: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .

This model was fit using MCMC.  To examine model diagnostics and check for degeneracy, use the mcmc.diagnostics() function.
summary(flomodel.02)

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

Formula:   flomarriage ~ edges + triangle

Iterations:  2 out of 20 

Monte Carlo MLE Results:
         Estimate Std. Error MCMC % p-value    
edges     -1.6761     0.3485      0  <1e-04 ***
triangle   0.1469     0.5668      0   0.796    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 166.4  on 120  degrees of freedom
 Residual Deviance: 108.1  on 118  degrees of freedom
 
AIC: 112.1    BIC: 117.6    (Smaller is better.) 

Now, how should we interpret coefficients?

The conditional log-odds of two actors having a tie is:

\[ \small{ -1.67\times\mbox{change in the number of ties}+0.14\times\mbox{change in number of triangles} } \]


  • For a tie that will create no triangles, the conditional log-odds is: \(-1.67\).
  • if one triangle: \(-1.67 + 0.14 =-1.53\)
  • if two triangles: \(-1.67 +0.14\times2=-1.39\)
  • the corresponding probabilities are 0.16, 0.18, and 0.20.

Let’s take a closer look at the ergm object itself:

class(flomodel.02) # this has the class ergm
[1] "ergm"
names(flomodel.02) # the ERGM object contains lots of components.
 [1] "coef"          "sample"        "sample.obs"    "iterations"   
 [5] "MCMCtheta"     "loglikelihood" "gradient"      "hessian"      
 [9] "covar"         "failure"       "network"       "newnetworks"  
[13] "newnetwork"    "coef.init"     "est.cov"       "coef.hist"    
[17] "stats.hist"    "steplen.hist"  "control"       "etamap"       
[21] "formula"       "target.stats"  "target.esteq"  "constrained"  
[25] "constraints"   "reference"     "estimate"      "offset"       
[29] "drop"          "estimable"     "null.lik"      "mle.lik"      
flomodel.02$coef # you can extract/inspect individual components
     edges   triangle 
-1.6760880  0.1468579 

Nodal covariates: effects on mean degree

We can test whether edge probabilities are a function of wealth. This is a nodal covariate, so we use the ergm-term nodecov.

wealth <- flomarriage %v% 'wealth' # %v% references vertex attributes
wealth
 [1]  10  36  55  44  20  32   8  42 103  48  49   3  27  10 146  48
summary(wealth) # summarize the distribution of wealth
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   3.00   17.50   39.00   42.56   48.25  146.00 
plot(flomarriage, vertex.cex=wealth/25, main="Florentine marriage by wealth", cex.main=0.8) # network plot with vertex size proportional to wealth

summary(flomarriage~edges+nodecov('wealth')) 
         edges nodecov.wealth 
            20           2168 
# observed statistics for the model
flomodel.03 <- ergm(flomarriage~edges+nodecov('wealth'))
Evaluating log-likelihood at the estimate. 
summary(flomodel.03)

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

Formula:   flomarriage ~ edges + nodecov("wealth")

Iterations:  4 out of 20 

Monte Carlo MLE Results:
                Estimate Std. Error MCMC % p-value    
edges          -2.594929   0.536056      0  <1e-04 ***
nodecov.wealth  0.010546   0.004674      0  0.0259 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 166.4  on 120  degrees of freedom
 Residual Deviance: 103.1  on 118  degrees of freedom
 
AIC: 107.1    BIC: 112.7    (Smaller is better.) 

Yes, there is a significant positive wealth effect on the probability of a tie.

How do we interpret the coefficients here? Note that the wealth effect operates on both nodes in a dyad. The conditional log-odds of a tie between two actors is:

\[ \small{ -2.59\times\mbox{change in the number of ties} + 0.01\times\mbox{the wealth of node 1} + 0.01\times\mbox{the wealth of node 2} } \]

\[ \small{ -2.59\times\mbox{change in the number of ties} + 0.01\times\mbox{the sum of the wealth of the two nodes} } \]


  • for a tie between two nodes with minimum wealth, the conditional log-odds is:
    \(-2.59 + 0.01*(3+3) = -2.53\)
  • for a tie between two nodes with maximum wealth:
    \(-2.59 + 0.01*(146+146) = 0.33\)
  • for a tie between the node with maximum wealth and the node with minimum wealth:
    \(-2.59 + 0.01*(146+3) = -1.1\)
  • The corresponding probabilities are 0.07, 0.58, and 0.25.

Note: This model specification does not include a term for homophily by wealth. It just specifies a relation between wealth and mean degree. To specify homophily on wealth, you would use the ergm-term absdiff see section 3 below for more information on ergm-terms

Nodal covariates: Homophily

Let’s try a larger network, a simulated mutual friendship network based on one of the schools from the Add Health study. Here, we’ll examine the homophily in friendships by grade and race. Both are discrete attributes so we use the ergm-term nodematch.

data(faux.mesa.high) 
mesa <- faux.mesa.high
mesa
 Network attributes:
  vertices = 205 
  directed = FALSE 
  hyper = FALSE 
  loops = FALSE 
  multiple = FALSE 
  bipartite = FALSE 
  total edges= 203 
    missing edges= 0 
    non-missing edges= 203 

 Vertex attribute names: 
    Grade Race Sex 

No edge attributes
par(mfrow=c(1,1)) # Back to 1-panel plots
plot(mesa, vertex.col='Grade')
legend('bottomleft',fill=7:12,legend=paste('Grade',7:12),cex=0.75)

fauxmodel.01 <- ergm(mesa ~edges + nodematch('Grade',diff=T) + nodematch('Race',diff=T))
Observed statistic(s) nodematch.Race.Black and nodematch.Race.Other are at their smallest attainable values. Their coefficients will be fixed at -Inf.
Evaluating log-likelihood at the estimate. 
summary(fauxmodel.01)

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

Formula:   mesa ~ edges + nodematch("Grade", diff = T) + nodematch("Race", 
    diff = T)

Iterations:  8 out of 20 

Monte Carlo MLE Results:
                     Estimate Std. Error MCMC % p-value    
edges                 -6.2328     0.1742      0  <1e-04 ***
nodematch.Grade.7      2.8740     0.1981      0  <1e-04 ***
nodematch.Grade.8      2.8788     0.2391      0  <1e-04 ***
nodematch.Grade.9      2.4642     0.2647      0  <1e-04 ***
nodematch.Grade.10     2.5692     0.3770      0  <1e-04 ***
nodematch.Grade.11     3.2921     0.2978      0  <1e-04 ***
nodematch.Grade.12     3.8376     0.4592      0  <1e-04 ***
nodematch.Race.Black     -Inf     0.0000      0  <1e-04 ***
nodematch.Race.Hisp    0.0679     0.1737      0  0.6959    
nodematch.Race.NatAm   0.9817     0.1842      0  <1e-04 ***
nodematch.Race.Other     -Inf     0.0000      0  <1e-04 ***
nodematch.Race.White   1.2685     0.5371      0  0.0182 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 28987  on 20910  degrees of freedom
 Residual Deviance:  1928  on 20898  degrees of freedom
 
AIC: 1952    BIC: 2047    (Smaller is better.) 

 Warning: The following terms have infinite coefficient estimates:
  nodematch.Race.Black nodematch.Race.Other 

Note that two of the coefficients are estimated as -Inf (the nodematch coefficients for race Black and Other). Why is this?

table(mesa %v% 'Race') # Frequencies of race

Black  Hisp NatAm Other White 
    6   109    68     4    18 
mixingmatrix(mesa, "Race")
Note:  Marginal totals can be misleading
 for undirected mixing matrices.
      Black Hisp NatAm Other White
Black     0    8    13     0     5
Hisp      8   53    41     1    22
NatAm    13   41    46     0    10
Other     0    1     0     0     0
White     5   22    10     0     4

The problem is that there are very few students in the Black and Other race categories, and these few students form no within-group ties. The empty cells are what produce the -Inf estimates.

Note that we would have caught this earlier if we had looked at the \(g(y)\) stats at the beginning:

summary(mesa ~edges + nodematch('Grade',diff=T) + nodematch('Race',diff=T))
               edges    nodematch.Grade.7    nodematch.Grade.8 
                 203                   75                   33 
   nodematch.Grade.9   nodematch.Grade.10   nodematch.Grade.11 
                  23                    9                   17 
  nodematch.Grade.12 nodematch.Race.Black  nodematch.Race.Hisp 
                   6                    0                   53 
nodematch.Race.NatAm nodematch.Race.Other nodematch.Race.White 
                  46                    0                    4 

Moral: It’s a good idea to check the descriptive statistics of a model in the observed network before fitting the model.

See also the ergm-term nodemix for fitting mixing patterns other than homophily on discrete nodal attributes.

Directed ties

Let’s try a model for a directed network, and examine the tendency for ties to be reciprocated (“mutuality”). The ergm-term for this is mutual. We’ll fit this model to the third wave of the classic Sampson Monastery data, and we’ll start by taking a look at the network.

data(samplk) 
ls() # directed data: Sampson's Monks
 [1] "faux.mesa.high" "fauxmodel.01"   "flobusiness"    "flomarriage"   
 [5] "flomodel.01"    "flomodel.02"    "flomodel.03"    "mesa"          
 [9] "samplk1"        "samplk2"        "samplk3"        "wealth"        
samplk3
 Network attributes:
  vertices = 18 
  directed = TRUE 
  hyper = FALSE 
  loops = FALSE 
  multiple = FALSE 
  bipartite = FALSE 
  total edges= 56 
    missing edges= 0 
    non-missing edges= 56 

 Vertex attribute names: 
    cloisterville group vertex.names 

No edge attributes
plot(samplk3)

summary(samplk3~edges+mutual)
 edges mutual 
    56     15 

The plot now shows the direction of a tie, and the \(g(y)\) statistics for this model in this network are 56 total ties, and 15 mutual dyads (so 30 of the 56 ties are mutual ties).

sampmodel.01 <- ergm(samplk3~edges+mutual)
Starting maximum likelihood estimation via MCMLE:
Iteration 1 of at most 20: 
The log-likelihood improved by 0.000677 
Step length converged once. Increasing MCMC sample size.
Iteration 2 of at most 20: 
The log-likelihood improved by 0.001099 
Step length converged twice. Stopping.
Evaluating log-likelihood at the estimate. Using 20 bridges: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .

This model was fit using MCMC.  To examine model diagnostics and check for degeneracy, use the mcmc.diagnostics() function.
summary(sampmodel.01)

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

Formula:   samplk3 ~ edges + mutual

Iterations:  2 out of 20 

Monte Carlo MLE Results:
       Estimate Std. Error MCMC % p-value    
edges   -2.1542     0.2145      0  <1e-04 ***
mutual   2.3006     0.4784      0  <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 424.2  on 306  degrees of freedom
 Residual Deviance: 267.9  on 304  degrees of freedom
 
AIC: 271.9    BIC: 279.4    (Smaller is better.) 

There is a strong and significant mutuality effect. The coefficients for the edges and mutual terms roughly cancel for a mutual tie, so the conditional odds of a mutual tie are about even, and the probability is about 50%. By contrast a non-mutual tie has a conditional log-odds of -2.16, or 10% probability.

Triangle terms in directed networks can have many different configurations, given the directional ties. Many of these configurations are coded up as ergm-terms (and we’ll talk about these more below).

Missing data

It is important to distinguish between the absence of a tie, and the absence of data on whether a tie exists. You should not code both of these as “0”. The \(ergm\) package recognizes and handles missing data appropriately, as long as you identify the data as missing. Let’s explore this with a simple example.

Let’s start with estimating an ergm on a network with two missing ties, where both ties are identified as missing.

missnet <- network.initialize(10,directed=F)
missnet[1,2] <- missnet[2,7] <- missnet[3,6] <- 1
missnet[4,6] <- missnet[4,9] <- missnet[5,6] <- NA
summary(missnet)
Network attributes:
  vertices = 10
  directed = FALSE
  hyper = FALSE
  loops = FALSE
  multiple = FALSE
  bipartite = FALSE
 total edges = 6 
   missing edges = 3 
   non-missing edges = 3 
 density = 0.06666667 

Vertex attributes:
  vertex.names:
   character valued attribute
   10 valid vertex names

No edge attributes

Network adjacency matrix:
   1 2 3  4  5  6 7 8  9 10
1  0 1 0  0  0  0 0 0  0  0
2  1 0 0  0  0  0 1 0  0  0
3  0 0 0  0  0  1 0 0  0  0
4  0 0 0  0  0 NA 0 0 NA  0
5  0 0 0  0  0 NA 0 0  0  0
6  0 0 1 NA NA  0 0 0  0  0
7  0 1 0  0  0  0 0 0  0  0
8  0 0 0  0  0  0 0 0  0  0
9  0 0 0 NA  0  0 0 0  0  0
10 0 0 0  0  0  0 0 0  0  0
# plot missnet with missing edge colored red. 
tempnet <- missnet
tempnet[4,6] <- tempnet[4,9] <- tempnet[5,6] <- 1
missnetmat <- as.matrix(missnet)
missnetmat[is.na(missnetmat)] <- 2
plot(tempnet,label = network.vertex.names(tempnet),edge.col = missnetmat)

summary(missnet~edges)
edges 
    3 
summary(ergm(missnet~edges))
Evaluating log-likelihood at the estimate. 

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

Formula:   missnet ~ edges

Iterations:  5 out of 20 

Monte Carlo MLE Results:
      Estimate Std. Error MCMC %  p-value    
edges  -2.5649     0.5991      0 0.000109 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 58.22  on 42  degrees of freedom
 Residual Deviance: 21.61  on 41  degrees of freedom
 
AIC: 23.61    BIC: 25.35    (Smaller is better.) 

The coefficient equals -2.56, which corresponds to a probability of 7.14%. Our network has 3 ties, out of the 42 non-missing nodal pairs (10 choose 2 minus 3): 3/42 = 7.14%. So our estimate represents the probability of a tie in the observed sample.

Now let’s assign those missing ties the value “0” and see what happens.

missnet_bad <- missnet
missnet_bad[4,6] <- missnet_bad[4,9] <- missnet_bad[5,6] <- 0
summary(missnet_bad)
Network attributes:
  vertices = 10
  directed = FALSE
  hyper = FALSE
  loops = FALSE
  multiple = FALSE
  bipartite = FALSE
 total edges = 3 
   missing edges = 0 
   non-missing edges = 3 
 density = 0.06666667 

Vertex attributes:
  vertex.names:
   character valued attribute
   10 valid vertex names

No edge attributes

Network adjacency matrix:
   1 2 3 4 5 6 7 8 9 10
1  0 1 0 0 0 0 0 0 0  0
2  1 0 0 0 0 0 1 0 0  0
3  0 0 0 0 0 1 0 0 0  0
4  0 0 0 0 0 0 0 0 0  0
5  0 0 0 0 0 0 0 0 0  0
6  0 0 1 0 0 0 0 0 0  0
7  0 1 0 0 0 0 0 0 0  0
8  0 0 0 0 0 0 0 0 0  0
9  0 0 0 0 0 0 0 0 0  0
10 0 0 0 0 0 0 0 0 0  0
summary(ergm(missnet_bad~edges))
Evaluating log-likelihood at the estimate. 

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

Formula:   missnet_bad ~ edges

Iterations:  5 out of 20 

Monte Carlo MLE Results:
      Estimate Std. Error MCMC % p-value    
edges  -2.6391     0.5976      0  <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 62.38  on 45  degrees of freedom
 Residual Deviance: 22.04  on 44  degrees of freedom
 
AIC: 24.04    BIC: 25.85    (Smaller is better.) 

The coefficient is smaller now because the missing ties are counted as “0”, and translates to a conditional tie probability of 6.67%. It’s a small difference in this case (and a small network, with little missing data).

MORAL: If you have missing data on ties, be sure to identify them by assigning the “NA” code. This is particularly important if you’re reading in data as an edgelist, as all dyads without edges are implicitly set to “0” in this case.

3. Model terms available for ergm estimation and simulation

Model terms are the expressions (e.g. “triangle”) used to represent predictors on the right-hand size of equations used in:

Many ERGM terms are simple counts of configurations (e.g., edges, nodal degrees, stars, triangles), but others are more complex functions of these configurations (e.g., geometrically weighted degrees and shared partners). In theory, any configuration (or function of configurations) can be a term in an ERGM. In practice, however, these terms have to be constructed before they can be used—that is, one has to explicitly write an algorithm that defines and calculates the network statistic of interest. This is another key way that ERGMs differ from traditional linear and general linear models.

The terms that can be used in a model also depend on the type of network being analyzed: directed or undirected, one-mode or two-mode (“bipartite”), binary or valued edges.

Terms provided with ergm

For a list of available terms that can be used to specify an ERGM, type:

help('ergm-terms')

A table of commonly used terms can be found here

A more complete discussion of many of these terms can be found in the ‘Specifications’ paper in the Journal of Statistical Software v24(4)

Finally, note that models with only dyad independent terms are estimated in statnet using a logistic regression algorithm to maximize the likelihood. Dyad dependent terms require a different approach to estimation, which, in statnet, is based on a Monte Carlo Markov Chain (MCMC) algorithm that stochastically approximates the Maximum Likelihood.

Coding new ergm-terms

We have recently released a new package (ergm.userterms) that makes it much easier to write one’s own ergm-terms. The package is available on CRAN, and installing it will include the tutorial (ergmuserterms.pdf). Alternatively, the tutorial can be found in the Journal of Statistical Software 52(2), and some introductory slides from the workshop we teach on coding ergm-terms can be found here.

Note that writing up new ergm terms requires some knowledge of C and the ability to build R from source (although the latter is covered in the tutorial, the many environments for building R and the rapid changes in these environments make these instructions obsolete quickly).

4. Network simulation: the simulate command and network.list objects

Once we have estimated the coefficients of an ERGM, the model is completely specified. It defines a probability distribution across all networks of this size. If the model is a good fit to the observed data, then networks drawn from this distribution will be more likely to “resemble” the observed data. To see examples of networks drawn from this distribution we use the simulate command:

flomodel.03.sim <- simulate(flomodel.03,nsim=10)
class(flomodel.03.sim) 
[1] "network.list"
summary(flomodel.03.sim)
Number of Networks: 10 
Model: flomarriage ~ edges + nodecov("wealth") 
Reference: ~Bernoulli 
Constraints: ~. 
Parameters:
         edges nodecov.wealth 
   -2.59492903     0.01054591 

Stored network statistics:
      edges nodecov.wealth
 [1,]    22           2259
 [2,]    20           2091
 [3,]    23           2360
 [4,]    20           2257
 [5,]    20           1996
 [6,]    19           1956
 [7,]    16           1670
 [8,]    15           1870
 [9,]    17           2036
[10,]    22           2565
length(flomodel.03.sim)
[1] 10
flomodel.03.sim[[1]]
 Network attributes:
  vertices = 16 
  directed = FALSE 
  hyper = FALSE 
  loops = FALSE 
  multiple = FALSE 
  bipartite = FALSE 
  total edges= 22 
    missing edges= 0 
    non-missing edges= 22 

 Vertex attribute names: 
    priorates totalties vertex.names wealth 

No edge attributes
plot(flomodel.03.sim[[1]], label= flomodel.03.sim[[1]] %v% "vertex.names")

Voila. Of course, yours will look somewhat different.

Simulation can be used for many purposes: to examine the range of variation that could be expected from this model, both in the sufficient statistics that define the model, and in other statistics not explicitly specified by the model. Simulation will play a large role in analyizing egocentrically sampled data in section 7 below. And if you take the tergm workshop, you will see how we can use simulation to examine the temporal implications of a model based on a single cross-sectional egocentrically sampled dataset.

For now, we will examine one of the primary uses of simulation in the ergm package: using simulated data from the model to evaluate goodness of fit to the observed data.

5. Examining the quality of model fit – GOF

There are two types of goodness of fit (GOF) tests in ergm. The first, and one you’ll always want to run, is used to evaluate how well the estimates are reproducing the terms that are in the model. These are maximum likelihood estimates, so they should reproduce the observed sufficient statistics well, and if they don’t it’s an indication that something may be wrong in the estimation process. Assuming all is well here, you can move on to the next step.

The second type of GOF test is used to see how well the model fits other emergent patterns in your network data, patterns that are not explicitly represented by the terms in the model. ERGMs are cross-sectional; they don’t directly model the process of tie formation and dissolution (that would be a temporal ergm (TERGM), see the tergm package). But ERGMs can be seen as generative models in another sense: they represent the process that governs the emergent global pattern of ties from a local perspective.

To see this, it’s worth digging a bit deeper into the MCMC estimation process. When you estimate an ERGM in statnet, the MCMC algorithm at each step draws a dyad at random, and evaluates the probability of a tie from the perspective of these two nodes. That probability is governed by the ergm-terms in the model, and the current estimates of the coefficients on these terms. Once the estimates converge, simulations from the model will produce networks that are centered on the observed model statistics (as we saw above), but the networks will also have other emergent global properties, even though these global properties are not represented by explicit terms in the model. If the local processes represented by the model terms capture the true underlying process, the model should reproduce these global properties as well.

So the second of whether a local model “fits the data” is to evaluate how well it reproduces observed global network properties that are not in the model. We do this by choosing network statistics that are not in the model, and comparing the value observed in the original network to the distribution of values we get in simulated networks from our model.

Both types of tests can be conducted by using the gof function. The gof function is a bit different than the summary, ergm, and simulate functions, in that it currently only takes 4 possible arguments: model, degree, esp (edgwise share partners), and distance (geodesic distances). “model” uses gof to evaluate the fit to the terms in the model, and the other 3 terms are used to evaluate the fit to emergent global network properties, at either the node level (degree), the edge level (esp), or the dyad level (distance). Note that these 3 global terms represent distributions, not single number summary statistics.

Let’s start by looking at how to assess the fit of the model to the terms in the model, using model 3 from the flomarriage example (flomarriage ~ edges+nodecov('wealth')).

flo.03.gof.model <- gof(flomodel.03 ~ model)
flo.03.gof.model

Goodness-of-fit for model statistics 

                obs min    mean  max MC p-value
edges            20  10   19.54   30       1.00
nodecov.wealth 2168 773 2096.15 3239       0.94
plot(flo.03.gof.model)

Looks pretty good. Now let’s look at the fit to the 3 global network terms that are not in the model.

flo.03.gof.global <- gof(flomodel.03 ~ degree + esp + distance)
flo.03.gof.global

Goodness-of-fit for degree 

  obs min mean max MC p-value
0   1   0 1.39   6       1.00
1   4   0 3.82  10       1.00
2   2   0 3.92   9       0.48
3   6   0 3.11   7       0.14
4   2   0 2.08   6       1.00
5   0   0 0.84   4       0.94
6   1   0 0.52   3       0.78
7   0   0 0.24   2       1.00
8   0   0 0.07   1       1.00
9   0   0 0.01   1       1.00

Goodness-of-fit for edgewise shared partner 

     obs min  mean max MC p-value
esp0  12   3 12.14  20        1.0
esp1   7   0  5.56  17        0.8
esp2   1   0  1.48   8        1.0
esp3   0   0  0.27   4        1.0
esp4   0   0  0.03   1        1.0

Goodness-of-fit for minimum geodesic distance 

    obs min  mean max MC p-value
1    20  10 19.48  32       1.00
2    35  12 33.09  60       0.90
3    32   7 26.40  41       0.66
4    15   0 11.59  26       0.66
5     3   0  3.87  14       1.00
6     0   0  0.93   8       1.00
7     0   0  0.17   4       1.00
8     0   0  0.02   1       1.00
Inf  15   0 24.45  89       1.00
plot(flo.03.gof.global)