Last updated 07-01-2014

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 3.1.2 and network 1.9.0 (as of 6/17/2014)
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 edge. 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 
summary(flomodel.01) # The fitted model object

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

Formula:   flomarriage ~ edges

Iterations:  20 

Monte Carlo MLE Results:
      Estimate Std. Error MCMC % p-value    
edges   -1.609      0.245     NA  <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 166  on 120  degrees of freedom
 Residual Deviance: 108  on 119  degrees of freedom
 
AIC: 110    BIC: 113    (Smaller is better.) 

plot of chunk unnamed-chunk-10

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) 
Iteration 1 of at most 20: 
Convergence test P-value: 1.6e-29 
The log-likelihood improved by 0.001997 
Iteration 2 of at most 20: 
Convergence test P-value: 7e-09 
The log-likelihood improved by 0.0005728 
Iteration 3 of at most 20: 
Convergence test P-value: 1.4e-02 
The log-likelihood improved by 0.0001287 
Iteration 4 of at most 20: 
Convergence test P-value: 2.2e-01 
The log-likelihood improved by < 0.0001 
Iteration 5 of at most 20: 
Convergence test P-value: 8.9e-01 
Convergence detected. Stopping.
The log-likelihood improved by < 0.0001 

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:  20 

Monte Carlo MLE Results:
         Estimate Std. Error MCMC % p-value    
edges      -1.675      0.346      0  <1e-04 ***
triangle    0.159      0.578      0    0.78    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 166  on 120  degrees of freedom
 Residual Deviance: 108  on 118  degrees of freedom
 
AIC: 112    BIC: 118    (Smaller is better.) 

Now, how should we interpret coefficients?

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

\[ \small{ -1.68\times\mbox{change in the number of ties}+0.16\times\mbox{change in number of triangles} } \]


  • For a tie that will create no triangles, the conditional log-odds is: \(-1.68\).
  • if one triangle: \(-1.68 + 0.16 =-1.52\)
  • if two triangles: \(-1.68 +0.16\times2=-1.36\)
  • 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"       "mc.se"         "network"      
[13] "newnetwork"    "coef.init"     "initialfit"    "coef.hist"    
[17] "stats.hist"    "etamap"        "formula"       "target.stats" 
[21] "constrained"   "constraints"   "control"       "reference"    
[25] "estimate"      "offset"        "drop"          "estimable"    
[29] "null.lik"      "mle.lik"      
flomodel.02$coef # you can extract/inspect individual components
   edges triangle 
 -1.6749   0.1591 

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. 
    3.0    17.5    39.0    42.6    48.2   146.0 
plot(flomarriage, vertex.cex=wealth/25, main="Florentine marriage by wealth", cex.main=0.8) # network plot with vertex size proportional to wealth

plot of chunk unnamed-chunk-14

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

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

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

Iterations:  20 

Monte Carlo MLE Results:
               Estimate Std. Error MCMC % p-value    
edges          -2.59493    0.53606     NA  <1e-04 ***
nodecov.wealth  0.01055    0.00467     NA   0.026 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 166  on 120  degrees of freedom
 Residual Deviance: 103  on 118  degrees of freedom
 
AIC: 107    BIC: 113    (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)

plot of chunk unnamed-chunk-16

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.
summary(fauxmodel.01)

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

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

Iterations:  20 

Monte Carlo MLE Results:
                     Estimate Std. Error MCMC % p-value    
edges                 -6.2343     0.1741     NA  <1e-04 ***
nodematch.Grade.7      2.8749     0.1981     NA  <1e-04 ***
nodematch.Grade.8      2.8785     0.2391     NA  <1e-04 ***
nodematch.Grade.9      2.4621     0.2647     NA  <1e-04 ***
nodematch.Grade.10     2.5703     0.3770     NA  <1e-04 ***
nodematch.Grade.11     3.2930     0.2977     NA  <1e-04 ***
nodematch.Grade.12     3.8384     0.4592     NA  <1e-04 ***
nodematch.Race.Black     -Inf         NA     NA      NA    
nodematch.Race.Hisp    0.0692     0.1737     NA   0.690    
nodematch.Race.NatAm   0.9832     0.1842     NA  <1e-04 ***
nodematch.Race.Other     -Inf         NA     NA      NA    
nodematch.Race.White   1.2694     0.5370     NA   0.018 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 28987  on 20910  degrees of freedom
 Residual Deviance:  1899  on 20898  degrees of freedom
 
AIC: 1923    BIC: 2018    (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] "metadata"       "samplk1"        "samplk2"        "samplk3"       
[13] "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)

plot of chunk unnamed-chunk-20

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)
Iteration 1 of at most 20: 
Convergence test P-value: 8.9e-01 
Convergence detected. Stopping.
The log-likelihood improved by < 0.0001 

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:  20 

Monte Carlo MLE Results:
       Estimate Std. Error MCMC % p-value    
edges    -2.155      0.218      0  <1e-04 ***
mutual    2.299      0.480      0  <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 424  on 306  degrees of freedom
 Residual Deviance: 268  on 304  degrees of freedom
 
AIC: 272    BIC: 279    (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 = 0.06667 

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)

plot of chunk unnamed-chunk-22

summary(missnet~edges)
edges 
    3 
summary(ergm(missnet~edges))

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

Formula:   missnet ~ edges

Iterations:  20 

Monte Carlo MLE Results:
      Estimate Std. Error MCMC % p-value    
edges   -2.565      0.599     NA 0.00011 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 58.2  on 42  degrees of freedom
 Residual Deviance: 21.6  on 41  degrees of freedom
 
AIC: 23.6    BIC: 25.4    (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 = 0.06667 

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))

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

Formula:   missnet_bad ~ edges

Iterations:  20 

Monte Carlo MLE Results:
      Estimate Std. Error MCMC % p-value    
edges   -2.639      0.598     NA  <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 62.4  on 45  degrees of freedom
 Residual Deviance: 22.0  on 44  degrees of freedom
 
AIC: 24    BIC: 25.9    (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 
      -2.59493        0.01055 

Stored network statistics:
      edges nodecov.wealth
 [1,]    23           2327
 [2,]    19           1975
 [3,]    15           1731
 [4,]    14           1766
 [5,]    26           2534
 [6,]    23           2618
 [7,]    22           2341
 [8,]    21           2389
 [9,]    24           2661
[10,]    26           2647
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= 23 
    missing edges= 0 
    non-missing edges= 23 

 Vertex attribute names: 
    priorates totalties vertex.names wealth 

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

plot of chunk unnamed-chunk-25

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

ERGMs can be seen as generative models when they represent the process that governs the global patterns of tie prevalence from a local perspective: the perspective of the nodes involved in the particular micro-configurations represented by the ergm-terms in the model. The locally generated processes in turn aggregate up to produce characteristic global network properties, even though these global properties are not explicit terms in the model.

One test of whether a local model “fits the data” is therefore how well it reproduces the observed global network properties that are not in the model. We do this by choosing a network statistic that is not in the model, and comparing the value of this statistic observed in the original network to the distribution of values we get in simulated networks from our model, using the gof function.

The gof function is a bit different than the summary, ergm, and simulate functions, in that it currently only takes 3 ergm-terms as arguments: degree, esp (edgwise share partners), and distance (geodesic distances). Each of these terms captures an aggregate network distribution, at either the node level (degree), the edge level (esp), or the dyad level (distance).

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

Goodness-of-fit for degree 

  obs min mean max MC p-value
0   1   0 1.55   6       1.00
1   4   0 3.28   7       0.90
2   2   0 4.19  10       0.40
3   6   0 3.19   7       0.08
4   2   0 2.08   6       1.00
5   0   0 0.94   3       0.88
6   1   0 0.43   2       0.68
7   0   0 0.24   2       1.00
8   0   0 0.09   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   5 11.64  19       1.00
esp1   7   0  6.44  17       0.96
esp2   1   0  1.50  10       1.00
esp3   0   0  0.07   1       1.00
esp4   0   0  0.01   1       1.00

Goodness-of-fit for minimum geodesic distance 

    obs min  mean max MC p-value
1    20   7 19.66  31       1.00
2    35   5 33.17  64       0.90
3    32   2 25.10  43       0.50
4    15   0 10.13  24       0.50
5     3   0  3.03  17       0.82
6     0   0  0.88  11       1.00
7     0   0  0.21   6       1.00
8     0   0  0.05   2       1.00
Inf  15   0 27.77 103       0.96
plot(flomodel.03.gof)

plot of chunk unnamed-chunk-26plot of chunk unnamed-chunk-26plot of chunk unnamed-chunk-26

mesamodel.02 <- ergm(mesa~edges)
mesamodel.02.gof <- gof(mesamodel.02~degree + esp + distance, nsim=10)
plot(mesamodel.02.gof)

plot of chunk unnamed-chunk-27plot of chunk unnamed-chunk-27plot of chunk unnamed-chunk-27

For a good example of model exploration and fitting for the Add Health Friendship networks, see Goodreau, Kitts & Morris, Demography 2009.
For more technical details on the approach, see Hunter, Goodreau and Handcock JASA 2008

6. Diagnostics: troubleshooting and checking for model degeneracy

The computational algorithms in ergm use MCMC to estimate the likelihood function when dyad dependent terms are in the model. Part of this process involves simulating a set of networks to use as a sample for approximating the unknown component of the likelihood: the \(k(\theta)\) term in the denominator.

When a model is not a good representation of the observed network, these simulated networks may be far enough away from the observed network that the estimation process is affected. In the worst case scenario, the simulated networks will be so different that the algorithm fails altogether.

For more detailed discussion of model degeneracy in the ERGM context, see the papers by Mark Handcock referenced below.

In the worst case scenario, we end up not being able to obtain coefficent estimates, so we can’t use the GOF function to identify how the model simulations deviate from the observed data. In this case, however, we can use the MCMC diagnostics to observe what is happening with the simulation algorithm, and this (plus some experience and intuition about the behavior of ergm-terms) can help us improve the model specification.

Below we show a simple example of a model that converges, and one that doesn’t, and how to use the MCMC diagnostics to improve a model that isn’t converging.

What it looks like when a model converges properly

We will first consider a simulation where the algorithm works using the program defaults, and observe the behavior of the MCMC estimation algorithm using the mcmc.diagnostics function.

summary(flobusiness~edges+degree(1))
  edges degree1 
     15       3 
fit <- ergm(flobusiness~edges+degree(1))
Iteration 1 of at most 20: 
Convergence test P-value: 0e+00 
The log-likelihood improved by 0.06583 
Iteration 2 of at most 20: 
Convergence test P-value: 4e-248 
The log-likelihood improved by 0.01558 
Iteration 3 of at most 20: 
Convergence test P-value: 4.8e-49 
The log-likelihood improved by 0.00338 
Iteration 4 of at most 20: 
Convergence test P-value: 3.9e-09 
The log-likelihood improved by 0.0005943 
Iteration 5 of at most 20: 
Convergence test P-value: 6.6e-05 
The log-likelihood improved by 0.0002866 
Iteration 6 of at most 20: 
Convergence test P-value: 3.5e-01 
The log-likelihood improved by < 0.0001 
Iteration 7 of at most 20: 
Convergence test P-value: 2.9e-01 
The log-likelihood improved by < 0.0001 
Iteration 8 of at most 20: 
Convergence test P-value: 1.3e-01 
The log-likelihood improved by < 0.0001 
Iteration 9 of at most 20: 
Convergence test P-value: 2.6e-01 
The log-likelihood improved by < 0.0001 
Iteration 10 of at most 20: 
Convergence test P-value: 8.6e-01 
Convergence detected. Stopping.
The log-likelihood improved by < 0.0001 

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

Iterations = 10000:1009900
Thinning interval = 100 
Number of chains = 1 
Sample size per chain = 10000 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

           Mean   SD Naive SE Time-series SE
edges   -0.0147 3.74   0.0374         0.0408
degree1  0.0086 1.63   0.0163         0.0164

2. Quantiles for each variable:

        2.5% 25% 50% 75% 97.5%
edges     -7  -3   0   3     7
degree1   -3  -1   0   1     3


Are sample statistics significantly different from observed?
             edges degree1 Overall (Chi^2)
diff.      -0.0147  0.0086              NA
test stat. -0.3604  0.5243          0.2964
P-val.      0.7185  0.6001          0.8623

Sample statistics cross-correlations:
          edges degree1
edges    1.0000 -0.4357
degree1 -0.4357  1.0000

Sample statistics auto-correlation:
Chain 1 
            edges   degree1
Lag 0    1.000000  1.000000
Lag 100  0.087320  0.016873
Lag 200  0.008309  0.006461
Lag 300 -0.003954  0.008045
Lag 400 -0.012858 -0.023378
Lag 500  0.008232 -0.003913

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

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

  edges degree1 
-0.1701  1.7295 

P-values (lower = worse):
  edges degree1 
0.86495 0.08372 
Loading required package: latticeExtra
Loading required package: RColorBrewer

plot of chunk unnamed-chunk-28

This is what you want to see in the MCMC diagnostics: the MCMC sample statistics are varying randomly around the observed values at each step (so the chain is “mixing” well) and the difference between the observed and simulated values of the sample statistics have a roughly bell-shaped distribution, centered at 0. The sawtooth pattern visible on the degree term deviation plot is due to the combination of discrete values and small range in the statistics: the observed number of degree 1 nodes is 3, and only a few discrete values are produced by the simulations. So the sawtooth pattern is is an inherent property of the statistic, not a problem with the fit.

There are many control parameters for the MCMC algorithm (“help(control.ergm)”), and we’ll play with some of these below. To see what the algorithm is doing at each step, you can drop the sampling interval down to 1:

fit <- ergm(flobusiness~edges+degree(1), 
control=control.ergm(MCMC.interval=1)

This runs a version with every network returned, and might be useful if you are trying to debug a bad model fit.

What it looks like when a model fails

Now let us look at a more problematic case, using a larger network:

data('faux.magnolia.high')
magnolia <- faux.magnolia.high
plot(magnolia, vertex.cex=.5)