ergm is used to fit linear exponential random network models, in which
the probability of a given network, $y$, on a set of nodes is
$\exp(\theta{\cdot}g(y))/c(\theta)$, where
$g(y)$ is a vector of network statistics,
$\theta$ is a parameter vector of the same length and $c(\theta)$ is the
normalizing constant for the distribution.
ergm can return either a maximum pseudo-likelihood
estimate or an approximate maximum likelihood estimator based on a Monte Carlo scheme.ergm(formula, theta0="MPLE",
MPLEonly=FALSE, MLestimate=!MPLEonly, seed=NULL,
burnin=10000, MCMCsamplesize=10000, interval=100, maxit=3,
constraints=~.,
control=control.ergm(),
verbose=FALSE, ...)theta0="MPLE").TRUE if the maximum
pseudo-likelihood estimate is to be computed and returned.
Note that MPLEonly=TRUE will render moot most other parameters in this list.TRUE if only the Monte Carlo maximum likelihood estimate
is to be computed and returned.formula argument. Multiple constraints
may be given, separated by control.ergm.NULL to
use whatever the state of the random number generator is at the time
of the call.TRUE, the program will print out additional information,
including goodness of fit statistics.ergm returns an object of class ergm that is a list
consisting of the following elements:sample is roughly a random sample from the distribution
of network statistics specified by the model with the parameter equal
to MCMCtheta. If MPLEonly=TRUE then
MCMCtheta equals the MPLE.glm object.degeneracy.value. Mainly
for internal use.formula entered into the ergm function.ergm callergm call
(part of the control.ergm function output).print.ergm for details on how
an ergm object is printed. Note that the
method summary.ergm returns a summary of the
relevant parts of the ergm object in concise summary
format.ergm function allows the user to explore a large number
of potential models for their network data. The
terms currently supported by the program,
and a brief description of each is given in the documentation
ergm-terms.
In the formula for the model, the model terms are various function-like
calls, some of which require arguments, separated by + signs.
For a more detailed understanding of the model terms, see and Morris, Handcock and Hunter (2008).In the implementation of ergm, the model is initialized
in R, then all the model information is passed to a C program
that generates the sample of network statistics using MCMC.
This sample is then returned to R, which implements a
simple Newton-Raphson algorithm to approximate the MLE.
An alternative style of maximum likelihood estimation is to use a stochastic
approximation algorithm. This can be chosen with the
control.ergm(style="Robbins-Monro") option.
The default mechanism for proposing new networks for the MCMC sample
space is the Metropolis-Hastings algorithm, which simply chooses a dyad
at random and proposes to toggle that edge; each possible dyad is
equally likely. The proposaltype
option allow many more complex proposals to be specified.
We have developed and implemented a wide range of algorithms. These are
described in the documentation for proposaltype.For example, we have included proposal functions that condition on
maintaining the absolute degree distribution for the observed
network. Each proposal network will have exactly the same number
of nodes with each degree as does the original network; this
means that if the proposal network removes an edge between a
node of degree 3 and a node of degree 5, it must also add
an edge between a node of degree 2 and a node of degree 4.
Note that one or both of the latter nodes may be the same as
the former nodes.
The package is designed so that the user can add additional proposal types.
constraints term bd.Let's consider the simple cases first. Suppose you want to condition on the total number of degrees regardless of attributes. That is, if you had a survey that asked respondents to name three alters and no more, then you might want to limit your maximal outdegree to three without regard to any of the alters' attributes. The argument is then:
constraints=~bd(maxout=3)
Similar calls are used to restrict the number of indegrees
(maxin), the minimum number of outdegrees
(minout), and the minimum number of indegrees
(minin).
You can also set ego specific limits. For example:
constraints=bd(maxout=rep(c(3,4),c(36,35)))
limits the first 36 to 3 and the other 35 to 4 outdegrees.
Multiple restrictions can be combined. bd is very flexible.
In general, the bd term can contain up to five arguments:
bd(attribs=attribs, maxout=maxout, maxin=maxin, minout=minout, minin=minin)
Omitted arguments are unrestricted, and arguments of length 1
are replicated out to all nodes (as above). If an individual
entry in maxout,..., minin is NA then
no restriction of that kind is applied to that actor.
In general, attribs is a matrix of the attributes on
which we are conditioning. The dimensions of attribs
are n_nodes rows by attrcount columns, where
attrcount is the number of distinct attribute values
on which we want to condition (i.e., a separate column is
required for attribs[n, i], therefore, is TRUE
if node n has attribute value i, and FALSE otherwise.
(Note that, since each column represents only a single value
of a single attribute, the values of this matrix are all
Boolean (TRUE or FALSE).) It is important to
note that attribs is a matrix of nodal attributes,
not alter attributes.
So, for instance, if we wanted to construct an attribs matrix
with two columns, one each for male and female attribute
values (we are conditioning on these values of the attribute
n_node-long vector of 0s and 1s (men and women),
then our code would look as follows:
# male column: bit vector, TRUE for males attrsex1 <- (ads.sex == 0) # female column: bit vector, TRUE for females attrsex2 <- (ads.sex == 1) # now create attribs matrix attribs <- matrix(ncol=2,nrow=71, data=c(attrsex1,attrsex2))
maxout is a matrix of alter attributes, with the same
dimensions as the attribs matrix. maxout is n_nodes
rows by attrcount columns. The value of maxout[n,i],
therefore, is the maximum number of outdegrees permitted
from node n to nodes with the attribute i (where a NA
means there is no maximum).
For example: if we wanted to create a maxout matrix to work
with our attribs matrix above, with a maximum from every
node of five outedges to males and five outedges to females,
our code would look like this:
# every node has maximum of 5 outdegrees to male alters maxoutsex1 <- c(rep(5,71)) # every node has maximum of 5 outdegrees to female alters maxoutsex2 <- c(rep(5,71)) # now create maxout matrix maxout <- cbind(maxoutsex1,maxoutsex2)
The maxin, minout, and minin matrices
are constructed exactly like the maxout matrix,
except for the maximum allowed indegree, the minimum allowed
outdegree, and the minimum allowed indegree, respectively.
Note that in an undirected network, we only look at the outdegree
matrices; maxin and minin will both be ignored
in this case.
attribs[n][0] = 1 # just the ego values
maxout[n][0] = minout[n][0] = observed outdegree of n in network
maxin[n][0] = minin[n][0] = observed indegree of n in network
Bender-deMoll S, Morris M, Moody J (2008).
{Prototype Packages for Managing and Animating Longitudinal
Network Data:
Boer P, Huisman M, Snijders T, Zeggelink E (2003). {StOCNET: an open software system for the advanced statistical analysis of social networks}. Groningen: ProGAMMA / ICS, version 1.4 edition.
Butts CT (2006).
{
Butts CT (2007).
{
Butts CT (2008).
{
Butts CT, with help~from David~Hunter, Handcock MS (2007).
{
Goodreau SM, Handcock MS, Hunter DR, Butts CT, Morris M (2008a).
{A
Goodreau SM, Kitts J, Morris M (2008{{b}}).
{Birds of a Feather, or Friend of a Friend? Using Exponential
Random Graph Models to Investigate Adolescent Social Networks.}
{Demography},
Handcock, M. S. (2003)
Assessing Degeneracy in Statistical Models of Social Networks,
Working Paper #39,
Center for Statistics and the Social Sciences,
University of Washington.
Handcock MS (2003{{b}}).
{
Handcock MS, Hunter DR, Butts CT, Goodreau SM, Morris M (2003{{a}}).
{
Handcock MS, Hunter DR, Butts CT, Goodreau SM, Morris M (2003{{b}}).
{
Hunter, D. R. and Handcock, M. S. (2006) Inference in curved exponential family models for networks, Journal of Computational and Graphical Statistics.
Hunter DR, Handcock MS, Butts CT, Goodreau SM, Morris M (2008{{b}}).
{
Krivitsky PN, Handcock MS (2007).
{
Morris M, Handcock MS, Hunter DR (2008).
{Specification of Exponential-Family Random Graph Models:
Terms and Computational Aspects.}
{Journal of Statistical Software},
# import synthetic network that looks like a molecule data(molecule) # Add a attribute to it to mimic the atomic type molecule %v% "atomic type" <- c(1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3) # # create a plot of the social network # colored by atomic type # plot(molecule, vertex.col="atomic type",vertex.cex=3)
# measure tendency to match within each atomic type gest <- ergm(molecule ~ edges + kstar(2) + triangle + nodematch("atomic type"), MCMCsamplesize=10000) summary(gest)
# compare it to differential homophily by atomic type
gest <- ergm(molecule ~ edges + kstar(2) + triangle
+ nodematch("atomic type",diff=TRUE),
MCMCsamplesize=10000)
summary(gest)