R: Exponential-Family Random Graph Models

ergm {ergm} | R Documentation

## Exponential-Family Random Graph Models

### Description

`ergm` is used to fit exponential-family random graph models (ERGMs), in which the probability of a given network, y, on a set of nodes is h(y) \exp\{η(θ) \cdot g(y)\}/c(θ), where h(y) is the reference measure (usually h(y)=1), g(y) is a vector of network statistics for y, η(θ) is a natural parameter vector of the same length (with η(θ)=θ for most terms), and c(θ) is the normalizing constant for the distribution. `ergm` can return a maximum pseudo-likelihood estimate, an approximate maximum likelihood estimate based on a Monte Carlo scheme, or an approximate contrastive divergence estimate based on a similar scheme. (For an overview of the package, see `ergm-package`.)

### Usage

    ergm(
      formula,
      response = NULL,
      reference = ~Bernoulli,
      constraints = ~.,
      obs.constraints = ~. - observed,
      offset.coef = NULL,
      target.stats = NULL,
      eval.loglik = getOption("ergm.eval.loglik"),
      estimate = c("MLE", "MPLE", "CD"),
      control = control.ergm(),
      verbose = FALSE,
      ...,
      basis = ergm.getnetwork(formula)
    )

    is.ergm(object)

    ## S3 method for class 'ergm'
    nobs(object, ...)

    ## S3 method for class 'ergm'
    print(x, digits = max(3, getOption("digits") - 3), ...)

    ## S3 method for class 'ergm'
    vcov(object, sources = c("all", "model", "estimation"), ...)

### Arguments

`formula` | An R `formula` object, of the form `y ~ <model terms>`, where `y` is a `network` object or a matrix that can be coerced to a `network` object. For the details on the possible `<model terms>`, see `ergmTerm` and Morris, Handcock and Hunter (2008) for binary ERGM terms and Krivitsky (2012) for valued ERGM terms (terms for weighted edges). To create a `network` object in R, use the `network()` function, then add nodal attributes to it using the `%v%` operator if necessary. Enclosing a model term in `offset()` fixes its value to one specified in `offset.coef`. (A second argument—a logical or numeric index vector—can be used to select which of the parameters within the term are offsets.)
`response` | Either a character string, a formula, or `NULL` (the default), to specify the response attributes and whether the ERGM is binary or valued. Interpreted as follows: `NULL` Model simple presence or absence, via a binary ERGM. character string The name of the edge attribute whose value is to be modeled. Type of ERGM will be determined by whether the attribute is `logical` (`TRUE`/`FALSE`) for binary or `numeric` for valued. a formula must be of the form `NAME~EXPR|TYPE` (with `|` being literal). `EXPR` is evaluated in the formula's environment with the network's edge attributes accessible as variables. The optional `NAME` specifies the name of the edge attribute into which the results should be stored, with the default being a concise version of `EXPR`. Normally, the type of ERGM is determined by whether the result of evaluating `EXPR` is logical or numeric, but the optional `TYPE` can be used to override by specifying a scalar of the type involved (e.g., `TRUE` for binary and `1` for valued).
`reference` | A one-sided formula specifying the reference measure (h(y)) to be used. See help for ERGM reference measures implemented in the ergm package.
`constraints` | A formula specifying one or more constraints on the support of the distribution of the networks being modeled, using syntax similar to the `formula` argument, on the right-hand side. Multiple constraints may be given, separated by “+” and “-” operators. (See ERGM constraints for the explanation of their semantics.) Together with the model terms in the formula and the reference measure, the constraints define the distribution of networks being modeled. It is also possible to specify a proposal function directly either by passing a string with the function's name (in which case, arguments to the proposal should be specified through the `prop.args` argument to `control.ergm`) or by giving it on the LHS of the constraints formula, in which case it will override the one chosen automatically. The default is `~.`, for an unconstrained model. See the ERGM constraints documentation for the constraints implemented in the ergm package. Other packages may add their own constraints. Note that not all possible combinations of constraints and reference measures are supported. However, for relatively simple constraints (i.e., those that simply permit or forbid specific dyads or sets of dyads from changing), arbitrary combinations should be possible.
`obs.constraints` | A one-sided formula specifying one or more constraints or other modification in addition to those specified by `constraints` that had affected the observation process for the network, using syntax similar to the `formula` argument. Multiple constraints may be given, separated by “+” operators. This allows the domain of the integral in the numerator of the partially obseved network face-value likelihoods of Handcock and Gile (2010) and Karwa et al. (2017) to be specified explicitly. The default is `~observed`, to constrain the integral to only integrate over the missing dyads. (It is dropped automatically if the network is completely observed.) It is also possible to specify a proposal function directly by passing a string with the function's name. In that case, arguments to the proposal should be specified through the `obs.prop.args` argument to `control.ergm`. See the ERGM constraints documentation for the constraints implemented in the ergm package. Other packages may add their own constraints. Note that not all possible combinations of constraints and reference measures are supported.
`offset.coef` | A vector of coefficients for the offset terms.
`target.stats` | vector of "observed network statistics," if these statistics are for some reason different than the actual statistics of the network on the left-hand side of `formula`. Equivalently, this vector is the mean-value parameter values for the model. If this is given, the algorithm finds the natural parameter values corresponding to these mean-value parameters. If `NULL`, the mean-value parameters used are the observed statistics of the network in the formula.
`eval.loglik` | Logical: For dyad-dependent models, if TRUE, use bridge sampling to evaluate the log-likelihoood associated with the fit. Has no effect for dyad-independent models. Since bridge sampling takes additional time, setting to FALSE may speed performance if likelihood values (and likelihood-based values like AIC and BIC) are not needed. Can be set globally via `option(ergm.eval.loglik=...)`, which is set to `TRUE` when the package is loaded. (See `options?ergm`.)
`estimate` | If "MPLE," then the maximum pseudolikelihood estimator is returned. If "MLE" (the default), then an approximate maximum likelihood estimator is returned. For certain models, the MPLE and MLE are equivalent, in which case this argument is ignored. (To force MCMC-based approximate likelihood calculation even when the MLE and MPLE are the same, see the `force.main` argument of `control.ergm`. If "CD" (EXPERIMENTAL), the Monte-Carlo contrastive divergence estimate is returned. )
`control` | A list of control parameters for algorithm tuning, typically constructed with `control.ergm()`. Its documentation gives the the list of recognized control parameters and their meaning. The more generic utility `snctrl()` (StatNet ConTRoL) also provides argument completion for the available control functions and limited argument name checking.
`verbose` | A logical or an integer to control the amount of progress and diagnostic information to be printed. `FALSE`/`0` produces minimal output, wit higher values producing more detail. Note that very high values (5+) may significantly slow down processing.
`...` | Additional arguments, to be passed to lower-level functions.
`basis` | a value (usually a `network`) to override the LHS of the formula.
`object` | an `ergm` object.
`x, digits` | See `print()`. Automatically called when an object of class `ergm` is printed. Currently, summarizes the size of the MCMC sample, the θ vector governing the selection of the sample, and the Monte Carlo MLE. The optional `digits` argument specifies the significant digits for coefficients
`sources` | For the `vcov` method, specify whether to return the covariance matrix from the ERGM model, the estimation process, or both combined.

### Value

`ergm` returns an object of class `ergm` that is a list consisting of the following elements:

`coef` | The Monte Carlo maximum likelihood estimate of θ, the vector of coefficients for the model parameters.
`sample` | The n\times p matrix of network statistics, where n is the sample size and p is the number of network statistics specified in the model, generated by the last iteration of the MCMC-based likelihood maximization routine. These statistics are centered with respect to the observed statistics or `target.stats`, unless missing data MLE is used.
`sample.obs` | As `sample`, but for the constrained sample.
`iterations` | The number of Newton-Raphson iterations required before convergence.
`MCMCtheta` | The value of θ used to produce the Markov chain Monte Carlo sample. As long as the Markov chain mixes sufficiently well, `sample` is roughly a random sample from the distribution of network statistics specified by the model with the parameter equal to `MCMCtheta`. If `estimate="MPLE"` then `MCMCtheta` equals the MPLE.
`loglikelihood` | The approximate change in log-likelihood in the last iteration. The value is only approximate because it is estimated based on the MCMC random sample.
`gradient` | The value of the gradient vector of the approximated loglikelihood function, evaluated at the maximizer. This vector should be very close to zero.
`covar` | Approximate covariance matrix for the MLE, based on the inverse Hessian of the approximated loglikelihood evaluated at the maximizer.
`failure` | Logical: Did the MCMC estimation fail?
`network` | Network passed on the left-hand side of `formula`. If `target.stats` are passed, it is replaced by the network returned by `san()`.
`newnetworks` | A list of the final networks at the end of the MCMC simulation, one for each thread.
`newnetwork` | The first (possibly only) element of `newnetworks`.
`coef.init` | The initial value of θ.
`est.cov` | The covariance matrix of the model statistics in the final MCMC sample.
`coef.hist, steplen.hist, stats.hist, stats.obs.hist` | For the MCMLE method, the history of coefficients, Hummel step lengths, and average model statistics for each iteration..
`control` | The control list passed to the call.
`etamap` | The set of functions mapping the true parameter theta to the canonical parameter eta (irrelevant except in a curved exponential family model)
`formula` | The original `formula` entered into the `ergm` function.
`target.stats` | The target.stats used during estimation (passed through from the Arguments)
`target.esteq` | Used for curved models to preserve the target mean values of the curved terms. It is identical to target.stats for non-curved models.
`constrained` | The list of constraints implied by the constraints used by original `ergm` call
`constraints` | Constraints used during estimation (passed through from the Arguments)
`reference` | The reference measure used during estimation (passed through from the Arguments)
`estimate` | The estimation method used (passed through from the Arguments).
`offset` | vector of logical telling which model parameters are to be set at a fixed value (i.e., not estimated).
`drop` | If `control$drop=TRUE`, a numeric vector indicating which terms were dropped due to to extreme values of the corresponding statistics on the observed network, and how: `0` The term was not dropped. `-1` The term was at its minimum and the coefficient was fixed at `-Inf`. `+1` The term was at its maximum and the coefficient was fixed at `+Inf`.
`estimable` | A logical vector indicating which terms could not be estimated due to a `constraints` constraint fixing that term at a constant value.
`null.lik` | Log-likelihood of the null model. Valid only for unconstrained models.
`mle.lik` | The approximate log-likelihood for the MLE. The value is only approximate because it is estimated based on the MCMC random sample.

### Methods (by generic)

- `nobs`: Return the number of informative dyads of a model fit.

- `print`:

- `vcov`: extracts the variance-covariance matrix of parameter estimates.

### Notes on model specification

Although each of the statistics in a given model is a summary statistic for the entire network, it is rarely necessary to calculate statistics for an entire network in a proposed Metropolis-Hastings step. Thus, for example, if the triangle term is included in the model, a census of all triangles in the observed network is never taken; instead, only the change in the number of triangles is recorded for each edge toggle.

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 mechanism for proposing new networks for the MCMC sampling scheme, which is a Metropolis-Hastings algorithm, depends on two things: The `constraints`, which define the set of possible networks that could be proposed in a particular Markov chain step, and the weights placed on these possible steps by the proposal distribution. The former may be controlled using the `constraints` argument described above. The latter may be controlled using the `prop.weights` argument to the `control.ergm` function.

The package is designed so that the user could conceivably add additional proposal types.

### References

Admiraal R, Handcock MS (2007). networksis: Simulate bipartite graphs with fixed marginals through sequential importance sampling. Statnet Project, Seattle, WA. Version 1. https://statnet.org.

Bender-deMoll S, Morris M, Moody J (2008). Prototype Packages for Managing and Animating Longitudinal Network Data: dynamicnetwork and rSoNIA. Journal of Statistical Software, 24(7). doi: 10.18637/jss.v024.i07

Butts CT (2007). sna: Tools for Social Network Analysis. R package version 2.3-2. https://cran.r-project.org/package=sna.

Butts CT (2008). network: A Package for Managing Relational Data in R. Journal of Statistical Software, 24(2). doi: 10.18637/jss.v024.i02

Butts C (2015). network: The Statnet Project (https://statnet.org). R package version 1.12.0, https://cran.r-project.org/package=network.

Goodreau SM, Handcock MS, Hunter DR, Butts CT, Morris M (2008a). A statnet Tutorial. Journal of Statistical Software, 24(8). doi: 10.18637/jss.v024.i08

Goodreau SM, Kitts J, Morris M (2008b). Birds of a Feather, or Friend of a Friend? Using Exponential Random Graph Models to Investigate Adolescent Social Networks. Demography, 45, in press.

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. https://csss.uw.edu/research/working-papers/assessing-degeneracy-statistical-models-social-networks

Handcock MS (2003b). degreenet: Models for Skewed Count Distributions Relevant to Networks. Statnet Project, Seattle, WA. Version 1.0, https://statnet.org.

Handcock MS and Gile KJ (2010). Modeling Social Networks from Sampled Data. Annals of Applied Statistics, 4(1), 5-25. doi: 10.1214/08-AOAS221

Handcock MS, Hunter DR, Butts CT, Goodreau SM, Morris M (2003a). ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks. Statnet Project, Seattle, WA. Version 2, https://statnet.org.

Handcock MS, Hunter DR, Butts CT, Goodreau SM, Morris M (2003b). statnet: Software Tools for the Statistical Modeling of Network Data. Statnet Project, Seattle, WA. Version 2, https://statnet.org.

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 (2008b). ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks. Journal of Statistical Software, 24(3). doi: 10.18637/jss.v024.i03

Karwa V, Krivitsky PN, and Slavkovi\'c AB (2017). Sharing Social Network Data: Differentially Private Estimation of Exponential-Family Random Graph Models. Journal of the Royal Statistical Society, Series C, 66(3):481–500. doi: 10.1111/rssc.12185

Krivitsky PN (2012). Exponential-Family Random Graph Models for Valued Networks. Electronic Journal of Statistics, 2012, 6, 1100-1128. doi: 10.1214/12-EJS696

Morris M, Handcock MS, Hunter DR (2008). Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects. Journal of Statistical Software, 24(4). doi: 10.18637/jss.v024.i04

Snijders, T.A.B. (2002), Markov Chain Monte Carlo Estimation of Exponential Random Graph Models. Journal of Social Structure. Available from https://www.cmu.edu/joss/content/articles/volume3/Snijders.pdf.

### See Also

`network`, `%v%`, `%n%`, `ergmTerm`, `ergmMPLE`, `summary.ergm()`

### Examples

    #
    # load the Florentine marriage data matrix
    #
    data(flo)
    #
    # attach the sociomatrix for the Florentine marriage data
    # This is not yet a network object.
    #
    flo
    #
    # Create a network object out of the adjacency matrix
    #
    flomarriage <- network(flo,directed=FALSE)
    flomarriage
    #
    # print out the sociomatrix for the Florentine marriage data
    #
    flomarriage[,]
    #
    # create a vector indicating the wealth of each family (in thousands of lira)
    # and add it as a covariate to the network object
    #
    flomarriage %v% "wealth" <- c(10,36,27,146,55,44,20,8,42,103,48,49,10,48,32,3)
    flomarriage
    #
    # create a plot of the social network
    #
    plot(flomarriage)
    #
    # now make the vertex size proportional to their wealth
    #
    plot(flomarriage, vertex.cex=flomarriage %v% "wealth" / 20, main="Marriage Ties")
    #
    # Use 'data(package = "ergm")' to list the data sets in a
    #
    data(package="ergm")
    #
    # Load a network object of the Florentine data
    #
    data(florentine)
    #
    # Fit a model where the propensity to form ties between
    # families depends on the absolute difference in wealth
    #
    gest <- ergm(flomarriage ~ edges + absdiff("wealth"))
    summary(gest)
    #
    # add terms for the propensity to form 2-stars and triangles
    # of families
    #
    gest <- ergm(flomarriage ~ kstar(1:2) + absdiff("wealth") + triangle)
    summary(gest)

    # 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"))
    summary(gest)

    # compare it to differential homophily by atomic type
    gest <- ergm(molecule ~ edges + kstar(2) + triangle
                            + nodematch("atomic type",diff=TRUE))
    summary(gest)

    # Extract parameter estimates as a numeric vector:
    coef(gest)
    # Sources of variation in parameter estimates:
    vcov(gest, sources="model")
    vcov(gest, sources="estimation")
    vcov(gest, sources="all") # the default

[Package ergm version 4.2.1 Index]