MEclustnet: A package for model-based clustering of nodes in a network, accounting for covariates.

Description

The main function of interest is MEclustnet which will fit a mixture of experts latent position cluster model to a binary network.

MEclustnet will fit a mixture of experts latent position cluster model to a binary network.

Usage

MEclustnet(Y, covars, link.vars = c(1:ncol(covars)),
  mix.vars = c(1:ncol(covars)), G = 2, d = 2, itermax = 10000,
  uphill = 100, burnin = 1000, thin = 10, rho.input = 1,
  verbose = TRUE, ...)

Arguments

Details

This function fits the mixture of experts latent position cluster model to a binary network via a Metropolis-within-Gibbs sampler. Covariates can influence either the link probabilities between nodes and/or the cluster memberships of nodes.

Value

An object of class MEclustnet, which is a list containing:

zstore

An n x d x store.dim array of sampled latent location matrices, where store.dim is the number of post burnin thinned iterations.

betastore

A store.dim x p matrix of sampled beta vectors, the logistic regression parameters of the link probabilities model.

Kstore

A store.dim x n matrix of sampled cluster membership vectors.

mustore

A G x d x store.dim array of sampled cluster mean latent location matrices.

sigma2store

A store.dim x G matrix of sampled cluster variances.

lambdastore

An n x G x store.dim array of sampled mixing proportion matrices.

taustore

A G x s x store.dim array of sampled tau vectors, the logistic regression parameters of the mixing proportions model, where s is the length of tau.

LLstore

A vector of length store.dim storing the loglikelihood from each stored iteration.

G

The number of clusters fitted

d

The dimension of the latent space

countbeta

Count of accepted beta values

counttau

Count of accepted tau values

MEclustnet functions

MEclustnet

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Examples

#################################################################
# An example from the Gormley and Murphy (2010) paper, using the Lazega lawyers friendship network.
#################################################################
# Number of iterations etc. are set to low values for illustrative purposes.
# Longer run times are likely to be required to achieve sufficient mixing.

library(latentnet)
data(lawyers.adjacency.friends)
data(lawyers.covariates)

link.vars = c(1)
mix.vars = c(1,4,5)

fit = MEclustnet(lawyers.adjacency.friends, lawyers.covariates,
link.vars, mix.vars, G=2, d=2, itermax = 500, burnin = 50, uphill = 1, thin=10)

# Plot the trace plot of the mean of dimension 1 for each cluster.
matplot(t(fit$mustore[,1,]), type="l", xlab="Iteration", ylab="Parameter")

# Compute posterior summaries
summ = summaryMEclustnet(fit, lawyers.adjacency.friends)
plot(summ$zmean, col=summ$Kmode, xlab="Dimension 1", ylab="Dimension 2", pch=summ$Kmode,
     main = "Posterior mean latent location for each node.")

# Plot the resulting latent space, with uncertainties
plotMEclustnet(fit, lawyers.adjacency.friends, link.vars, mix.vars)

#################################################################
# An example analysing a 2016 Twitter network of US politicians.
#################################################################
# Number of iterations etc. are set to low values for illustrative purposes.
# Longer run times are likely to be required to achieve sufficient mixing.

library(latentnet)
data(us.twitter.adjacency)
data(us.twitter.covariates)

link.vars = c(1)
mix.vars = c(1,5,7,8)

fit = MEclustnet(us.twitter.adjacency, us.twitter.covariates,
link.vars, mix.vars, G=4, d=2, itermax = 500, burnin = 50, uphill = 1, thin=10)

# Plot the trace plot of the mean of dimension 1 for each cluster.
matplot(t(fit$mustore[,1,]), type="l", xlab="Iteration", ylab="Parameter")

# Compute posterior summaries
summ = summaryMEclustnet(fit, us.twitter.adjacency)

plot(summ$zmean, col=summ$Kmode, xlab="Dimension 1", ylab="Dimension 2", pch=summ$Kmode,
     main = "Posterior mean latent location for each node.")

# Plot the resulting latent space, with uncertainties
plotMEclustnet(fit, us.twitter.adjacency, link.vars, mix.vars)

# Examine which politicians are in which clusters...
clusters = list()
for(g in 1:fit$G)
{
  clusters[[g]] = us.twitter.covariates[summ$Kmode==g,c("name", "party")]
}
clusters

Title Compute mixing proportions

Description

Function to compute the each observation's mixing proportions which are modeled as a logistic function of their covariates.

Usage

calclambda(tau, x.mix)

Arguments

Value

An n x G matrix of mixing proportions.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Calculate the log likelihood function of the data.

Description

This function calculates the log likelihood function of the data.

Usage

calcloglikelihood(pis, y)

Arguments

Value

The value of the log likelihood function.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Totals the number of observations in each cluster.

Description

Update the count of the number of observations in each cluster.

Usage

calcm(m, G, K)

Arguments

Value

Vector of length G containing the number of nodes in each cluster.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Calculate link probabilities.

Description

Function calculates link probabilities between nodes.

Usage

calcpis(beta, x.link, delta, n.tilde)

Arguments

Value

A vector of length n^2 - n providing the link probabilities between all pairs of nodes.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Reformat matrix of covariates.

Description

This function reformats the matrix of input covariates into the required format for the link probabilities and for the mixing proportions.

Usage

formatting.covars(covars, link.vars, mix.vars, n)

Arguments

Details

For the link regression model, the difference in the link.vars covariates, for all pairs of nodes is calculated. For the mixing proportions model, the required representation of the mix.vars required is formed, where for categorical/factor variables a dummy value representation is used.

Value

A list with

x.link

A matrix with n^2 rows and length(link.vars) columns, detailing the differences in covariates for all pairs of nodes.

x.mix

A matrix with n rows and number of columns equal to the number of variables detailed in mix.vars, where dummy variable representations will be used for categorical.factor covariates.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Examples

data(us.twitter.covariates)
link.vars = c(1)
mix.vars = c(1,5,7,8)
res = formatting.covars(us.twitter.covariates, link.vars, mix.vars, nrow(us.twitter.covariates))
dim(res$x.link)
dim(res$x.mix)

Account for invariance of configurations.

Description

This function accounts for the fact that configurations in the latent space are invariant to rotations, reflections and translations.

Usage

invariant(z, zMAP)

Arguments

Details

Procrustean rotations, reflections and translations (note: NOT dilations) are employed to best match z to zMAP.

Value

The transformed version of the input configuration z that best matches zMAP.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Label switching correction.

Description

This function corrects for the issue of label switching when fitting mixture models in a Bayesian setting.

Usage

labelswitch(mu, sigma2, lambda, tau, K, G, d, perms, muMAP, iter, uphill,
  burnin, thin, s, x.mix)

Arguments

Details

The muMAP matrix is used as the reference to which each new estimate the cluster means is matched to correct for any label switching which may have occurred during sampling. A sum of squares function is employed as the loss function.

Value

A list containing:list(mu, sigma2, lambda, tau, K)

mu

The label-corrected matrix of cluster means.

sigma2

The label-corrected vector of cluster covariances.

lambda

The label-corrected matrix of mixing proportions.

tau

The label-corrected matrix of logistic regression coefficients for the mixing proportions model.

K

The label-corrected vector of length n detailing the number of the cluster to which each node belongs.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Adjacency matrix detailing the presence or absence of advice links between the ‘Lazega Lawyers’.

Description

Data on whether or not 71 lawyers in a northeastern American law firm asked each other for advice.

Usage

lawyers.adjacency.advice

Format

A 71 x 71 binary matrix, with 0 down the diagonal.

Source

E. Lazega, The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership, Oxford University Press, Oxford, England, 2001.

Adjacency matrix detailing the presence or absence of coworker links between the ‘Lazega Lawyers’.

Description

Data on whether or not 71 lawyers in a northeastern American law firm work with each other.

Usage

lawyers.adjacency.coworkers

Format

A 71 x 71 binary matrix, with 0 down the diagonal.

Source

E. Lazega, The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership, Oxford University Press, Oxford, England, 2001.

Adjacency matrix detailing the presence or absence of friendship links between the ‘Lazega Lawyers’.

Description

Data on whether or not 71 lawyers in a northeastern American law firm are friends outside of work.

Usage

lawyers.adjacency.friends

Format

A 71 x 71 binary matrix, with 0 down the diagonal.

Source

E. Lazega, The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership, Oxford University Press, Oxford, England, 2001.

A matrix of covariates of the ‘Lazega Lawyers’.

Description

Covariates on each of 71 lawyers in a northeastern American law firm. Note the first column is a column of 1's.

Usage

lawyers.covariates

Format

A data frame with 71 observations on the following 8 variables.

Intercept

a column of 1s should always be the first column.

Seniority

a factor with levels 1 = partner, 2 = associate.

Gender

a factor with 1 = male, 2 = female.

Office

a factor with levels 1 = Boston, 2 = Hartford and 3 = Providence

Years

a numeric vector detailing years with the firm.

Age

a numeric vector detailing the age of each lawyer.

Practice

a factor with levels 1 = litigation and 2 = corporate.

School

a factor with levels 1 = Harvard or Yale, 2 = University of Connecticut and 3 = Other.

Source

E. Lazega, The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership, Oxford University Press, Oxford, England, 2001.

Plot latent position network.

Description

Function to plot the resulting fitted network, using first two dimensions only.

Usage

plotMEclustnet(fit, Y, link.vars, mix.vars)

Arguments

Details

This function will plot the posterior mean latent location for each node in the network. The colour of each node reflects the posterior modal cluster membership, and the ellipses are 50% posterior sets illustrating the uncertainty in the latent locations. The grey lines illustrate the observed links between the nodes.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Examples

#################################################################
# An example analysing a 2016 Twitter network of US politicians.
#################################################################
# Number of iterations etc. are set to low values for illustrative purposes.
# Longer run times are likely to be required to achieve sufficient mixing.

library(latentnet)
data(us.twitter.adjacency)
data(us.twitter.covariates)

link.vars = c(1)
mix.vars = c(1,5,7,8)

fit = MEclustnet(us.twitter.adjacency, us.twitter.covariates,
           link.vars, mix.vars, G=4, d=2, itermax = 500, burnin = 50, uphill = 1, thin=10)

# Plot the trace plot of the mean of dimension 1 for each cluster.
matplot(t(fit$mustore[,1,]), type="l", xlab="Iteration", ylab="Parameter")

# Compute posterior summaries
summ = summaryMEclustnet(fit, us.twitter.adjacency)

plot(summ$zmean, col=summ$Kmode, xlab="Dimension 1", ylab="Dimension 2", pch=summ$Kmode,
     main = "Posterior mean latent location for each node.")

# Plot the resulting latent space, with uncertainties
plotMEclustnet(fit, us.twitter.adjacency, link.vars, mix.vars)

# Examine which politicians are in which clusters...
clusters = list()
for(g in 1:fit$G)
{
  clusters[[g]] = us.twitter.covariates[summ$Kmode==g,c("name", "party")]
}
clusters

Summary of MEclustnet object.

Description

Summary of the output of the function MEclustnet which fits a mixture of experts latent position cluster model.

Usage

summaryMEclustnet(fit, Y)

Arguments

Value

A list with:

AICM

The value of the AICM criterion for the fitted model.

BICM

The value of the BICM criterion for the fitted model.

BICMCMC

The value of the BICMCMC criterion for the fitted model.

betamean

The posterior mean vector of the regression coefficients for the link probabilities model.

betasd

The standard deviation of the posterior distribution of beta.

taumean

A matrix with G rows, detailing the posterior mean of the regression coefficients for the mixing proportions model.

tausd

The standard deviation of the posterior distribution of tau.

mumean

A G x d matrix containing the posterior mean of the latent locations' mean.

meansd

The standard deviation of the posterior distribution of mu.

sigma2mean

A vector of length G containing the posterior mean of the latent locations' covariance.

sigma2sd

The standard deviation of the posterior distribution of the latent locations' covariance.

Kmode

A vector of length n detailing the posterior modal cluster membership for each node.

zmean

An n x d matrix containing the posterior mean latent location for each node.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Examples

#################################################################
# An example analysing a 2016 Twitter network of US politicians.
#################################################################
# Number of iterations etc. are set to low values for illustrative purposes.
# Longer run times are likely to be required to achieve sufficient mixing.

library(latentnet)
data(us.twitter.adjacency)
data(us.twitter.covariates)

link.vars = c(1)
mix.vars = c(1,5,7,8)

fit = MEclustnet(us.twitter.adjacency, us.twitter.covariates,
 link.vars, mix.vars, G=4, d=2, itermax = 500, burnin = 50, uphill = 1, thin=10)

# Plot the trace plot of the mean of dimension 1 for each cluster.
matplot(t(fit$mustore[,1,]), type="l", xlab="Iteration", ylab="Parameter")

# Compute posterior summaries
summ = summaryMEclustnet(fit, us.twitter.adjacency)

plot(summ$zmean, col=summ$Kmode, xlab="Dimension 1", ylab="Dimension 2", pch=summ$Kmode,
     main = "Posterior mean latent location for each node.")

# Plot the resulting latent space, with uncertainties
plotMEclustnet(fit, us.twitter.adjacency, link.vars, mix.vars)

# Examine which politicians are in which clusters...
clusters = list()
for(g in 1:fit$G)
{
  clusters[[g]] = us.twitter.covariates[summ$Kmode==g,c("name", "party")]
}
clusters

Update the cluster membership vector.

Description

A Gibbs update step for K, the cluster membership vector.

Usage

updateK(G, K, z, mu, sigma2, Id, lambda)

Arguments

Value

The cluster membership vector.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Update the logistic regression parameters in the link probabilities model.

Description

The Metropolis-Hastings update step for the logistic regression parameters in the link probabilities model, using a surrogate proposal distribution.

Usage

updatebeta(beta, p, x.link, delta, y, epsilon, psi, psi.inv, pis,
  countbeta, rho, n.tilde)

Arguments

Details

See appendix of the paper detailed below for details.

Value

A list:

beta

The returned version of the beta parameter vector.

countbeta

The count of the number of acceptances of beta to that point in the MCMC chain.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Update the mean of each cluster.

Description

A Gibbs step to update the mean of each cluster.

Usage

updatemu(G, z, K, m, sigma2, omega2, Id, mu, d)

Arguments

Value

The G x d matrix of cluster means.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Update variances in each cluster.

Description

A Gibbs step to update variances in each cluster.

Usage

updatesigma2(G, alpha, m, d, sigma02, z, K, mu, sigma2)

Arguments

Value

The G vector of cluster variances.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Update the logistic regression parameters in the mixing proportions model.

Description

The Metropolis-Hastings update step for the logistic regression parameters in the mixing proportions model, using a surrogate proposal distribution.

Usage

updatetau(G, x.mix, lambda, Sigmag, Sigmag.inv, K, gammag, tau, counttau,
  rho)

Arguments

Value

A list:

tau

The returned version of the tau parameter vector.

lambda

The returned version of the lambda matrix.

counttau

The count of the number of acceptances of tau to that point in the MCMC chain.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Update step for the latent locations.

Description

A Metropolis-Hastings update step for the latent locations.

Usage

updatez(n, z, x.link, delta, beta, y, mu, K, sigma2, Id, pis, iter, uphill,
  countz, delete, d, n.tilde)

Arguments

Value

A list:

z

The returned matrix of latent locations.

delta

Vector of Euclidean distances between locations in the latent space of all pairs of nodes.

pis

A vector of length n^2 - n providing the link probabilities between all pairs of nodes.

countz

Counter for z acceptance rate.

References

Isobel Claire Gormley and Thomas Brendan Murphy. (2010) A Mixture of Experts Latent Position Cluster Model for Social Network Data. Statistical Methodology, 7 (3), pp.385-405.

See Also

MEclustnet

Directed adjacency matrix detailing the presence or absence of Twitter friend/follower links between US politicians.

Description

Network data on whether or not 69 US politicians are friends/followers on Twitter.

Usage

us.twitter.adjacency

Format

A 69 x 69 binary matrix, with 0 down the diagonal.

Source

With thanks to Dr. Derek Greene. School of Computer Science, University College Dublin.

A matrix of covariates of the US politicians.

Description

Covariates on each of 69 US politicians. Note the first column is a column of 1's.

Usage

us.twitter.covariates

Format

A data frame with 69 observations on the following 8 variables.

‘⁠1⁠’

a column of 1s should always be the first column.

twitter_id

Twitter number.

twitter_name

Twitter name.

name

Actual name.

party

a factor with levels Democrat Republican

location

a factor with levels detailing location.

role

a factor with levels Candidate, Representative and Senator

gender

a factor with levels Female and Male

Source

E. Lazega, The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership, Oxford University Press, Oxford, England, 2001.