Maximum Likelihood Estimation of edge probabilities between blocks of a graph, under beta-SBM

Description

get_mle_BetaSBM obtains MLE for the probability of edges between blocks in a graph, used in calculating the goodness-of-fit test statistic for the beta-SBM (Karwa et al. (2023))

Usage

get_mle_BetaSBM(G, C)

Arguments

Value

A matrix of maximum likelihood estimates

References

Karwa et al. (2023). "Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels", Journal of the Royal Statistical Society Series B: Statistical Methodology, doi:10.1093/jrsssb/qkad084

See Also

goftest_BetaSBM() performs the goodness-of-fit test for the beta-SBM, where the MLE of the edge probabilities are required

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 <- 2
n2 <- 2
n3 <- 2

# Generating block assignments for each of the nodes
n <- n1 + n2 + n3
class <- rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat <- matrix(
  c(
    0.80, 0.50, 0.50,
    0.50, 0.80, 0.50,
    0.50, 0.50, 0.80
  ),
  ncol = 3,
  byrow = TRUE
)
pmat <- cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p <- pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] <- rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm <- adj + t(adj)

# graph from the adjacency matrix
G <- igraph::graph_from_adjacency_matrix(adjsymm, mode = "undirected", weighted = NULL)

# mle of the edge probabilities
get_mle_BetaSBM(G, class)

Maximum Likelihood Estimation of edge probabilities between blocks of a graph, under ERSBM

Description

get_mle_ERSBM obtains MLE for the probability of edges between blocks in a graph, used in calculating the goodness-of-fit test statistic for the ERSBM (Karwa et al. (2023))

Usage

get_mle_ERSBM(G, C)

Arguments

Value

A matrix of maximum likelihood estimates

References

Karwa et al. (2023). "Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels", Journal of the Royal Statistical Society Series B: Statistical Methodology, doi:10.1093/jrsssb/qkad084

See Also

goftest_ERSBM() performs the goodness-of-fit test for the ERSBM, where the MLE of the edge probabilities are required

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 = 2
n2 = 2
n3 = 2

# Generating block assignments for each of the nodes
n = n1 + n2 + n3
class = rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat = matrix(
  c(
    0.80, 0.05, 0.05,
    0.05, 0.80, 0.05,
    0.05, 0.05, 0.80
  ),
  ncol = 3,
  byrow = TRUE
)
pmat = cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p = pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] = rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm = adj + t(adj)

# graph from the adjacency matrix
G = igraph::graph_from_adjacency_matrix(adjsymm, mode = "undirected", weighted = NULL)

# mle of the edge probabilities
get_mle_ERSBM(G, class)

Monte Carlo goodness-of-fit test for a beta stochastic blockmodel (beta-SBM)

Description

goftest_BetaSBM performs chi square goodness-of-fit test for network data considering the model as beta-SBM (Karwa et al. (2023))

Usage

goftest_BetaSBM(A, K = NULL, C = NULL, numGraphs = 100)

Arguments

Value

A list with the elements

References

Karwa et al. (2023). "Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels", Journal of the Royal Statistical Society Series B: Statistical Methodology, doi:10.1093/jrsssb/qkad084

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 = 10
n2 = 10
n3 = 10

# Generating block assignments for each of the nodes
n = n1 + n2 + n3
class = rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat = matrix(
  c(
    0.80, 0.05, 0.05,
    0.05, 0.80, 0.05,
    0.05, 0.05, 0.80
  ),
  ncol = 3,
  byrow = TRUE
)
pmat = cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p = pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] = rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm = adj + t(adj)

# When class assignment is known
out = goftest_BetaSBM(adjsymm, C = class, numGraphs = 10)

chi_sq_seq = out$statistic
pvalue = out$p.value
print(pvalue)

# Plotting histogram of the sequence of the test statistics
hist(chi_sq_seq, 20, xlab = "chi-square test statistics", main = NULL)
abline(v = chi_sq_seq[1], col = "red", lwd = 5) # adding test statistic on the observed network
legend("topleft", legend = paste("observed GoF = ", chi_sq_seq[1]))

Monte Carlo goodness-of-fit test for an Erdős-Rényi stochastic blockmodel (ERSBM)

Description

goftest_ERSBM performs chi square goodness-of-fit test for network data considering the model as ERSBM (Karwa et al. (2023))

Usage

goftest_ERSBM(A, K = NULL, C = NULL, numGraphs = 100)

Arguments

Value

A list with the elements

References

Karwa et al. (2023). "Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels", Journal of the Royal Statistical Society Series B: Statistical Methodology, doi:10.1093/jrsssb/qkad084

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 = 10
n2 = 10
n3 = 10

# Generating block assignments for each of the nodes
n = n1 + n2 + n3
class = rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat = matrix(
  c(
    0.80, 0.05, 0.05,
    0.05, 0.80, 0.05,
    0.05, 0.05, 0.80
  ),
  ncol = 3,
  byrow = TRUE
)
pmat = cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p = pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] = rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm = adj + t(adj)

# When class assignment is known
out = goftest_ERSBM(adjsymm, C = class, numGraphs = 10)

chi_sq_seq = out$statistic
pvalue = out$p.value
print(pvalue)

# Plotting histogram of the sequence of the test statistics
hist(chi_sq_seq, 20, xlab = "chi-square test statistics", main = NULL)
abline(v = chi_sq_seq[1], col = "red", lwd = 5) # adding test statistic on the observed network
legend("topleft", legend = paste("observed GoF = ", chi_sq_seq[1]))

Computation of the chi-square test statistic for goodness-of-fit, under beta-SBM

Description

graphchi_BetaSBM obtains the value of the chi-square test statistic required for the goodness-of-fit of a beta-SBM (Karwa et al. (2023))

Usage

graphchi_BetaSBM(G, C, p_mle)

Arguments

Value

A numeric value

See Also

goftest_BetaSBM() performs the goodness-of-fit test for the beta-SBM, where the values of the chi-square test statistics are required

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 = 2
n2 = 2
n3 = 2

# Generating block assignments for each of the nodes
n = n1 + n2 + n3
class = rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat = matrix(
  c(
    0.80, 0.5, 0.5,
    0.5, 0.80, 0.5,
    0.5, 0.5, 0.80
  ),
  ncol = 3,
  byrow = TRUE
)
pmat = cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p = pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] = rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm = adj + t(adj)

# graph from the adjacency matrix
G = igraph::graph_from_adjacency_matrix(adjsymm, mode = "undirected", weighted = NULL)

# mle of the edge probabilities
p.hat = get_mle_BetaSBM (G, class)

# chi-square test statistic values
graphchi_BetaSBM(G, class, p.hat)

Computation of the chi-square test statistic for goodness-of-fit, under ERSBM

Description

graphchi_ERSBM obtains the value of the chi-square test statistic required for the goodness-of-fit of a ERSBM (Karwa et al. (2023))

Usage

graphchi_ERSBM(G, C, p_mle)

Arguments

Value

A numeric value

See Also

goftest_ERSBM() performs the goodness-of-fit test for the ERSBM, where the values of the chi-square test statistics are required

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 = 2
n2 = 2
n3 = 2

# Generating block assignments for each of the nodes
n = n1 + n2 + n3
class = rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat = matrix(
  c(
    0.8, 0.5, 0.5,
    0.5, 0.8, 0.5,
    0.5, 0.5, 0.8
  ),
  ncol = 3,
  byrow = TRUE
)
pmat = cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p = pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] = rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm = adj + t(adj)

# graph from the adjacency matrix
G = igraph::graph_from_adjacency_matrix(adjsymm, mode = "undirected", weighted = NULL)

# mle of the edge probabilities
p.hat = get_mle_ERSBM(G, class)

# chi-square test statistic values
graphchi_ERSBM(G, class, p.hat)

Sampling a graph through a Markov move (basis) for beta-SBM

Description

sample_a_move_BetaSBM to sample a graph in the same fiber; sampling according to the beta-SBM (Karwa et al. (2023))

Usage

sample_a_move_BetaSBM(C, G_current)

Arguments

Value

A graph

References

Karwa et al. (2023). "Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels", Journal of the Royal Statistical Society Series B: Statistical Methodology, doi:10.1093/jrsssb/qkad084

See Also

goftest_BetaSBM() performs the goodness-of-fit test for the beta-SBM, where graphs are being sampled

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 = 5
n2 = 5
n3 = 5

# Generating block assignments for each of the nodes
n = n1 + n2 + n3
class = rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat = matrix(
  c(
    10, 0.05, 0.05,
    0.05, 10, 0.05,
    0.05, 0.05, 10
  ),
  ncol = 3,
  byrow = TRUE
)
pmat = cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p = pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] = rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm = adj + t(adj)

# graph from the adjacency matrix
G = igraph::graph_from_adjacency_matrix(adjsymm, mode = "undirected", weighted = NULL)

# sampling a Markov move for the beta-SBM
G_sample = sample_a_move_BetaSBM(class, G)

# plotting the sampled graph
plot(G_sample, main = "The sampled graph after one Markov move for beta-SBM")

Sampling a graph through a Markov move (basis) for ERSBM

Description

sample_a_move_ERSBM to sample a graph in the same fiber; sampling according to the ERSBM (Karwa et al. (2023))

Usage

sample_a_move_ERSBM(C, G_current)

Arguments

Value

A graph

References

Karwa et al. (2023). "Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels", Journal of the Royal Statistical Society Series B: Statistical Methodology, doi:10.1093/jrsssb/qkad084

See Also

goftest_ERSBM() performs the goodness-of-fit test for the ERSBM, where graphs are being sampled

Examples

RNGkind(sample.kind = "Rounding")
set.seed(1729)

# We model a network with 3 even classes
n1 = 5
n2 = 5
n3 = 5

# Generating block assignments for each of the nodes
n = n1 + n2 + n3
class = rep(c(1, 2, 3), c(n1, n2, n3))

# Generating the adjacency matrix of the network
# Generate the matrix of connection probabilities
cmat = matrix(
  c(
    10, 0.05, 0.05,
    0.05, 10, 0.05,
    0.05, 0.05, 10
  ),
  ncol = 3,
  byrow = TRUE
)
pmat = cmat / n

# Creating the n x n adjacency matrix
adj <- matrix(0, n, n)
for (i in 2:n) {
  for (j in 1:(i - 1)) {
    p = pmat[class[i], class[j]] # We find the probability of connection with the weights
    adj[i, j] = rbinom(1, 1, p) # We include the edge with probability p
  }
}

adjsymm = adj + t(adj)

# graph from the adjacency matrix
G = igraph::graph_from_adjacency_matrix(adjsymm, mode = "undirected", weighted = NULL)

# sampling a Markov move for the ERSBM
G_sample = sample_a_move_ERSBM(class, G)

# plotting the sampled graph
plot(G_sample, main = "The sampled graph after one Markov move for ERSBM")

Zachary Karate Club Data

Description

Zachary’s Karate club data is a classic, well-studied social network of friendships between 34 members of a Karate club at a US university, collected by Wayne Zachary in 1977. Each node represents a member of the club, and each edge represents a tie between two members of the club. The network is undirected. An often discussed problem using this dataset is to find the two groups of people into which the karate club split after an argument between two teachers. Refer to the example in README.

Usage

zachary

Format

Two 34 by 34 matrices:

ZACHE

symmetric, binary 34 by 34 adjacency matrix.

ZACHC

symmetric, valued 34 by 34 matrix, indicating the relative strength of the associations

Source

(Zachary, 1977), http://vlado.fmf.uni-lj.si/pub/networks/data/Ucinet/UciData.htm#zachary.