Penalized Optimization Framework for Community Detection in Networks with Covariates.

Description

Semidefinite programming for optimizing the inner product between combined network and the solution matrix.

Usage

ADMM(
  Adj,
  Covariate,
  lambda,
  K,
  alpha,
  rho,
  TT,
  tol,
  quiet = NULL,
  report_interval = NULL,
  r = NULL
)

Arguments

Details

ADMM is proposed in Covariate Regularized Community Detection in Sparse Graphs of Yan & Sarkar (2021). ADMM relies on semidefinite programming (SDP) relaxations for detecting the community structure in sparse networks with covariates.

Value

References

Yan, B., & Sarkar, P. (2021). Covariate Regularized Community Detection in Sparse Graphs. Journal of the American Statistical Association, 116(534), 734-745.
doi:10.1080/01621459.2019.1706541

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
caseno = 4; Nrange = 10; Nmin = 10; prob1 = 0.9; p = n*4;
Q = matrix(runif(p*K, 0, 1), nrow = p, ncol = K)
Q = sweep(Q,2,colSums(Q),`/`)
W = matrix(0, nrow = n, ncol = K);
for(jj in 1:n) {
  if(runif(1) <= prob1) {W[jj, 1:K] = Pi[jj, ];}
  else W[jj, sample(K, 1)] = 1;
}
W = t(W)
D0 = Q %*% W
X = matrix(0, n, p)
N = switch(caseno, rep(100, n), rep(100, n), round(runif(n)*Nrange+ Nmin),
  round(runif(n)* Nrange+Nmin))
for (i in 1: ncol(D0)){
  X[i, ] = rmultinom(1, N[i], D0[, i])
}
ADMM(Adj, X, lambda = 0.2, K = K, alpha = 0.5, rho = 2, TT = 100, tol = 5)

Covariate Assisted Spectral Clustering.

Description

CASC clusters graph nodes by applying spectral clustering with the assistance from node covariates.

Usage

CASC(Adj, Covariate, K, alphan = 5, itermax = 100, startn = 10)

Arguments

Details

CASC is a community detection algorithm for networks with node covariates, proposed in Covariate-assisted spectral clustering of Binkiewicz, et al. (2017). CASC applies k-means on the first K leading eigenvectors of the balanced matrix between the Laplacian matrix and the covariate matrix.

Value

References

Binkiewicz, N., Vogelstein, J. T., & Rohe, K. (2017). Covariate-assisted spectral clustering. Biometrika, 104(2), 361-377.
doi:10.1093/biomet/asx008

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
caseno = 4; Nrange = 10; Nmin = 10; prob1 = 0.9; p = n*4;
Q = matrix(runif(p*K, 0, 1), nrow = p, ncol = K)
Q = sweep(Q,2,colSums(Q),`/`)
W = matrix(0, nrow = n, ncol = K);
for(jj in 1:n) {
  if(runif(1) <= prob1) {W[jj, 1:K] = Pi[jj, ];}
  else W[jj, sample(K, 1)] = 1;
}
W = t(W)
D0 = Q %*% W
X = matrix(0, n, p)
N = switch(caseno, rep(100, n), rep(100, n), round(runif(n)*Nrange+ Nmin),
  round(runif(n)* Nrange+Nmin))
for (i in 1: ncol(D0)){
  X[i, ] = rmultinom(1, N[i], D0[, i])
}
CASC(Adj, X, 2)

Covariate Assisted Spectral Clustering on Ratios of Eigenvectors.

Description

Using ratios-of-eigenvectors to detect underlying communities in networks with node covariates.

Usage

CASCORE(
  Adj,
  Covariate,
  K,
  alpha = NULL,
  alphan = 5,
  itermax = 100,
  startn = 10
)

Arguments

Details

CASCORE is fully established in Network-Adjusted Covariates for Community Detection of Hu & Wang (2023). CASCORE detects the latent community structure under the covariate assisted degree corrected stochastic block model (CADCSBM), and it allows the disagreement between the community structures indicated in the graph and the covariates, respectively. K-means is applied on the entry-wise ratios between first leading eigenvector and each of the other K leading eigenvectors of the combined matrix of the adjacency matrix and the covariate matrix, to reveal the underlying memberships.

Value

.

References

Hu, Y., & Wang, W. (2023) Network-AdjustedCovariatesforCommunity Detection,
https://arxiv.org/abs/2306.15616

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
caseno = 4; Nrange = 10; Nmin = 10; prob1 = 0.9; p = n*4;
Q = matrix(runif(p*K, 0, 1), nrow = p, ncol = K)
Q = sweep(Q,2,colSums(Q),`/`)
W = matrix(0, nrow = n, ncol = K);
for(jj in 1:n) {
  if(runif(1) <= prob1) {W[jj, 1:K] = Pi[jj, ];}
  else W[jj, sample(K, 1)] = 1;
}
W = t(W)
D0 = Q %*% W
X = matrix(0, n, p)
N = switch(caseno, rep(100, n), rep(100, n), round(runif(n)*Nrange+ Nmin),
  round(runif(n)* Nrange+Nmin))
for (i in 1: ncol(D0)){
  X[i, ] = rmultinom(1, N[i], D0[, i])
}
CASCORE(Adj, X, 2)

Covariates-based Clustering.

Description

Covariates-based Clustering is a spectral clustering method that focuses solely on the covariates structure of a network. It employs k-means on the first K leading eigenvectors of the weighted cogariates matrix of a graph, with each eigenvector normalized to have unit magnitude.

Usage

Cov_based(Adj, K, tau = NULL, itermax = NULL, startn = NULL)

Arguments

Value

A label vector.

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
Cov_based(Adj, 2)

Network-based Clustering.

Description

Network-based Clustering is a spectral clustering method that focuses solely on the topological structure of a network. It employs k-means on the first K leading eigenvectors of the weighted adjacency matrix of a graph, with each eigenvector normalized to have unit magnitude.

Usage

Net_based(Adj, K, tau = NULL, itermax = NULL, startn = NULL)

Arguments

Value

A label vector.

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
Net_based(Adj, 2)

Spectral Clustering On Ratios-of-Eigenvectors.

Description

Using ratios-of-eigenvectors to detect underlying communities.

Usage

SCORE(G, K, itermax = NULL, startn = NULL)

Arguments

Details

SCORE is fully established in Fast community detection by SCORE of Jin (2015). SCORE uses the entry-wise ratios between the first leading eigenvector and each of the other K-1 leading eigenvectors for clustering. It is noteworthy that SCORE only works on connected graphs, in other words, it does not allow for isolated vertices.

Value

References

Jin, J. (2015). Fast community detection by score. The Annals of Statistics 43 (1), 57–89.
doi:10.1214/14-AOS1265

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
library(igraph)
is.igraph(Adj) # [1] FALSE
ix = components(graph.adjacency(Adj))
componentLabel = ix$membership
giantLabel = which(componentLabel == which.max(ix$csize))
Giant = Adj[giantLabel, giantLabel]
SCORE(Giant, 2)

Normalized Principle Component Analysis.

Description

Normalized Principle Component Analysis (nPCA), also known as spectral clustering on the graph Laplacian, is a classical spectral clustering method that applies k-means on the first K leading (unit-norm) eigenvectors of the degree-corrected normalized graph laplacian.

Usage

nPCA(Adj, K, tau = NULL, itermax = 100, startn = 10)

Arguments

Value

References

Chung, F. R., & Graham, F. C. (1997). Spectral graph theory (Vol. 92). American Mathematical Soc..

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
nPCA(Adj, 2)

Ordinary Principle Component Analysis.

Description

Ordinary Principle Component Analysis (oPCA), also known as spectral clustering on the adjacency matrix is a classical spectral clustering method that applies k-means on the first K leading (unit-norm) eigenvectors of the adjacency matrix of a graph.

Usage

oPCA(Adj, K, itermax = 100, startn = 10)

Arguments

Value

References

Chung, F. R., & Graham, F. C. (1997). Spectral graph theory (Vol. 92). American Mathematical Soc..

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
oPCA(Adj, 2)