network.comm.num function

Consistent Estimation of the Number of Communities via Regularized Network Embedding.

The main function for Consistent Estimation of the Number of Communities via Regularized Network Embedding.

network.comm.num(A, sample.index.n, lambda, Z.int, B.int,
                        a=3, kappa=1, alpha=1, eps=5e-2, niter=20,
               niter.Z=5, update.B="ADMM",local.oppro=FALSE, merge.all=TRUE,
               ad.BIC=FALSE, Fully.Connected=TRUE, trace=FALSE,
               line.search=TRUE, ad.BIC.B=FALSE)

Arguments

• A: An observed n * n adjacency matrix of undirected graph.
• sample.index.n: A 3 * (n*(n-1)/2) matrix, all pairs of integers from 1 to n.
• lambda: A list, the sequences of the tuning parameters ($\lambda_1$, $\lambda_2$, and $\lambda_3$).
• Z.int: A n * r matrix, the initial values of embedding vectors corresponding to n nodes.
• B.int: A n * r matrix, the initial values of community centers corresponding to n nodes.
• a: A float value, regularization parameter in MCP, the default setting is 3.
• kappa: A float value, the penalty parameter in ADMM algorithm, the default setting is 1.
• alpha: A float value, the step size of coordinate descent algorithm updating Z, the default setting is 1.
• eps: A float value, algorithm termination threshold.
• niter: Int, maximum number of cycles of the overall ADMM algorithm.
• niter.Z: Int, maximum number of cycles of coordinate descent algorithm updating Z.
• update.B: The optimization algorithm updating B, which can be selected "ADMM" (default) and "AMA".
• local.oppro: The logical variable, whether to use local approximations when updating Z, the default setting is F.
• merge.all: Whether to merge pairs of nodes indirectly connected (but without the direct edge) in the estimated community membership matrix.
• ad.BIC: Whether to use the adjusted BIC, the default setting is F.
• Fully.Connected: Whether to use the all pairs (i,j) in fusion penalty, the default setting is T. If F, the pairs (i,j) in fusion penalty will be determined by the observed n * n adjacency matrix A.
• trace: Whether to output the intermediate process of the algorithm.
• line.search: Linear search or not, the default setting is T.
• ad.BIC.B: Whether the BIC criterion contains terms involving the B matrix, the default setting is F.

Returns

A list including all estimated parameters and the BIC values with all choices of given tuning parameters, and the selected optional parameters. Opt_Z: A n * r matrix, the estimated embedding vectors corresponding to n nodes; Opt_B: A n * r matrix, the estimated community centers corresponding to n nodes; Opt_K: Int, the estimated number of communities; Opt_member: A n-dimensional vector, describing the membership of n nodes; Opt_cluster.matrix: A n * n membership matrix, whose (i,j)-element is 1, if nodes i and j belong to the same community, and 0, otherwise.

Examples

library(cencrne)
data(example.data)
A                   = example.data$A
K.true              = example.data$K.true
Z.true              = example.data$Z.true
B.true              = example.data$B.true
P.true              = example.data$P.true
Theta.true          = example.data$Theta.true
cluster.matrix.true = example.data$cluster.matrix.true

n       = dim(A)[1]
lam.max = 3
lam.min = 0.5
lam1.s  = 2/log(n)
lam2.s  = sqrt(8*log(n)/n)
lam3.s  = 1/8/log(n)/sqrt(n)
lambda  = genelambda.obo(nlambda1=3,lambda1_max=lam.max*lam1.s,lambda1_min=lam.min*lam1.s,
                         nlambda2=10,lambda2_max=lam.max*lam2.s,lambda2_min=lam.min*lam2.s,
                         nlambda3=1,lambda3_max=lam.max*lam3.s,lambda3_min=lam.min*lam3.s)

sample.index.n = rbind(combn(n,2),1:(n*(n-1)/2))
int.list       = gen.int(A)
Z.int          = int.list$Z.int
B.int          = int.list$B.int
res            = network.comm.num(A, sample.index.n, lambda, Z.int, B.int)

K.hat = res$Opt_K # the estimated number of communities
Z.hat = res$Opt_Z # the estimated embedding vectors corresponding to n nodes
cluster.matrix.hat = res$Opt_cluster.matrix # the n * n estimated membership matrix
evaluation(Z.hat, Z.true, cluster.matrix.hat, cluster.matrix.true,
           P.true, Theta.true, K.hat, K.true)

References

Ren, M., Zhang S. and Wang J. (2022). Consistent Estimation of the Number of Communities via Regularized Network Embedding.

Author(s)

Mingyang Ren, Sanguo Zhang, Junhui Wang. Maintainer: Mingyang Ren renmingyang17@mails.ucas.ac.cn.