Package 'cencrne'

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

Description

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

Usage

evaluation(Z.hat, Z.true, cluster.matrix.hat, cluster.matrix.true,
                  P.true, Theta.true, K.hat=4, K.true=4)

Arguments

Z.hat	A n * r matrix, the estimated embedding vectors corresponding to n nodes.
Z.true	A n * r matrix, the true embedding vectors corresponding to n nodes.
cluster.matrix.hat	A n * n estimated membership matrix, whose (i,j)-element is 1, if nodes i and j are estimated to belong to the same community, and 0, otherwise.
cluster.matrix.true	A n * n true membership matrix, whose (i,j)-element is 1, if nodes i and j belong to the same community, and 0, otherwise.
P.true	A n * n true probability matrix.
Theta.true	A n * n true matrix: Z.true %*% t(Z.true).
K.hat	The true number of communities.
K.true	The estimated number of communities.

Value

A vector including five evaluation index. prop. 1: the estimated and actual number of communities are equal; 0: not equal.

Author(s)

Mingyang Ren.

---

Some example data

Description

Some example data

Format

A list including: A: An observed n * n adjacency matrix of undirected graph, n=360. K.true: The estimated number of communities. Z.true: A n * r matrix, the true embedding vectors corresponding to n nodes, n=360, r=5. B.true: A n * r matrix, the true community centers corresponding to n nodes. P.true: A n * n true probability matrix. Theta.true: A n * n true matrix: Z.true %*% t(Z.true). cluster.matrix.true: A n * n true membership matrix, whose (i,j)-element is 1, if nodes i and j belong to the same community, and 0, otherwise.

Source

Simulated data

Examples

data(example.data)

---

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

Description

The function generating the initial values.

Usage

gen.int(A, R=8, K.max0=8,rand.seed=123,
               lambda3=0, a=3, kappa=1, alpha=1,
               eps = 1e-2, niter = 20, niter.Z=5)

Arguments

A	An observed n * n adjacency matrix of undirected graph.
R	Int, the relatively large dimension of embedding vectors given in advance.
K.max0	The relatively large upper bound of the number of communities given in advance to generate initial values of B.
rand.seed	The random seed of generating initial value.
lambda3	A float value, the tuning parameter for sparsity of Z.
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.

Value

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.

Author(s)

Mingyang Ren.

References

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

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]
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

---

Generate tuning parameters

Description

Generating a sequence of the tuning parameters (lambda1, lambda2, and lambda3).

Usage

genelambda.obo(nlambda1=10,lambda1_max=1,lambda1_min=0.05,
                      nlambda2=10,lambda2_max=1,lambda2_min=0.01,
                      nlambda3=10,lambda3_max=5,lambda3_min=0.5)

Arguments

nlambda1	The numbers of lambda 1.
lambda1_max	The maximum values of lambda 1.
lambda1_min	The minimum values of lambda 1.
nlambda2	The numbers of lambda 2.
lambda2_max	The maximum values of lambda 2.
lambda2_min	The minimum values of lambda 2.
nlambda3	The numbers of lambda 3.
lambda3_max	The maximum values of lambda 3.
lambda3_min	The minimum values of lambda 3.

Value

A sequence of the tuning parameters (lambda1, lambda2, and lambda3).

Author(s)

Mingyang Ren

Examples

lambda <- genelambda.obo(nlambda1=5,lambda1_max=0.5,lambda1_min=0.1, nlambda2=15,lambda2_max=1.5,
                         lambda2_min=0.1, nlambda3=10,lambda3_max=3.5,lambda3_min=0.5)
lambda

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

Description

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

Usage

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.

Value

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.

Author(s)

Mingyang Ren, Sanguo Zhang, Junhui Wang. Maintainer: Mingyang Ren [email protected].

References

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

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)

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