Type: Package
Title: A Stochastic Block Model for Multilevel Networks
Version: 0.2.4
Description: Simulation, inference and clustering of multilevel networks using a Stochastic Block Model framework as described in Chabert-Liddell, Barbillon, Donnet and Lazega (2021) <doi:10.1016/j.csda.2021.107179>. A multilevel network is defined as the junction of two interaction networks, the upper level or inter-organizational level and the lower level or inter-individual level. The inter-level represents an affiliation relationship.
License: GPL-3
Encoding: UTF-8
URL: https://github.com/Chabert-Liddell/MLVSBM
BugReports: https://github.com/Chabert-Liddell/MLVSBM/issues
RoxygenNote: 7.1.2
Depends: R (≥ 3.5.0)
Imports: R6, blockmodels, ape, magrittr, cluster
Suggests: testthat (≥ 2.1.0), covr, knitr, rmarkdown, ggplot2 (≥ 3.3.2), ggforce, spelling, cowplot, reshape2, dplyr
VignetteBuilder: knitr
Language: en-US
NeedsCompilation: no
Packaged: 2022-08-04 17:05:33 UTC; stc
Author: Saint-Clair Chabert-Liddell ORCID iD [aut, cre]
Maintainer: Saint-Clair Chabert-Liddell <academic@chabert-liddell.com>
Repository: CRAN
Date/Publication: 2022-08-05 10:10:05 UTC

Pipe operator

Description

See magrittr::%>% for details.

Usage

lhs %>% rhs

Compare two clustering with the Adjusted Rand Index

Description

Compare two clustering with the Adjusted Rand Index

Usage

ARI(x, y)

Arguments

x

A vector of integers, the clusters labels

y

A vector of integers of the same length as x, the clusters labels

Value

A number between 0 (random clustering) and 1 (identical clustering)

Examples

ARI(x = c(1, 2, 1), y = c(2, 2, 1))

An R6 Class object, a fitted multilevel network once $dovem() is done

Description

An R6 Class object, a fitted multilevel network once $dovem() is done

An R6 Class object, a fitted multilevel network once $dovem() is done

Public fields

vbound

The vector of variational bound for monitoring convergence

Active bindings

affiliation_matrix

Get the affiliation matrix

adjacency_matrix

Get the list of adjacency matrices

nb_nodes

Get the list of the number of nodes

nb_clusters

Get the list of the number of blocks

parameters

Get the list of the model parameters

membership

Get the list of the variational parameters

independent

Are the levels independent?

distribution

Emission distribution of each level

directed

Are the levels directed?

entropy

Get the entropy of the model

bound

Get the variational bound of the model

df_mixture

Get the degrees of freedom of the mixture parameters

df_connect

Get the degrees of freedom of the connection parameters

connect

Get the number of possible observed connections

ICL

Get the ICL model selection criterion of the model

full_penalty

Get the penalty used to compute the ICL

Z

Get the list of block memberships (vector form)

X_hat

Get the list of the matrices of probability connection predictions

map

Get the list of block memberships (matrix form)

penalty

Get the ICL penalty

likelihood

Compute the likelihood of both levels

complete_likelihood

Get the complete likelihood of the model

Methods

Public methods

Method new()

Constructor for the FitMLVSBM class

Usage
FitMLVSBM$new(
  Q = list(I = 1, O = 1),
  A = NA,
  X = NA,
  M = list(I = NA, O = NA),
  directed = NA,
  distribution = list("bernoulli", "bernoulli"),
  independent = FALSE
)
Arguments
Q

List of number of blocks

A

Affiliation matrix

X

List of adjacency matrices

M

List of Mask matrices

directed

List of boolean

distribution

List of string

independent

Boolean

Returns

A FitMLVSBM object

Method update_alpha()

Update the connection parameters for the M step

Usage
FitMLVSBM$update_alpha(safeguard = 2 * .Machine$double.eps)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method update_pi()

Update the upper level mixture parameter for the M step

Usage
FitMLVSBM$update_pi(safeguard = 0.001)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method update_gamma()

Update the lower level mixture parameter for the M step

Usage
FitMLVSBM$update_gamma(safeguard = 1e-06)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method init_clustering()

init_clustering Initial clustering for VEM algorithm

Usage
FitMLVSBM$init_clustering(
  safeguard = 2 * .Machine$double.eps,
  method = "hierarchical",
  Z = NULL
)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

method

Algorithm used to initiate the clustering, either "spectral", "hierarchical" or "merge_split" (if Z is provided)

Z

Initial clustering if provided

Method clear()

Reset all parameters

Usage
FitMLVSBM$clear()

Method m_step()

m_step Compute the M step of the VEM algorithm

Usage
FitMLVSBM$m_step(safeguard = 1e-06)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method ve_step()

Compute the VE step of the VEM algorithm

Usage
FitMLVSBM$ve_step(threshold = 1e-06, fixPointIter = 10, safeguard = 1e-06)
Arguments
threshold

The convergence threshold

fixPointIter

The maximum number of fixed point iterations

safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method do_vem()

Launch a Variational EM algorithm

Usage
FitMLVSBM$do_vem(
  init = "hierarchical",
  threshold = 1e-06,
  maxIter = 1000,
  fixPointIter = 100,
  safeguard = 1e-06,
  Z = NULL
)
Arguments
init

The method for self$init_clustering

threshold

The convergence threshold

maxIter

The max number of VEM iterations

fixPointIter

The max number of fixed point iterations for VE step

safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Z

Initial clustering if provided

Method permute_empty_class()

permute_empty_class Put empty blocks numbers at the end

Usage
FitMLVSBM$permute_empty_class()

Method plot()

Plot of FitMLVSBM objects

Usage
FitMLVSBM$plot(type = c("matrix"), ...)
Arguments
type

A string for the type of plot, just "matrix" for now

Returns

a ggplot2 object

Method show()

print method

Usage
FitMLVSBM$show(type = "Multilevel Stochastic Block Model")
Arguments
type

character to tune the displayed name

Method print()

print method

Usage
FitMLVSBM$print()

Method clone()

The objects of this class are cloneable with this method.

Usage
FitMLVSBM$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

An R6 Class object for unilevel network

Description

a fitted level of a unilevel network once $do_vem() is done

Public fields

vbound

vector of variational bound for convergence monitoring

Active bindings

adjacency

Get the adjacency matrix

mask

Get the mask matrix for dealing with NA

nb_nodes

Get the number of nodes of the level

nb_clusters

Get the number of blocks

distribution

Get the distribution used for the connections

directed

Get if the level is directed or not

mixture_parameter

Access the block proportions

connectivity_parameter

Access the connectivity matrix

membership

Access the variational parameters

entropy

Get the entropy of the model

bound

Get the variational bound of the model

df_mixture

Get the degree of freedom of the block proportion

df_connect

Get the degree of freedom of the connection parameters

connect

Get the number of observed dyads

ICL

Get the ICL model selection criterion

penalty

Get the penalty used for computing the ICL

Z

Access the vector of block membership (clustering)

X_hat

Get the connection probability matrix

X_likelihood

adjacency part of the log likelihood

Z_likelihood

block part of the log likelihood

likelihood

complete log likelihood

Methods

Public methods

Method new()

Constructor for FitSBM R6 class

Usage
FitSBM$new(
  Q = 1,
  X = NULL,
  M = NULL,
  directed = FALSE,
  distribution = "bernoulli"
)
Arguments
Q

Number of blocks

X

Adjacency matrix

M

Mask matrix

directed

boolean

distribution

string (only "bernoulli")

Returns

A new FitSBM object

Method update_alpha()

Update the connection parameter for the M step

Usage
FitSBM$update_alpha(safeguard = 1e-06)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method update_pi()

Update the upper level mixture parameter for the M step

Usage
FitSBM$update_pi(safeguard = 1e-06)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method init_clustering()

init_clustering Initial clustering for VEM algorithm

Usage
FitSBM$init_clustering(safeguard = 1e-06, method = "hierarchical", Z = NULL)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

method

Algorithm used to initiate the clustering, either "spectral", "hierarchical" or "merge_split" (if Z is provided)

Z

Initial clustering if provided

Method m_step()

m_step Compute the M step of the VEM algorithm

Usage
FitSBM$m_step(safeguard = 1e-06)
Arguments
safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method ve_step()

Compute the VE step of the VEM algorithm

Usage
FitSBM$ve_step(threshold = 1e-06, fixPointIter = 100, safeguard = 1e-06)
Arguments
threshold

The convergence threshold

fixPointIter

The maximum number of fixed point iterations

safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Method do_vem()

Launch a Variational EM algorithm

Usage
FitSBM$do_vem(
  init = "hierarchical",
  threshold = 1e-06,
  maxIter = 1000,
  fixPointIter = 100,
  safeguard = 1e-06,
  Z = NULL
)
Arguments
init

The method for self$init_clustering

threshold

The convergence threshold

maxIter

The max number of VEM iterations

fixPointIter

The max number of fixed point iterations for VE step

safeguard

Parameter live in a compact [safeguard, 1-safeguard]

Z

Initial clustering if provided

Method permute_empty_class()

permute_empty_class Put empty blocks numbers at the end

Usage
FitSBM$permute_empty_class()

Method clear()

Reset all parameters

Usage
FitSBM$clear()

Method clone()

The objects of this class are cloneable with this method.

Usage
FitSBM$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

R6Class for multilevel object

Description

Store all simulation parameters and list of fittedmodels. Methods for global inference and model selection are included.

Active bindings

nb_nodes

List of the umber of nodes for each levels

simulation_parameters

List of parameters of the MLVSBM

affiliation_matrix

Access the affiliation matrix

adjacency_matrix

Access the list of adjacency_matrix

memberships

Access the list of the clusterings

fittedmodels

Get the list of selected fitted FitMLVSBM objects

ICL

A summary table of selected fitted models and ICL model selection criterion

ICL_sbm

Summary table of ICL by levels

tmp_fittedmodels

A list of all fitted FitMLVSBM objects

fittedmodels_sbm

A list of selected fitted FitSBM objects of each levels

max_clusters

Access the list of maximum model size

min_clusters

Access the list of minimum model size

directed

Access the list of boolean for levels direction

directed

Access the list of the distribution used for each levels

Methods

Public methods

Method estimate_level()

Usage
MLVSBM$estimate_level(
  level = "lower",
  Q_min = 1,
  Q_max = 10,
  Z = NULL,
  init = "hierarchical",
  depth = 1,
  nb_cores = NULL
)

Method estimate_sbm_neighbours()

Usage
MLVSBM$estimate_sbm_neighbours(
  level = "lower",
  Q = NULL,
  Q_min = 1,
  Q_max = 10,
  fit = NULL,
  nb_cores = NULL,
  init = NULL
)

Method estimate_sbm_from_neighbours()

Usage
MLVSBM$estimate_sbm_from_neighbours(
  level = "lower",
  Q = NULL,
  fits = NULL,
  nb_cores = NULL
)

Method estimate_sbm()

Usage
MLVSBM$estimate_sbm(level = "lower", Q = Q, Z = NULL, init = "hierarchical")

Method mcestimate()

Usage
MLVSBM$mcestimate(Q, Z = NULL, init = "hierarchical", independent = FALSE)

Method estimate_from_neighbours()

Usage
MLVSBM$estimate_from_neighbours(
  Q,
  models = NULL,
  independent = FALSE,
  nb_cores = nb_cores
)

Method estimate_neighbours()

Usage
MLVSBM$estimate_neighbours(
  Q,
  fit = NULL,
  independent = independent,
  nb_cores = NULL
)

Method merge_split_membership()

Usage
MLVSBM$merge_split_membership(
  fitted = private$fitted[[length(private$fitted)]]
)

Method mc_ms_estimate()

Usage
MLVSBM$mc_ms_estimate(Z = NA, independent = FALSE, nb_cores = NULL)

Method estimate_one()

Usage
MLVSBM$estimate_one(
  Q,
  Z = NULL,
  independent = FALSE,
  init = "hierarchical",
  nb_cores = NULL
)

Method estimate_all_bm()

Usage
MLVSBM$estimate_all_bm(
  Q = NULL,
  Z = NULL,
  independent = FALSE,
  clear = TRUE,
  nb_cores = NULL
)

Method new()

Constructor for R6 class MLVSBM

Usage
MLVSBM$new(
  n = NULL,
  X = NULL,
  A = NULL,
  Z = NULL,
  directed = NULL,
  sim_param = NULL,
  distribution = list("bernoulli", "bernoulli")
)
Arguments
n

A list of size 2, the number of nodes

X

A list of 2 adjacency matrices

A

The affiliation matrix

Z

A list of 2 vectors, the blocks membership

directed

A list of 2 booleans

sim_param

A list of MLVSBM parameters for simulating networks

distribution

The distributions of the interactions ("bernoulli")

Returns

A MLVSBM object

Method findmodel()

Find a fitted model of a given size

Usage
MLVSBM$findmodel(nb_clusters = NA, fit = NA)
Arguments
nb_clusters

A list of the size of the model

fit

if fit = "best" return the best model according to the ICL

Returns

A FitMLVSBM object

Method clearmodels()

delete all fitted models

Usage
MLVSBM$clearmodels()

Method addmodel()

Added a FitMLVSBM object to the list of fitted model

Usage
MLVSBM$addmodel(fit)
Arguments
fit

The FitMLVSBM object to be added

Method simulate()

Usage
MLVSBM$simulate()

Method clone()

The objects of this class are cloneable with this method.

Usage
MLVSBM$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

Title

Description

Title

Usage

build_fold_matrix(X, K)

Arguments

X

An adjacency matrix

K

An integer, the number of folds

Value

A matrix of the same size than X with class integer as coefficient

Extract model coefficients

Description

Extracts model coefficients from objects with class FitMLVSBM

Usage

## S3 method for class 'FitMLVSBM'
coef(object, ...)

Arguments

object

an R6 object of class FitMLVSBM

...

additional parameters for S3 compatibility. Not used

Value

List of parameters.

Perform a Hierarchical Clustering

Description

Perform a Hierarchical Clustering

Usage

hierarClust(X, K)

Arguments

X

An Adjacency Matrix

K

the number of wanted clusters

Value

A vector : The clusters labels

Merge a list of clusters

Description

Merge a list of clusters

Usage

merge_clust(Z, Q)

Arguments

Z

a vector of cluster memberships

Q

the number of original clusters

Value

A list of Q(Q-1)/2 clustering of Q-1 clusters

Create a MLVSBM object from observed data

Description

Create a MLVSBM object from observed data

Usage

mlvsbm_create_network(
  X,
  A,
  directed = NULL,
  distribution = list("bernoulli", "bernoulli")
)

Arguments

X

A list of 2 squares binary matrices, the first one being the individual or lower level the second one being the organizational or upper level

A

A matrix the affiliation matrix with individuals in rows and organizations in columns

directed

A list of 2 boolean are the upper and lower level directed or not. Default will check if the matrix are symmetric or not.

distribution

A list for the distribution of X, only "bernoulli" is implemented

Value

An unfitted MLVSBM object corresponding to the multilevel network

Examples

ind_adj <- matrix(stats::rbinom(n = 10**2, size = 1, prob = .2),
                  nrow = 10, ncol = 10)
org_adj <- matrix(stats::rbinom(n = 10**2, size = 1, prob = .3),
                  nrow = 10, ncol = 10)
affiliation <- diag(1, 10)
my_mlvsbm <- mlvsbm_create_network(X = list(I = ind_adj, O = org_adj),
                                   directed = list(I = FALSE, O = FALSE),
                                   A = affiliation)

Infer a multilevel network (MLVSBM object), the original object is modified

Description

The inference use a greedy algorithm to navigate between model size. For a given model size, the inference is done via a variational EM algorithm. The returned model is the one with the highest ICL criterion among all visited models.

By default the algorithm fits a single level SBM for each level, before inferring the multilevel network. This step can be skipped by specifying an initial clustering with the init_clustering. Also, a given model size can be force by setting the parameters nb_clusters to a given value.

Usage

mlvsbm_estimate_network(
  mlv,
  nb_clusters = NULL,
  init_clustering = NULL,
  nb_cores = NULL,
  init_method = "hierarchical"
)

Arguments

mlv

A MLVSBM object, the network to be inferred.

nb_clusters

A list of 2 integers, the model size. If left to NULL, the algorithm will navigate freely. Otherwise it will navigate between the specified model size and its neighbors.

init_clustering

A list of 2 vectors of integers of the same length as the number of node of each level. If specified, the algorithm will start from this clustering, then navigate freely.

nb_cores

An integer, the number of cores to use. Default to 1 for Windows and detectCores()/2 for Linux and MacOS

init_method

One of "hierarchical" (the default) or "spectral", "spectral" might be more efficient but can lead to some numeric errors. Not used when int_clustering is given.

Value

A FitMLVSBM object, the best inference of the network

Examples

my_mlvsbm <- MLVSBM::mlvsbm_simulate_network(
  n = list(I = 10, O = 20), # Number of nodes for the lower level and the upper level
  Q = list(I = 2, O = 2), # Number of blocks for the lower level and the upper level
  pi = c(.3, .7), # Block proportion for the upper level, must sum to one
  gamma = matrix(c(.9, .2,   # Block proportion for the lower level,
                   .1, .8), # each column must sum to one
                 nrow = 2, ncol = 2, byrow = TRUE),
  alpha = list(I = matrix(c(.8, .2,
                            .2, .1),
                          nrow = 2, ncol = 2, byrow = TRUE), # Connection matrix
               O = matrix(c(.99, .3,
                            .3, .1),
                          nrow = 2, ncol = 2, byrow = TRUE)),# between blocks
  directed = list(I = FALSE, O = FALSE), # Are the upper and lower level directed or not ?
  affiliation = "preferential") # How the affiliation matrix is generated
fit <- MLVSBM::mlvsbm_estimate_network(mlv = my_mlvsbm, nb_cores = 1)

Compute the complete log likelihood of a multilevel network for a given clustering of the nodes.

Description

This function is useful to compute the likelihood for clusters obtained by different methods.

Usage

mlvsbm_log_likelihood(mlv, clustering)

Arguments

mlv

A MLVSBM object, the network data

clustering

A list of 2 vectors of integers of the same length as the number of node of each level.

Value

A numeric, the log likelihood of the multilevel network for the given clustering.

Examples

my_mlvsbm <- MLVSBM::mlvsbm_simulate_network(
  n = list(I = 40, O = 20), # Number of nodes for the lower level and the upper level
  Q = list(I = 2, O = 2), # Number of blocks for the lower level and the upper level
  pi = c(.3, .7), # Block proportion for the upper level, must sum to one
  gamma = matrix(c(.9, .2,   # Block proportion for the lower level,
                   .1, .8), # each column must sum to one
                 nrow = 2, ncol = 2, byrow = TRUE),
  alpha = list(I = matrix(c(.8, .2,
                            .2, .1),
                          nrow = 2, ncol = 2, byrow = TRUE), # Connection matrix
               O = matrix(c(.99, .3,
                            .3, .1),
                          nrow = 2, ncol = 2, byrow = TRUE)),# between blocks
  directed = list(I = FALSE, O = FALSE), # Are the upper and lower level directed or not ?
  affiliation = "preferential") # How the affiliation matrix is generated
mlvsbm_log_likelihood(mlv = my_mlvsbm, clustering = my_mlvsbm$memberships)

Create a simulated multilevel network (MLVSBM object)

Description

Create a simulated multilevel network (MLVSBM object)

Usage

mlvsbm_simulate_network(
  n,
  Q,
  pi,
  gamma,
  alpha,
  directed,
  affiliation = "uniform",
  distribution = list("bernoulli", "bernoulli"),
  no_empty_org = FALSE,
  no_isolated_node = FALSE
)

Arguments

n

A list of 2 positive integers, the number of individuals and organizations.

Q

A list of 2 positive integers, the number of clusters of individuals and organizations.

pi

A vector of probabilities of length Q_O, the mixture parameter for the organizations.

gamma

A Q_I \times Q_O matrix with each column summing to one, the mixture parameters for the individuals

alpha

A list of 2 matrices, a Q_I \times Q_I matrix giving the connectivity probabilities of the individuals and a Q_O \times Q_O matrix giving the connectivity probabilities of the organizations.

directed

A list of 2 logical. Is the individual level a directed network ? Is the inter-organizational level a directed network?

affiliation

The distribution under which the affiliation matrix is simulated in c("uniform", "preferential").

distribution

A list for the distribution of X, only "bernoulli" is implemented.

no_empty_org

A logical with FALSE as default, should every organizations have at least one affiliated individual? Needs to have n_I \geq n_O.

no_isolated_node

A logical, if TRUE then the network is simulated again until all nodes are connected.

Value

An MLVSBM object, a simulated multilevel network with levels, affiliations and memberships.

Examples

my_mlvsbm <- MLVSBM::mlvsbm_simulate_network(
  n = list(I = 10, O = 20), # Number of nodes for the lower level and the upper level
  Q = list(I = 2, O = 2), # Number of blocks for the lower level and the upper level
  pi = c(.3, .7), # Block proportion for the upper level, must sum to one
  gamma = matrix(c(.9, .2,   # Block proportion for the lower level,
                   .1, .8), # each column must sum to one
                 nrow = 2, ncol = 2, byrow = TRUE),
  alpha = list(I = matrix(c(.8, .2,
                            .2, .1),
                          nrow = 2, ncol = 2, byrow = TRUE), # Connection matrix
               O = matrix(c(.99, .3,
                            .3, .1),
                          nrow = 2, ncol = 2, byrow = TRUE)),# between blocks
  directed = list(I = FALSE, O = FALSE)) # Are the upper and lower level directed

Multilevel SBM Plot

Description

basic matrix plot method for a FitMLVSBM object

Usage

## S3 method for class 'FitMLVSBM'
plot(x, type = c("matrix"), ...)

Arguments

x

an R6 object of class FitMLVSBM

type

A string for the type of plot, just "matrix" for now

...

additional parameters. block ordering with order = c("affiliation", "degree", "natural")

Details

Basic matrix plot method for a FitMLVSBM object

Value

a ggplot2 object

Model Predictions

Description

Make predictions from an SBM.

Usage

## S3 method for class 'FitMLVSBM'
predict(object, ...)

Arguments

object

an R6 object of class FitMLVSBM

...

additional parameters for S3 compatibility. Not used

Value

A list with the following entries:

dyads

A list of matrix with the probability of each dyads

nodes

A list of vectors with the clustering of each nodes

Simulation an adjacency matrix

Description

Simulation an adjacency matrix

Usage

simulate_adjacency(
  Z,
  n,
  alpha,
  directed,
  distribution = "bernoulli",
  no_isolated_node = FALSE
)

Arguments

Z

A vector of integer of size n, the label

n

An integer, the number of rows or columns of the matrix

alpha

A \max(Z)\times \max(Z) matrix, the connectivity parameters

directed

A boolean, Is the network directed or not ?

distribution

The distribution of the indices: only "bernoulli"

no_isolated_node

A boolean, may row and column of adjacency matrices sum to 0

Value

A nxn adjacency matrix

Simulate of matrix of affiliation

Description

Simulate of matrix of affiliation

Usage

simulate_affiliation(n, m, affiliation = "uniform", no_empty_org = FALSE)

Arguments

n

An integer, the number of individuals

m

An integer, the number of organizations

affiliation

The type of affiliation between c("uniform", "preferential")

no_empty_org

A Boolean. Force all columns to have at least a 1. Must have n>m.

Value

A n \times m affiliation matrix, with a unique 1 on each rows

Perform a spectral clustering

Description

Perform a spectral clustering

Usage

spcClust(X, K)

Arguments

X

an Adjacency matrix

K

the number of clusters

Value

A vector : The clusters labels

Merge a list of clusters

Description

Merge a list of clusters

Usage

split_clust(X, Z, Q)

Arguments

X

an adjacency matrix

Z

a vector of cluster memberships

Q

The number of maximal clusters

Value

A list of Q clustering of Q+1 clusters

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