Discovering Community Structure in Multiplex Networks via a Co-Regularized Robust Tensor-Based Spectral Approach

Abstract

Complex networks arise in various fields, such as biology, sociology and communication, to model interactions among entities. Entities in many real-world systems exhibit different types of interactions, which requires modeling these type of systems properly. Multiplex networks are used to model these systems, as they can reflect the nodes’ pair-wise interactions as multiple distinct types of links across layers. Community detection is a widely studied application in network analysis as it provides insights into the structure and organization of the network. Even though multiple algorithms have been developed in the community detection field, many of them have a limited performance in the presence of noise. In this article, we develop a novel algorithm that combines tensor low-rank representation, spectral clustering and distance regularization to improve the accuracy in discovering communities in multiplex networks. The low-rank representation leads to reducing the noise and errors existing in the network and the optimization of an accurate consensus set of eigenvectors that reveals the communities in the network. Moreover, the proposed approach balances the agreement between the eigenvectors of each layer, i.e., individual subspaces, and the consensus set of eigenvectors, i.e., common subspaces, by minimizing the projection distance between them. The common and individual subspaces are computed efficiently through Tucker decomposition and modified spectral clustering, respectively. Finally, multiple experiments are conducted on real and simulated networks to evaluate the proposed approach and compare it to state-of-the-art algorithms. The proposed approach shows its robustness and efficiency in discovering the communities in multiplex networks.

1. Introduction

Real-world systems such as communications, transportation and biology, to name a few, can be presented as complex networks that reflect the intercommunication between their objects [1]. Real-world systems were first represented by simple graphs, and this representation did not take into account multiple or different types of interactions between the system’s objects. Consequently, simple graphs are considered inappropriate for modeling systems that exhibit different types of interactions between their objects. Multiplex networks have emerged to tackle this issue, which represent a multi-layer network consisting of the same nodes across layers, while the edges in the separate layers encode the different types of interactions [2,3].

Community detection is an important application that has been of crucial interest to researchers in analyzing networks [4]. The purpose of community detection is to reveal the set of communities in the network and reduce its dimensionality. A community is typically defined as a batch of strongly connected nodes that are weakly connected to the rest of the nodes in the network [5]. The majority of the early literature in community detection was on discovering communities in monoplex or single-layer networks [6]. Recently, various approaches have evolved to discover communities in multiplex networks [7]. Earlier multiplex network community detection approaches interpreted the multiplex network as a monoplex network by aggregating the different layers into a single-layer, then discovering the community structure by applying traditional simple graph community detection techniques [8]. However, these techniques are not appropriate to reveal the true community structure, as they do not consider the proprieties carried in each layer. To cope with the disadvantages of the aggregation-based methods, other multiplex community detection techniques have been recently proposed.

In [9], a unified non-negative matrix factorization algorithm that aimed to reveal the set of within- and across-layer clusters in fully-connected multi-layer networks was introduced, where the nodes were connected within and across the layers. The authors in [10] presented an approach to investigate a multi-view network structure based on a spectral clustering technique. However, they did not consider the effect of the individual layers on the common subspace. In [11], the authors presented a one parameter family of matrix power means (PM) to investigate the communities in multi-layer networks. The PM method merges the Laplacians from the separate layers and then analyzes them in the stochastic block model. In [12], two methods that aim to investigate the Laplacian rank constrained graph, which is considered as the centroid of the constructed graph for each layer with different confidences, were proposed. In the first method, the layers’ weights are learnt by introducing a hyper-parameter, whereas the second method is self-weighted. A top-bottom network partitioning strategy was proposed in [13], employing cross-layer edge clustering coefficients to assess the possibility of separating the communities. In [14], a robust multi-view spectral clustering (RMSC) technique was proposed. The proposed RMSC uncovered the clusters in the network based on decomposing the transition probability matrix into low rank and sparse components. In [15], the clusters in multi-layer networks were detected based on the Grassmann manifold theory through a modified spectral clustering approach (SCML). The proposed SCLM revealed the community structure of each layer by applying normalized spectral clustering for each layer individually. On the other hand, a centroid-based co-regularization (CentroidCoreg) for spectral clustering was proposed in [16], detecting the community structure based on a common subspace across the layers. In [17], the proposed method relied on the probability distribution of each layer to cluster the multiplex network. In [18], an approach that relied on graph factorization to explore the consensus, as well as the complementary information coming from the different views (2CMV), was presented. In [19], the authors proposed an algorithm (DiMMA) that captured the diversity and complexity of the data from multiple perspectives in order to provide a more accurate and reliable clustering solution. This was achieved by incorporating the distance information from the different views. Nevertheless, the abovementioned techniques are less robust in real-world conditions, as they do not perform well in noisy networks. In [20], the authors proposed to detect the clusters in multiplex networks through joint non-negative matrix factorization (NMF), where multiple formulations of the objective function were introduced: (i) collective symmetric NMF (CSNMF), (ii) collective projective NMF (CPNMF) and (iii) collective symmetric non-negative matrix trifactorization (CSNMTF). In these approaches, a set of non-negative low-dimensional feature representations were found first for the individual layers. The individual non-negative embeddings were then fused into a common feature representation through collective factorization.

In this article, we propose a new approach to discover the communities in multiplex networks called the Co-regularized Robust Tensor-based Spectral Approach (Mx-CRTSA). In particular, we develop an objective function that merges multiple terms, including tensor decomposition, spectral clustering and a centroid-based co-regularization. The goal of the proposed objective function is to optimize a common centroid set of eigenvectors to reveal the communities in the network after recovering a clean representation of the multiplex network. The introduced model is solved through an alternating minimization scheme, where the clean representation is recovered through alternating direction methods of multipliers and the consensus set of eigenvectors through Tucker decomposition. A summary of the proposed Mx-CRTSA is presented in Figure 1. The contributions of Mx-CRTSA are outlined as follows:

1.

The proposed Mx-CRTSA constructs an adjacency tensor form the multiplex network and minimizes its rank through nuclear norm minimization while pruning the noise by adopting the L${}_{2,1}$ for error minimization.

2.

The proposed Mx-CRTSA balances the agreement between the eigenvectors of each layer, i.e., the individual subspace, and a consensus set of eigenvectors, i.e., common or centroid subspace. This is achieved by minimizing the projection distance between the individual and common subspaces.

3.

A co-regularization term is considered in the objective function that includes a weighting parameter, which is optimized efficiently along with the consensus subspace through high order orthogonal iteration (HOOI), which can provide a more accurate and robust solution than other matrix factorization methods. With the aid of HOOI, the contribution of the individual subspaces to the common subspace is optimized.

4.

To assess the performance, robustness and effectiveness of the proposed Mx-CRTSA, multiple experiments are conducted to highlight its potential in clustering multiplex networks. The results indicate its superiority compared to other existing techniques.

The structure of this paper is as follows: The background is given in Section 2. The Mx-CRTSA approach is presented and solved in Section 3. The results are discussed in Section 4. Lastly, the conclusions are outlined in Section 5. A summary of the notation used in this paper is presented in

Table 1. List of notations.

Symbol	Description
$\mathcal{A}$, A, a	Tensor, matrix, vector
${\parallel \mathbf{A}\parallel }_F$	Frobenius norm of the matrix A, ${\parallel \mathbf{A}\parallel }_F^2={\sum }_{i,j}{\mathit{a}}_{\mathit{ij}}^{\mathit{2}}$
${\parallel \mathcal{A}\parallel }_F^2$	Frobenius norm of the tensor $\mathcal{A}$, ${\parallel \mathcal{A}\parallel }_F^2{\sum }_{i,j,k}{\mathit{a}}_{\mathit{ijk}}^{\mathit{2}}$
${\parallel \mathbf{A}\parallel }_{*}$	Nuclear norm of A, ${\parallel \mathbf{A}\parallel }_{*}={\sum }_i{\sigma }_i\left(\mathbf{A}\right)$
${\sigma }_i\left(\mathbf{A}\right)$	$i^{th}$ singular value of A
$\mathcal{A}(j,:,:)$	$j^{th}$ horizontal slice of $\mathcal{A}$
$\mathcal{A}(:,j,:)$	$j^{th}$ lateral slice of $\mathcal{A}$
$\mathcal{A}(:,:,j)$	$j^{th}$ frontal slice of $\mathcal{A}$
${\mathcal{A}}^{\left(j\right)}$	$j^{th}$ frontal slice of $\mathcal{A}$
$\mathcal{A}(:,i,j)$	Mode-1 fiber of $\mathcal{A}$
$\mathcal{A}(i,:,j)$	Mode-2 fiber of $\mathcal{A}$
$\mathcal{A}(i,j,:)$	Mode-3 fiber of $\mathcal{A}$
${\mathbf{A}}_{+}$	The positive values of matrix A

.

2. Background

2.1. Tensor Operations

In this section, the tensor operations that are used throughout this paper are presented. Suppose we have a 3-mode tensor, $\mathcal{A}\in {\mathbb{R}}^{n_1\times n_2\times n_3}$, the basic tensor operations can be then listed as follows:

• Block circulant matrix, $bcirc\left(\mathcal{A}\right)$: Converts the tensor $\mathcal{A}$ to a matrix by merging all frontal slices vertically and horizontally, which results in a matrix of the dimension $n_3{n}_1\times n_3{n}_2$, defined as:

  $$bcirc\left(\mathcal{A}\right)=\left[\begin{array}{cccc}{\mathcal{A}}^{\left(1\right)}& {\mathcal{A}}^{\left(n_3\right)}& \dots & {\mathcal{A}}^{\left(2\right)}\\ {\mathcal{A}}^{\left(2\right)}& {\mathcal{A}}^{\left(1\right)}& \dots & {\mathcal{A}}^{\left(3\right)}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathcal{A}}^{\left(n_3\right)}& {\mathcal{A}}^{(n_3-1)}& \dots & {\mathcal{A}}^{\left(1\right)}\end{array}\right],$$

  (1)

  where ${\mathcal{A}}^{\left(i\right)}$ represents the $i^{th}$ frontal slice of $\mathcal{A}$.
• Block vectorizing, $bvec\left(\mathcal{A}\right)$: Converts the tensor $\mathcal{A}$ to a block matrix by aggregating all frontal slices vertically and this results in a matrix of the dimension $n_3{n}_1\times n_3$ defined as follows:

  $$bvec\left(\mathcal{A}\right)=\left[\begin{array}{c}{\mathcal{A}}^{\left(1\right)}\\ {\mathcal{A}}^{\left(2\right)}\\ ⋮\\ {\mathcal{A}}^{\left(n_3\right)}\end{array}\right].$$

  (2)
  Furthermore, the opposite operation of the block vectorizing that transforms the aggregated matrix into a tensor can be defined as:

  $$bvfold\left(bvec\right(\mathcal{A}\left)\right)=\mathcal{A}.$$

  (3)
• Block diagonal matrix, $bdiag\left(\mathcal{A}\right)$: Converts the tensor $\mathcal{A}$ to a block diagonal matrix. This is executed by stacking all frontal slices on the diagonal of the matrix, leading to a matrix of the dimension $n_3{n}_1\times n_3{n}_2$ that can be defined as follows:

  $$bdiag\left(\mathcal{A}\right)=\left[\begin{array}{c}{\mathcal{A}}^{\left(1\right)}\\ & \ddots \\ & & {\mathcal{A}}^{\left(n_3\right)}\end{array}\right].$$

  (4)
  On the other hand, the block diagonal opposite operation $bdfold\left(bdiag\right(\mathcal{A}\left)\right)=\mathcal{A}$ recovers the tensor from the block diagonal matrix.
• Tensor Product (t-Product): Assuming that we have two 3-mode tensors, namely ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$. Then, the t-product between ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ can be calculated as follows:

  $${\mathcal{A}}_3={\mathcal{A}}_1*{\mathcal{A}}_2=bvfold\left\{bcirc\left({\mathcal{A}}_1\right)bvec\left({\mathcal{A}}_2\right)\right\}.$$

  (5)
• Tensor-Singular Value Decomposition (T-SVD):

  $$\mathcal{A}={\mathcal{Q}}_{left}*{\mathcal{S}}_f*{\mathcal{Q}}_{right}^{\top },$$

  (6)

  where ${\mathcal{Q}}_{left}$ and ${\mathcal{Q}}_{right}$ are the left and right orthonormal singular vectors of the tensor $\mathcal{A}$, respectively. ${\mathcal{S}}_f$ is the singular value core tensor.
• T-SVD-Based Tensor Nuclear Norm (T-SVD-TNN): The SVD of $\mathcal{A}$ via TNN in the frequency domain and can be calculated as follows:

  $${\parallel \mathcal{A}\parallel }_{⊛}=\sum _{i=1}^{min(n_1,n_2)}\sum _{j=1}^{n_3}\left|{\mathcal{A}}_f(i,i,j)\right|,$$

  (7)

  where ${\mathcal{A}}_f$ is the SVD of the tensor $\mathcal{A}$ in the frequency domain.
• Tucker Decomposition: Provides orthogonal subspace decomposition along each mode of the tensor $\mathcal{A}$ [21]:

  $$\mathcal{A}\approx \mathcal{B}{\times }_1{\mathrm{U}}_{\left(1\right)}{\times }_2{\mathrm{U}}_{\left(2\right)}{\times }_3{\mathrm{U}}_{\left(3\right)},$$

  (8)

  where $\mathcal{B}\in {\mathbb{R}}^{r_1\times r_2\times r_3}$ is the core tensor, ${\{{\mathrm{U}}_{\left(i\right)}\in {\mathbb{R}}^{n_i\times r_i}\}}_{i=1}^3$ are the low-dimensional orthogonal matrices and $r_i$ is the rank of $\left\{{\mathrm{U}}_{\left(i\right)}\right\}$ with $r_i\le n_i$. Tucker decomposition is solved through an iterative optimization procedure known as HOOI, which provides the best projection matrices [22].

2.2. Graph Theory

A simple network or graph, G, is defined by the set of nodes, V, the set of edges, E, and the adjacency matrix, A. The adjacency matrix $\mathbf{A}\in {\mathbb{R}}^{n\times n}$ is an undirected, symmetric and non-negative matrix that reflects the pair-wise affinity between nodes, where $a_{ij}\in [0,1]$ for weighted networks and $a_{ij}\in \{0,1\}$ for binary networks. A diagonal degree matrix, D, is defined from A, where $d_{ii}={\sum }_{j=1}^n{a}_{ij}$ [23,24]. ${\mathrm{A}}_{\mathrm{N}}={\mathrm{D}}^{-0.5}\mathrm{A}{\mathrm{D}}^{-0.5}$ defines a normalized version of the adjacency matrix, which is a positive semi-definite matrix.

An M-layer and n-vertex multiplex network, $\mathcal{G}$, is utilized as a group of simple graphs, $\left\{G^{\left(m\right)}\right\}$, where $G^{\left(m\right)}=(V^{\left(m\right)},E^{\left(m\right)},{\mathrm{A}}^{\left(m\right)})$ with $m\in \{1,\dots ,M\}$. ${\mathrm{A}}^{\left(m\right)}\in {\mathbb{R}}^{n\times n}$ refers to the adjacency matrix of the $m^{th}$ layer.

2.3. Spectral Clustering

Spectral clustering [23] is a famous approach in graph clustering to optimize the solution of:

$$\underset{\mathrm{U}}{max}tr\left({\mathrm{U}}^{\top }{\mathrm{A}}_N\mathrm{U}\right),\mathrm{s}.\mathrm{t}{\mathrm{U}}^{\top }\mathrm{U}=\mathrm{I},$$

(9)

where $tr$ refers to the matrix trace operator. As the matrix ${\mathrm{A}}_N$ is positive semi-definite, Equation (9) can be reformulated using the Frobenius norm as:

$$\underset{\mathrm{U}}{max}{\parallel {\mathrm{U}}^{\top }{\mathrm{A}}_N\mathrm{U}\parallel }_F^2,\mathrm{s}.\mathrm{t}{\mathrm{U}}^{\top }\mathrm{U}=\mathrm{I}.$$

(10)

Both maximization objective functions in Equations (9) and (10) can be solved by eigenvalue decomposition of ${\mathrm{A}}_N$, where $\mathrm{U}\in {\mathbb{R}}^{n\times c}$ is the matrix of the eigenvectors set that corresponds to the largest c eigenvalues of ${\mathrm{A}}_N$. After the indicator matrix U is computed, k-means is used to reveal the vertices community assignment [23].

2.4. Projection Distance

Let ${\mathrm{U}}_1\in {\mathbb{R}}^{n\times c}$ and ${\mathrm{U}}_2\in {\mathbb{R}}^{n\times c}$ be two orthonormal basis sets that correspond to two subspaces, $\mathit{span}\left({\mathbf{U}}_{\mathit{1}}\right)$ and $\mathit{span}\left({\mathbf{U}}_{\mathit{2}}\right)$, respectively. The distance between the two subspaces can be quantified by the principal angles between $\mathit{span}\left({\mathbf{U}}_{\mathit{1}}\right)$ and $\mathit{span}\left({\mathbf{U}}_{\mathit{2}}\right)$. Then, the squared projection distance between $\mathit{span}\left({\mathbf{U}}_{\mathit{1}}\right)$ and $\mathit{span}\left({\mathbf{U}}_{\mathit{2}}\right)$ can be formulated as [25]:

$$\begin{array}{cc}\hfill d_{proj}^2(\mathit{span}\left({\mathbf{U}}_1\right),\mathit{span}\left({\mathbf{U}}_2\right))& =\sum _{i=1}^c{sin}^2\left({\theta }_i\right)\hfill \\ & =c-\sum _{i=1}^c{cos}^2\left({\theta }_i\right)\hfill \\ & =c-tr\left({\mathbf{U}}_1{\mathbf{U}}_1^{\top }{\mathbf{U}}_2{\mathbf{U}}_2^{\top }\right)\hfill \\ & =\frac{1}{2}\left[2c-2tr\left({\mathbf{U}}_1{\mathbf{U}}_1^{\top }{\mathbf{U}}_2{\mathbf{U}}_2^{\top }\right)\right]\hfill \\ & =\frac{1}{2}\left[tr\left({\mathbf{U}}_1^{\top }{\mathbf{U}}_1\right)+tr\left({\mathbf{U}}_2^{\top }{\mathbf{U}}_2\right)-2tr\left({\mathbf{U}}_1{\mathbf{U}}_1^{\top }{\mathbf{U}}_2{\mathbf{U}}_2^{\top }\right)\right]\hfill \\ & =\frac{1}{2}{\parallel {\mathbf{U}}_1{\mathbf{U}}_1^{\top }-{\mathbf{U}}_2{\mathbf{U}}_2^{\top }\parallel }_F^2,\hfill \end{array}$$

(11)

where ${\theta }_i$ is the $i^{\mathit{th}}$ principal angle between $\mathit{span}\left({\mathbf{U}}_{\mathit{1}}\right)$ and $\mathit{span}\left({\mathbf{U}}_{\mathit{2}}\right)$.

3. Discovering Community Structure in Multiplex Networks: Co-regularized Robust Tensor-Based Spectral Approach (Mx-CRTSA)

In this paper, we approach the problem of discovering community structure in noisy multiplex networks and propose grouping the nodes by efficiently incorporating the information furnished by the different layers. More precisely, we propose reducing the effect of noise and outliers in the network by extracting a set of clean graphs from the original multiplex network. These clean graphs conserve the significant characteristics of the multiplex network and can be used to compute the lower dimensional subspaces by exploiting concepts from subspace analysis. The contribution of these individual subspaces is then optimized to find a representative common subspace of the network.

3.1. Proposed Objective Function

Given a noisy M-layer multiplex network, an adjacency tensor, $\mathcal{A}\in {\mathbb{R}}^{n\times n\times M}$, is constructed from the network by stacking the normalized adjacencies, ${\left\{{\mathbf{A}}_N^{\left(m\right)}\right\}}_{m=1}^M\in {\mathbb{R}}^{n\times n}$, of the individual layers as the frontal slices of the tensor. The proposed Mx-CRTSA aims to detect the community structure of the multiplex network by first extracting a clean self-representation tensor from the noisy adjacency tensor. The extracted clean tensor is then used to compute the low-dimensional individual spectral embeddings or subspaces, ${\left\{{\mathbf{U}}_m\right\}}_{m=1}^M\in {\mathbb{R}}^{n\times c}$, and a common centroid spectral embedding across all layers, ${\mathbf{U}}^{*}\in {\mathbb{R}}^{n\times c}$, where ${\mathbf{U}}^{*}$ is enforced to be close to all ${\left\{{\mathbf{U}}_m\right\}}_{m=1}^M$. The proposed objective function is formulated as follows:

$$\begin{array}{cc}\hfill & \underset{\begin{array}{c}{\mathbf{U}}^{*},{\mathbf{U}}^{\left(m\right)},\mathcal{Z},\\ \mathbf{E},\mathbf{w}\end{array}}{min}{\parallel \mathcal{Z}\parallel }_{⊛}+\gamma {\parallel \mathbf{E}\parallel }_{2,1}-\lambda \sum _{m=1}^M\left[tr\left({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{Z}}_N^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}\right)+w_{m}tr\left({{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{*}\right)\right]\hfill \\ & \mathrm{s}.\mathrm{t}.{\mathbf{A}}^{\left(m\right)}={\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}^{\left(m\right)}+{\mathbf{E}}^{\left(m\right)},\mathbf{E}=[{\mathbf{E}}^{\left(1\right)};{\mathbf{E}}^{\left(2\right)};\dots ;{\mathbf{E}}^{\left(M\right)}],{\mathbf{Z}}^{\left(m\right)}={{\mathbf{Z}}^{\left(m\right)}}^{\top },\hfill \\ & {{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{\left(m\right)}=\mathbf{I},m=\{1,\dots ,M\},\mathcal{Z}=\Phi ({\mathcal{Z}}^{\left(1\right)},{\mathcal{Z}}^{\left(2\right)},\dots ,{\mathcal{Z}}^{\left(M\right)}),\hfill \\ & {{\parallel \mathbf{w}\parallel }_2=1,\mathbf{w}\ge 0,{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I},\hfill \end{array}$$

(12)

where

• ${\parallel \mathcal{Z}\parallel }_{⊛}$ is considered to recover the low-rank approximation tensor, $\mathcal{Z}$, from the adjacency tensor, $\mathcal{A}$. $\Phi (·)$ is adopted to construct the clean tensor from the recovered self-representation matrices, ${\left\{{\mathbf{Z}}^{\left(m\right)}\right\}}_{m=1}^M$, and rotate it so that the computational complexity is reduced while the tensor’s nuclear norm is minimized.
• ${\parallel \mathbf{E}\parallel }_{2,1}$ represents the L${}_{2,1}$ of the error matrix $\mathbf{E}\in {\mathbb{R}}^{nM\times n}$, which is included to remove the noise from the adjacency tensor, $\mathcal{A}$. The matrix E is formed by arranging the individual error matrices, ${\left\{{\mathbf{E}}^{\left(m\right)}\right\}}_{m=1}^M$, vertically.
• The last term consists of two sub-terms. The first sub-term, ${\sum }_{m=1}^{M}tr\left({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{Z}}_N^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}\right)$, refers to the normalized cut of each layer which is included to find the individual spectral embedding of the separate layers, ${\left\{{\mathbf{U}}^{\left(m\right)}\right\}}_{m=1}^M$, where ${\mathbf{Z}}_N^{\left(m\right)}$ is the normalized clean adjacency matrix of the $m^{th}$ layer. The second sub-term, $w_{m}tr\left({{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{*}\right)$, represents the co-regularization across layers and corresponds to the projection distance between the individual subspaces, ${\left\{{\mathbf{U}}^{\left(m\right)}\right\}}_{m=1}^M$, and the centroid or common subspace, ${\mathbf{U}}^{*}$. By minimizing the projection distance, the common subspace is constrained to be close to all ${\left\{{\mathbf{U}}^{\left(m\right)}\right\}}_{m=1}^M$, while the node connectivity in each layer is still preserved. The degree of the consistency between the individual subspaces and the common subspace is controlled by the weighting vector, $\mathbf{w}={[w_1,w_2,\dots ,w_m]}^T$, which also will be optimized.
• $\gamma$ and $\lambda$ are regularization parameters to balance the trade-off between the different terms in the proposed objective function.

3.2. Optimization of the Proposed Objective Function

The proposed objective function is solved through an alternating minimization scheme. More precisely, we solve for $\mathcal{Z}$ and E while fixing ${\left\{{\mathbf{U}}^{\left(m\right)}\right\}}_{m=1}^M$, ${\mathbf{U}}^{*}$ and w through the alternating direction method of multipliers (ADMM). Then, the individual subspaces, ${\left\{{\mathbf{U}}^{\left(m\right)}\right\}}_{m=1}^M$, and the common subspace, ${\mathbf{U}}^{*}$, across all layers, along with w, are estimated efficiently using eigendecomposition and HOOI, respectively.

To solve the proposed objective function described in Equation (12), an auxiliary variable, $\mathcal{H}$, is introduced to allow variables separability. By introducing $\mathcal{H}$, the objective function is reformulated as follows:

$$\begin{array}{cc}\hfill & \underset{\begin{array}{c}{\mathbf{U}}^{*},{\mathbf{U}}^{\left(m\right)},\mathcal{Z},\\ \mathbf{E},\mathbf{w},\mathcal{H}\end{array}}{min}{\parallel \mathcal{H}\parallel }_{⊛}+\gamma {\parallel \mathbf{E}\parallel }_{2,1}-\lambda \sum _{m=1}^M\left[tr\left({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{Z}}_N^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}\right)+w_{m}tr\left({{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{*}\right)\right]\hfill \\ & \mathrm{s}.\mathrm{t}.{\mathbf{A}}^{\left(m\right)}={\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}^{\left(m\right)}+{\mathbf{E}}^{\left(m\right)},\mathbf{E}=[{\mathbf{E}}^{\left(1\right)};{\mathbf{E}}^{\left(2\right)};\cdots ;{\mathbf{E}}^{\left(M\right)}],{\mathbf{Z}}^{\left(m\right)}={{\mathbf{Z}}^{\left(m\right)}}^{\top }\hfill \\ & {{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{\left(m\right)}=\mathbf{I},m=\{1,\dots ,M\},\mathcal{Z}=\mathcal{H},\mathcal{H}=\Phi ({\mathcal{H}}^{\left(1\right)},{\mathcal{H}}^{\left(2\right)},\cdots ,{\mathcal{H}}^{\left(M\right)}),\hfill \\ & {\parallel \mathbf{w}\parallel }_2=1,\mathbf{w}\ge 0,{{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}.\hfill \end{array}$$

(13)

The augmented Lagrangian function can be then written as follows:

$$\begin{array}{cc}\hfill & \underset{\begin{array}{c}{\mathbf{U}}^{*},{\mathbf{U}}^{\left(m\right)},\mathcal{Z},\\ \mathbf{E},\mathbf{w},\mathcal{H}\end{array}}{min}{\parallel \mathcal{H}\parallel }_{⊛}+\gamma {\parallel \mathbf{E}\parallel }_{2,1}-\lambda \sum _{m=1}^M\left[tr\left({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{Z}}_N^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}\right)+w_{m}tr\left({{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{*}\right)\right]\hfill \\ & +\sum _{m=1}^M\left(\langle X_1^{\left(m\right)},{\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}^{\left(m\right)}-{\mathbf{E}}^{\left(m\right)}\rangle +\frac{{\beta }_1}{2}{\parallel {\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}^{\left(m\right)}-{\mathbf{E}}^{\left(m\right)}\parallel }_F^2\right)\hfill \\ & +\langle {\mathcal{X}}_2,\mathcal{Z}-\mathcal{H}\rangle +\frac{{\beta }_2}{2}{\parallel \mathcal{Z}-\mathcal{H}\parallel }_F^2\hfill \\ & \mathrm{s}.\mathrm{t}.\mathbf{E}=[{\mathbf{E}}^{\left(1\right)};{\mathbf{E}}^{\left(2\right)};\cdots ;{\mathbf{E}}^{\left(M\right)}],{\mathbf{Z}}^{\left(m\right)}={{\mathbf{Z}}^{\left(m\right)}}^{\top },{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{\left(m\right)}=\mathbf{I},m=\{1,\dots ,M\},\hfill \\ & \mathcal{H}=\Phi ({\mathcal{H}}^{\left(1\right)},{\mathcal{H}}^{\left(2\right)},\cdots ,{\mathcal{H}}^{\left(M\right)}),{\parallel \mathbf{w}\parallel }_2=1,\mathbf{w}\ge 0,{{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I},\hfill \end{array}$$

(14)

where the matrix ${\mathbf{X}}_1^{\left(m\right)}$ and tensor ${\mathcal{X}}_2$ are the Lagrangian multipliers. ${\beta }_1$ and ${\beta }_2$ are penalty parameters. $\langle ·\rangle$ is the trace inner product. To this end, Equation (14) can be solved by an alternating minimization scheme as follows:

• Updating $\mathcal{H}$: Fixing all the variables except $\mathcal{H}$ and considering all the terms that includes the variable $\mathcal{H}$ from Equation (14) gives:

  $$\begin{array}{cc}\hfill {\mathcal{H}}_{k+1}& =\underset{\mathcal{H}}{min}{\parallel \mathcal{H}\parallel }_{⊛}+\langle {\mathcal{X}}_{2_k},{\mathcal{H}}_k-{\mathcal{Z}}_k\rangle +\frac{{\beta }_2}{2}{\parallel {\mathcal{H}}_k-{\mathcal{Z}}_k\parallel }_F^2\hfill \\ & =\underset{\mathcal{H}}{min}{\parallel \mathcal{H}\parallel }_{⊛}+\frac{{\beta }_2}{2}{\parallel {\mathcal{H}}_k-{\mathcal{P}}_k\parallel }_F^2,\hfill \end{array}$$

  (15)

  where ${\mathcal{P}}_k={\mathcal{Z}}_k+\frac{{\mathcal{X}}_{2_k}}{{\beta }_2}$. The solution of Equation (15) can be found by applying tensor tubal-shrinkage operator [26]:

  $$\begin{array}{c}\hfill {\mathcal{H}}_{k+1}={\mathcal{F}}_{\tau }\left({\mathcal{P}}_k\right)={\mathcal{Q}}_{left}*{\mathcal{F}}_{\tau }\left({\mathcal{S}}_f\right)*{\mathcal{Q}}_{right}^{\top },\end{array}$$

  (16)

  where ${\mathcal{P}}_k={\mathcal{Q}}_{left}*{\mathcal{S}}_f*{\mathcal{Q}}_{right}^{\top }$, ${\mathcal{F}}_{\tau }\left({\mathcal{S}}_f\right)={\mathcal{S}}_f*\mathcal{D}$ and $\tau =\frac{M}{{\beta }_2}$. The diagonal tensor $\mathcal{D}\in {\mathbb{R}}^{n\times n\times M}$ contains the singular values in the frequency domain and can be calculated as $\mathcal{D}(i,i,j)={(1-\frac{\tau }{{\mathcal{S}}_f^{\left(j\right)}(i,i)})}_{+}$.
• Updating E: The sub-problem of E can be written as:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{E}}_{k+1}=\underset{\mathbf{E}}{min}\gamma {\parallel \mathbf{E}\parallel }_{2,1}+\sum _{m=1}^M(& \langle {\mathbf{X}}_{1_k}^{\left(m\right)},{\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}_k^{\left(m\right)}-{\mathbf{E}}_k^{\left(m\right)}\rangle \hfill \end{array}\begin{array}{c}\hfill +\frac{{\beta }_1}{2}{\parallel {\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}_k^{\left(m\right)}-{\mathbf{E}}_k^{\left(m\right)}\parallel }_F^2),\end{array}\end{array}$$

  (17)

  where Equation (17) can be written as:

  $$\begin{array}{cc}\hfill & {\mathbf{E}}_{k+1}=\underset{\mathbf{E}}{min}{\gamma \parallel \mathbf{E}\parallel }_{2,1}+\sum _{m=1}^M\frac{{\beta }_1}{2}{\parallel {\mathbf{E}}^{\left(m\right)}-{\mathbf{N}}_k^{\left(m\right)}\parallel }_F^2,\hfill \end{array}$$

  (18)

  where ${\mathbf{N}}_k^{\left(m\right)}={\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}_k^{\left(m\right)}+\frac{{{\mathbf{X}}_1}_k^{\left(m\right)}}{{\beta }_1}$. To simplify Equation (18), we construct a matrix N by stacking the layers of ${\mathbf{N}}^{\left(m\right)},m=\{1,\dots ,M\}$ vertically which leads to the following equation:

  $$\begin{array}{c}\hfill {\mathbf{E}}_{k+1}=\underset{\mathbf{E}}{min}{\gamma \parallel \mathbf{E}\parallel }_{2,1}+\frac{{\beta }_1}{2}{\parallel \mathbf{E}-\mathbf{N}\parallel }_F^2,\end{array}$$

  (19)

  where E can be computed as suggested in [27]:

  $$\begin{array}{c}\hfill {\mathbf{E}}_{k+1}=\left\{\begin{array}{cc}\frac{\parallel {\mathbf{N}}_{:,j}{\parallel }_2-\frac{\gamma }{{\beta }_1}}{\parallel {\mathbf{N}}_{:,j}{\parallel }_2}& ,\parallel {\mathbf{N}}_{:,j}{\parallel }_2>\frac{\gamma }{{\beta }_1}\hfill \\ 0& ,\mathrm{elsewhere}\hfill \end{array}\right.,\end{array}$$

  (20)

  where $\parallel {\mathbf{N}}_{:,j}{\parallel }_2$ refers to the $L_2$-norm of the $j^{th}$ column in N.
• Updating ${\mathbf{Z}}^{\left(m\right)}$: Considering all the terms with ${\mathbf{Z}}^{\left(m\right)}$ and fixing the rest of the variables, the sub-problem of ${\mathbf{Z}}_{k+1}^{\left(m\right)}$ is written as:

  $$\begin{array}{cc}\hfill & {\mathbf{Z}}_{k+1}^{\left(m\right)}=\underset{{\mathbf{Z}}^{\left(m\right)}}{min}\langle {{\mathbf{X}}_1}_k^{\left(m\right)},{\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}_k^{\left(m\right)}-{\mathbf{E}}_{k+1}^{\left(m\right)}\rangle +\frac{{\beta }_1}{2}{\parallel {\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}_k^{\left(m\right)}-{\mathbf{E}}_{k+1}^{\left(m\right)}\parallel }_F^2\hfill \\ & +\langle {\mathcal{X}}_2,{\mathcal{Z}}_k-{\mathcal{H}}_{k+1}\rangle +\frac{{\beta }_2}{2}{\parallel {\mathcal{Z}}_k-{\mathcal{H}}_{k+1}\parallel }_F^2-\lambda tr\left({{\mathbf{U}}_k^{\left(m\right)}}^{\top }{{\mathbf{Z}}_N}_k^{\left(m\right)}{\mathbf{U}}_k^{\left(m\right)}\right).\hfill \end{array}$$

  (21)
  The updated ${\mathbf{Z}}^{\left(m\right)}$ is calculated by finding the gradient of Equation (21) and setting it to zero [28] which results in a closed form solution of ${\mathbf{Z}}_{l+1}^{\left(m\right)}$ as:

  $$\begin{array}{cc}\hfill & {\mathbf{Z}}_{k+1}^{\left(m\right)}={\left({\beta }_2\mathbf{I}+{\beta }_1{{\mathbf{A}}^{\left(m\right)}}^{\top }{\mathbf{A}}^{\left(m\right)}\right)}^{-1}·(\lambda {{\mathbf{D}}^{\left(m\right)}}^{(-0.5)}{\mathbf{U}}_k^{\left(m\right)}{{\mathbf{U}}_k^{\left(m\right)}}^{\top }{{\mathbf{D}}^{\left(m\right)}}^{(-0.5)}+{\beta }_2{\mathbf{H}}_{k+1}^{\left(m\right)}\hfill \\ & -{\mathbf{X}}_{2_k}^{\left(m\right)}+{\beta }_1{\mathbf{A}}^{\left(m\right)}{\mathbf{T}}_k^{\left(m\right)}),\hfill \end{array}$$

  (22)

  where ${\mathbf{T}}_k^{\left(m\right)}={\mathbf{A}}^{\left(m\right)}-{\mathbf{E}}_{k+1}^{\left(m\right)}+\frac{{{\mathbf{X}}_1}_k^{\left(m\right)}}{{\beta }_1}$ and the matrix ${\mathbf{Z}}^{\left(m\right)}$ is found for each layer separately.
• Updating ${\mathbf{U}}^{\left(m\right)}$: Fixing all the variables except ${\mathbf{U}}^{\left(m\right)}$ in Equation (14), we obtain:

  $$\begin{array}{cc}\hfill & {\mathbf{U}}_{k+1}^{\left(m\right)}=\underset{{\mathbf{U}}^{\left(m\right)}}{min}-\lambda tr\left({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{Z}}_N^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}\right)+w_{m}tr{\left({{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{*}\right)}_{}\hfill \\ & \mathrm{s}.\mathrm{t}.{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{\left(m\right)}=\mathbf{I}.\hfill \end{array}$$

  (23)
  An equivalent form of Equation (23) is:

  $$\begin{array}{cc}\hfill & {\mathbf{U}}_{k+1}^{\left(m\right)}=\underset{{\mathbf{U}}^{\left(m\right)}}{max}\lambda tr{\left({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{Z}}_{mod}^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}\right)}_{}\hfill \\ & \mathrm{s}.\mathrm{t}.{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{\left(m\right)}=\mathbf{I},\hfill \end{array}$$

  (24)

  where ${\mathbf{Z}}_{mod}^{\left(m\right)}={{\mathbf{D}}^{\left(m\right)}}^{(-0.5)}{\mathbf{Z}}^{\left(m\right)}{{\mathbf{D}}^{\left(m\right)}}^{(-0.5)}+w_m{\mathbf{U}}^{*}{{\mathbf{U}}^{*}}^{\top }$. The resultant problem in Equation (24) is equivalent to the maximization problem in Equation (9), but with an adjusted or modified normalized adjacency matrix. Consequently, by the Rayleigh–Ritz theorem, Equation (24) can be solved through eigenvalue decomposition (EVD) of ${\mathbf{Z}}_{mod}^{\left(m\right)}$, where ${\mathbf{U}}^{\left(m\right)}$ contains the set of orthonormal eigenvectors that correspond to the largest c eigenvalues.
• Updating ${\mathbf{U}}^{*}$: The ${\mathbf{U}}^{*}$ sub-problem can be defined as:

  $$\begin{array}{cc}\hfill & {\mathbf{U}}_{k+1}^{*}=\underset{{\mathbf{U}}^{*},\mathbf{w}}{min}-\lambda tr{\left({{\mathbf{U}}^{*}}^{\top }\left(\sum _{m=1}^M{w}_m{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }\right){\mathbf{U}}^{*}\right)}_{}\hfill \\ & \mathrm{s}.\mathrm{t}.{\parallel \mathbf{w}\parallel }_2=1,\mathbf{w}\ge 0,{{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}.\hfill \end{array}$$

  (25)
  Equation (25) can be written as a trace maximization problem:

  $$\begin{array}{cc}\hfill & {\mathbf{U}}_{k+1}^{*}=\underset{{\mathbf{U}}^{*},\mathbf{w}}{max}\lambda tr{\left({{\mathbf{U}}^{*}}^{\top }\left(\sum _{m=1}^M{w}_m{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }\right){\mathbf{U}}^{*}\right)}_{}\hfill \\ & \mathrm{s}.\mathrm{t}.{\parallel \mathbf{w}\parallel }_2=1,\mathbf{w}\ge 0,{{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}.\hfill \end{array}$$

  (26)
  Equation (26) is rewritten in terms of Tucker decomposition as [4,29]:

  $$\begin{array}{cc}\hfill & {\mathbf{U}}_{k+1}^{*}=\underset{{\mathbf{U}}^{*},\mathbf{w}}{max}{\parallel \Gamma {\times }_1{{\mathbf{U}}^{*}}^{\top }{\times }_2{{\mathbf{U}}^{*}}^{\top }{\times }_3{\mathbf{w}}^{\top }\parallel }_F^2\hfill \\ & \mathrm{s}.\mathrm{t}.{\parallel \mathbf{w}\parallel }_2=1,\mathbf{w}\ge 0,{{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I},\hfill \end{array}$$

  (27)

  where ${\Gamma }^{\left(m\right)}={\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }$, $m=\{1,\dots ,M\}$. The common subspace between all layers of the multiplex network, ${\mathbf{U}}^{*}$, and the weighting vector across layers, ${\mathbf{w}}_m$, can be found efficiently using HOOI.
• Finally, the Lagrange multipliers, ${\mathbf{X}}_{1_{k+1}}^{\left(m\right)},{\mathcal{X}}_{2_{k+1}}$, and the penalty parameters, ${\beta }_{1_{k+1}}$, ${\beta }_{2_{k+1}}$, are computed as:

  $$\begin{array}{c}\hfill {\mathbf{X}}_{1_{k+1}}^{\left(m\right)}={\mathbf{X}}_{1_k}^{\left(m\right)}+{\beta }_{1_k}\left({\mathbf{A}}^{\left(m\right)}-{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}_{k+1}^{\left(m\right)}-{\mathbf{E}}_{k+1}^{\left(m\right)}\right),\end{array}$$

  (28)

  $$\begin{array}{c}\hfill {\mathcal{X}}_{2_{k+1}}={\mathcal{X}}_{2_k}+{\beta }_{2_{k+1}}\left({\mathcal{Z}}_{k+1}-{\mathcal{H}}_{k+1}\right),\end{array}$$

  (29)

  $$\begin{array}{c}\hfill {\beta }_{1_{k+1}}=min(\rho {\beta }_{1_k},{\beta }_{max}),\end{array}$$

  (30)

  $$\begin{array}{c}\hfill {\beta }_{2_{k+1}}=min(\rho {\beta }_{2_k},{\beta }_{max}),\end{array}$$

  (31)

  where $\rho$ is set to be 2. The proposed algorithm keeps iterating until it meets the following stopping criterion:

  $$\begin{array}{cc}\hfill \parallel {\mathbf{A}}^{\left(m\right)}& -{\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}^{\left(m\right)}-{\mathbf{E}}^{\left(m\right)}{\parallel }_{\infty }\le ϵ,\hfill \\ & {\parallel \mathcal{Z}-\mathcal{H}\parallel }_{\infty }\le ϵ.\hfill \end{array}$$

  (32)

A summary of the proposed algorithm is described in Algorithm 1.

Algorithm 1Mx-CRTSA

Input : $\mathcal{A}$, c, $\gamma$, $\lambda$, $\rho =2$, $ϵ=0.01$, ${\beta }_1={\beta }_2=0.01$, ${\gamma }_{max}={10}^{10}$. Output : Clustering labels. 1: Initialize $\mathcal{H}={\mathcal{X}}_2=0$, ${\mathbf{E}}^{\left(m\right)}={\mathbf{Z}}^{\left(m\right)}={\mathbf{X}}_1^{\left(m\right)}=0,\{m=1,\cdots ,M\}$. 2: while not converge do 3:    Update ${\mathcal{H}}_{k+1}$ through Equation (16). 4:    Update ${\mathbf{E}}_{k+1}^{\left(m\right)}$ through Equation (20). 5:    Update ${\mathbf{Z}}_{k+1}^{\left(m\right)}$ through Equation (22). 6:    Update ${\mathbf{U}}_{k+1}^{\left(m\right)}$ through Equation (24). 7:    Update ${\mathbf{U}}_{k+1}^{*}$ through Equation (27). 8:    Update ${\mathbf{X}}_{1_{k+1}}^{\left(m\right)}$, ${\mathcal{X}}_{2_{k+1}}$, ${\beta }_{1_{k+1}}$, and ${\beta }_{2_{k+1}}$ via Equations (28)–(31). 9: end while 10: Obtain community assignment by applying k-means to ${\mathbf{U}}^{*}$. 11: return Clustering labels.

3.3. Complexity of Mx-CRTSA

In Mx-CRTSA, multiple variables are updated in each iteration including $\mathcal{H}$, E, ${\mathbf{U}}^{\left(m\right)}$ and ${\mathbf{U}}^{*}$. The computational cost of $\mathcal{H}$ and E is $\mathcal{O}(M{n}^{2}log\left(n\right)+M^2{n}^2)$ and $\mathcal{O}\left(M{n}^2\right)$, respectively. The cost of computing ${\mathbf{U}}^{\left(m\right)}$ is the same as computing ${\mathbf{U}}^{*}$ and equals $\mathcal{O}\left(M{n}^3\right)$. Therefore, the total cost of the Mx-CRTSA algorithm is equal to $\mathcal{O}\left(L(M{n}^{2}log\left(n\right)+M^2{n}^2+M{n}^2+2M{n}^3)\right)\approx \mathcal{O}\left(ML{n}^3\right)$, where L, M and n refer to the number of iterations, layers and vertices, respectively.

4. Results and Discussion

Multiple experiments on simulated and real multiplex networks have been conducted to investigate the effectiveness and robustness of the proposed Mx-CRTSA compared to other community detection methods, including PM [11], RMSC [14], SCML [15], CentroidCoreg [16], CSNMF [20], CPNMF [20], CSNMTF [20], DiMMA [19] and 2CMV [18]. A comparison between the objective functions of the different methods considered in the experiments is illustrated in

Table 2. Comparison between the objective functions of the different methods considered in the experiments.

Method	Objective Function	Constraints
CentroidCoreg	${\displaystyle \underset{{\mathbf{U}}^{\left(m\right)},{\mathbf{U}}^{*}}{max}\sum _{m=1}^M\left[tr({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{L}}^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}+{\lambda }_{m}tr\left({\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{*}{{\mathbf{U}}^{*}}^{\top }\right))\right]}$	${{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}$, ${{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{\left(m\right)}=\mathbf{I}$.
SCML	${\displaystyle \underset{\mathbf{U}}{min}\sum _{m=1}^{M}tr\left({\mathbf{U}}^{\top }{\mathbf{L}}^{\left(m\right)}\mathbf{U}\right)-\lambda \sum _{m=1}^{M}tr\left(\mathbf{U}{{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }\right)}$	${{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}$.
CSNMF	${\displaystyle \underset{\mathbf{H}}{min}\parallel {\mathbf{A}}^{\left(m\right)}-{\mathbf{U}}^{*}{{\mathbf{U}}^{*}}^{\top }{\parallel }_F^2+{\parallel \mathbf{U}{{\mathbf{U}}^{*}}^{\top }-{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }\parallel }_F^2}$	$\mathbf{U}\ge 0$, ${{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}$.
CPNMF	${\displaystyle \underset{\mathbf{H}}{min}\parallel {\mathbf{A}}^{\left(m\right)}-{\mathbf{U}}^{*}{{\mathbf{U}}^{*}}^{\top }{\mathbf{A}}^{\left(m\right)}{\parallel }_F^2+{\parallel \mathbf{U}{{\mathbf{U}}^{*}}^{\top }-{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }\parallel }_F^2}$	$\mathbf{U}\ge 0$,  ${{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}$.
CSNMTF	${\displaystyle \underset{\mathbf{H}}{min}\parallel {\mathbf{A}}^{\left(m\right)}-{\mathbf{U}}^{*}{\mathbf{S}}^{\left(m\right)}{{\mathbf{U}}^{*}}^{\top }{\parallel }_F^2+{\parallel \mathbf{U}{{\mathbf{U}}^{*}}^{\top }-{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }\parallel }_F^2}$	$\mathbf{U}\ge 0$,  ${{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}$.
Mx-CRTSA	${\displaystyle \underset{\begin{array}{c}{\mathbf{U}}^{*},{\mathbf{U}}^{\left(m\right)},\mathcal{Z},\mathbf{E},\mathbf{w}\end{array}}{min}{\parallel \mathcal{Z}\parallel }_{⊛}+\gamma {\parallel \mathbf{E}\parallel }_{2,1}-\lambda \sum _{m=1}^M\left[tr\left({{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{Z}}_N^{\left(m\right)}{\mathbf{U}}^{\left(m\right)}\right)+w_{m}tr\left({{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{\left(m\right)}{{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{*}\right)\right]}$	${\mathbf{A}}^{\left(m\right)}={\mathbf{A}}^{\left(m\right)}{\mathbf{Z}}^{\left(m\right)}+{\mathbf{E}}^{\left(m\right)}$,${{\mathbf{U}}^{*}}^{\top }{\mathbf{U}}^{*}=\mathbf{I}$,${{\mathbf{U}}^{\left(m\right)}}^{\top }{\mathbf{U}}^{\left(m\right)}=\mathbf{I}$, ${\parallel \mathbf{w}\parallel }_2=1$,     $\mathbf{w}\ge 0$

. The performance of the different methods is quantified using accuracy (ACC), normalized mutual information (NMI), adjusted Rand index (AR) and F-score [30,31]. All the adopted quality metrics are normalized in the range of $[0,1]$, where higher values indicate more accurate clustering results. The experiments were carried out using MATLAB 2020a on a PC (Intel(R) Core(TM) i7 processor and 16 GB of RAM). For reproducibility purposes, the implemented MATLAB code of the proposed Mx-CRTSA approach is publicly available in our GitHub repository https://github.com/wardat99/Mx-CRTSA, accessed on 8 February 2023.

4.1. Simulated Multiplex Networks

In this experiment, the adopted simulated multiplex networks are developed in [32]. The structure of these networks is controlled by two parameters: ${\mu }_0$ and $p_0$. ${\mu }_0$ is the community mixing parameter that controls the network’s modularity, while $p_0$ is the value of the probability of the inter-layer dependency tensor which is used to control the dependency across the layers of the network. The values of ${\mu }_0$ and $p_0$ can vary in the range of $[0,1]$. As the value of ${\mu }_0$ increases, the noise level in the network increases.

The simulated multiplex networks are generated by varying the parameter ${\mu }_0$ to control the noise level in the multiplex network so that the robustness of the proposed Mx-CRTSA can be evaluated. In our experiment, we kept the value of $p_0=1$ and varied the value of ${\mu }_0$.

The effectiveness of the different algorithms, including Mx-CRTSA, in discovering the community structure in multiplex networks as ${\mu }_0$ changes in the range $[0.1,0.8]$ was validated by NMI and AR under multiple scenarios. In the first scenario, 2-layer multiplex networks with $n=128$ nodes and $c=4$ communities were generated. Examples of some of the networks tested in the first scenario for different values of ${\mu }_0$ are shown Figure 2a–d. In the second scenario, the number of layers was increased to three, while the number of nodes and clusters were fixed, i.e., $n=128$ and $c=4$, to explore the performance of the proposed Mx-CRTSA method with networks with a larger number of layers. Furthermore, some examples of the generated networks from scenario 2 are shown in Figure 3a–f. In the third scenario, 2-layer multiplex networks were generated with $n=128$ nodes and $c=10$ communities, to explore the performance of the proposed Mx-CRTSA with networks that consist of small-sized clusters. More examples of the generated networks from scenario 3 are shown in Figure 4a–d.

In all scenarios, the experiments were replicated 50 times and the variation in the average values of NMI and AR with respect to ${\mu }_0$ for the different methods are reported in Figure 5a–f. As the figure shows, the proposed Mx-CRTSA method scores the best results in all networks in terms of NMI and AR compared to the other methods. This experiment reflects the robustness and effectiveness of the proposed Mx-CRTSA compared to other methods under comparison. These results indicate that the proposed algorithm mitigates the impact of noise by estimating the clean low-rank representation tensor which leads to more accurate clustering results.

Scalability Analysis

To test the proposed Mx-CRTSA’s scalability, a set of simulated networks were generated where the number of nodes in the networks was increased from 32 to 2048 on a logarithmic scale. The multiplex networks were generated following [32] with $c=10$, $M=2$ and ${\mu }_0=0.1$. The comparison between the run time taken by Mx-CRTSA and the other methods is reported in Figure 6. As the figure demonstrates, Mx-CRTSA takes a longer time compared to most of the other methods and is comparable to CSNMTF and DiMMA. However, the Mx-CRTSA method obtains the best results over the compared methods in terms of community detection.

4.2. Real-World Multiplex Networks

Multiple well-known real-world multiplex networks were adopted to validate the performance of the proposed Mx-CRTSA, including:

• Nokia Research Center (NRC). (https://bitbucket.org/uuinfolab/20csur/src/master/algorithms/spectral/data/) (accessed on 15 January 2023) Mobile phone data of 136 users living and working in Switzerland with eight communities and three layers, including physical location, Bluetooth scans and phone calls [33].
• CKM physicians innovation. (https://manliodedomenico.com/data.php) (accessed on 15 January 2023) A network that reflects data on medical innovation, considering physicians in four towns in Illinois (Peoria, Bloomington, Quincy and Galesburg) with three layers and four communities [34].
• C. Elegans. A three-layer multiplex network of the Caenorhabditis Elegans connectome that reflects the synaptic junctions between the neurons including electric, chemical monadic and polyadic junctions. The network consists of 279 nodes and 12 communities representing the neuron groups [35,36].
• Cora. (https://people.cs.umass.edu/~mccallum/data.html) (accessed on 15 January 2023) A subset of the Cora bibliographic dataset consisting of 292 research papers with three communities, including natural language processing, data mining and robotics, and three layers, including, title similarity, abstract similarity and citation relationship.
• BBCSPORT. (http://mlg.ucd.ie/datasets/bbc.html) (accessed on 15 January 2023) A collection of BBC sport articles ($n=544$) with two layers and grouped into five clusters: athletics, cricket, football, rugby and tennis. [37].
• Wikipedia. (http://www.svcl.ucsd.edu/projects/crossmodal/) (accessed on 15 January 2023) A dataset containing 693 articles, where each article is considered a node with two layers and ten communities.
• COIL-20. (https://www.cs.columbia.edu/CAVE/software/softlib/) (accessed on 15 January 2023) An image dataset that consists of 1440 images collected from the Columbia object image library with three layers that reflect intensity, LBP and Gabor, and twenty communities.
• 100Leaves. (https://archive.ics.uci.edu/ml/datasets/One-hundred+plant+species+leaves+) (accessed on 15 January 2023) A plant species leaf database with 1600 samples, 100 communities and 3 layers, where the edges in the layers reflect shape descriptor, fine-scale margin and texture.

A description of the real-world multiplex networks is shown in

Table 3. Description of the real-world multiplex networks.

Network	Nodes	Layers	Clusters
NRC	136	3	8
CKM	246	3	4
C. Elegans	279	3	12
Cora	292	3	3
BBCSPORT	544	2	5
Wikipedia	693	2	10
COIL20	1440	3	20
100Leaves	1600	3	100

.

4.2.1. Real-World Multiplex Networks Clustering

In this section, the experimental results of the proposed Mx-CRTSA method on real multiplex networks are presented and compared to other well-known methods in clustering multiplex networks through several clustering quality measures. These quality measures include ACC, NMI, AR and F-score. The comparison results are summarized in

Table 4. Performance comparison between the different algorithms including the proposed Co-regularized Robust Tensor-based Spectral Approach (Mx-CRTSA) for the real-world multiplex networks. The best performance is in boldface.

Network	Metric	CentroidCoreg	PM	RMSC	SCML	CSNMF	CPNMF	CSNMTF	DiMMA	2CMV	Mx-CRTSA
NRC	ACC	$0.426$	$0.419$	$0.455$	$0.441$	$0.416$	$0.439$	$0.384$	$0.375$	$0.412$	$\mathbf{0}.\mathbf{541}$
$\gamma =0.03$	NMI	$0.352$	$0.368$	$0.371$	$0.387$	$0.333$	$0.148$	$0.275$	$0.297$	$0.336$	$\mathbf{0}.\mathbf{544}$
$\lambda =0.008$	AR	$0.208$	$0.153$	$0.216$	$0.218$	$0.153$	$0.027$	$0.121$	$0.147$	$0.194$	$\mathbf{0}.\mathbf{331}$
	F-score	$0.337$	$0.419$	$0.358$	$0.357$	$0.312$	$0.302$	$0.296$	$0.286$	$0.314$	$\mathbf{0}.\mathbf{448}$
CKM	ACC	$0.983$	$0.479$	$0.947$	$0.987$	$0.991$	$0.317$	$0.987$	$0.605$	$0.836$	$\mathbf{1}.\mathbf{00}$
$\gamma =0.04$	NMI	$0.948$	$0.032$	$0.821$	$0.958$	$0.821$	$0.017$	$0.963$	$0.472$	$0.778$	$\mathbf{1}.\mathbf{00}$
$\lambda =0.1$	AR	$0.975$	$0.010$	$0.825$	$0.980$	$0.851$	$0.009$	$0.979$	$0.215$	$0.794$	$\mathbf{1}.\mathbf{00}$
	F-score	$0.983$	$0.483$	$0.866$	$0.986$	$0.990$	$0.280$	$0.985$	$0.491$	$0.812$	$\mathbf{1}.\mathbf{00}$
C. Elegans	ACC	$0.419$	$0.369$	$0.301$	$0.401$	$0.322$	$0.369$	$0.333$	$0.290$	$0.286$	$\mathbf{0}.\mathbf{438}$
$\gamma =0.01$	NMI	$0.432$	$0.427$	$0.303$	$0.448$	$0.442$	$0.382$	$0.461$	$0.341$	$0.334$	$\mathbf{0}.\mathbf{509}$
$\lambda =0.02$	AR	$0.183$	$0.217$	$0.124$	$0.184$	$0.219$	$0.157$	$0.157$	$0.095$	$0.077$	$\mathbf{0}.\mathbf{270}$
	F-score	$\mathbf{0}.\mathbf{433}$	$0.332$	$0.266$	$0.402$	$0.275$	$0.313$	$0.293$	$0.207$	$0.201$	$0.362$
Cora	ACC	$0.982$	$0.979$	$0.503$	$0.979$	$0.603$	$0.442$	$0.508$	$0.455$	$0.507$	$\mathbf{1}.\mathbf{00}$
$\gamma =0.04$	NMI	$0.981$	$0.906$	$0.528$	$0.906$	$0.476$	$0.123$	$0.447$	$0.354$	$0.441$	$\mathbf{1}.\mathbf{00}$
$\lambda =0.1$	AR	$0.949$	$0.939$	$0.389$	$0.939$	$0.356$	$0.065$	$0.353$	$0.176$	$0.337$	$\mathbf{1}.\mathbf{00}$
	F-score	$0.966$	$0.959$	$0.592$	$0.959$	$0.620$	$0.473$	$0.579$	$0.533$	$0.569$	$\mathbf{1}.\mathbf{00}$
BBCSport	ACC	$0.768$	$0.527$	$0.902$	$0.742$	$0.279$	$0.237$	$0.292$	$0.459$	$0.721$	$\mathbf{0}.\mathbf{996}$
$\gamma =0.3$	NMI	$0.604$	$0.337$	$0.835$	$0.563$	$0.031$	$0.006$	$0.025$	$0.193$	$0.656$	$\mathbf{0}.\mathbf{989}$
$\lambda =0.02$	AR	$0.552$	$0.179$	$0.853$	$0.494$	$0.004$	$0.001$	$0.011$	$0.088$	$0.599$	$\mathbf{0}.\mathbf{996}$
	F-score	$0.659$	$0.474$	$0.889$	$0.616$	$0.235$	$0.216$	$0.234$	$0.406$	$0.699$	$\mathbf{0}.\mathbf{997}$
Wikipedia	ACC	$0.565$	$0.551$	$0.546$	$0.568$	$0.574$	$0.216$	$0.572$	$0.341$	$0.398$	$\mathbf{0}.\mathbf{583}$
$\gamma =0.04$	NMI	$0.477$	$0.488$	$0.346$	$0.493$	$0.527$	$0.030$	$0.525$	$0.309$	$0.425$	$\mathbf{0}.\mathbf{573}$
$\lambda =0.5$	AR	$0.399$	$0.405$	$0.223$	$0.412$	$0.404$	$0.003$	$\mathbf{0}.\mathbf{450}$	$0.058$	$0.215$	$0.437$
	F-score	$0.492$	$0.419$	$0.482$	$0.492$	$0.437$	$0.141$	$0.449$	$0.216$	$0.324$	$\mathbf{0}.\mathbf{505}$
COIL20	ACC	$0.698$	$0.551$	$0.732$	$0.681$	$0.648$	$0.096$	$0.662$	$0.657$	$0.633$	$\mathbf{0}.\mathbf{744}$
$\gamma =0.1$	NMI	$0.796$	$0.732$	$0.812$	$0.806$	$0.786$	$0.041$	$0.750$	$0.752$	$0.769$	$\mathbf{0}.\mathbf{831}$
$\lambda =0.1$	AR	$0.641$	$0.491$	$0.663$	$0.627$	$0.594$	$0.008$	$0.533$	$0.556$	$0.577$	$\mathbf{0}.\mathbf{676}$
	F-score	$0.660$	$0.522$	$0.681$	$0.647$	$0.616$	$0.049$	$0.558$	$0.580$	$0.600$	$\mathbf{0}.\mathbf{692}$
100leaves	ACC	$0.765$	$0.581$	$0.685$	$0.789$	$0.648$	$0.096$	$0.662$	$0.513$	$0.708$	$\mathbf{0}.\mathbf{816}$
$\gamma =0.06$	NMI	$0.902$	$0.813$	$0.856$	$0.905$	$0.786$	$0.041$	$0.750$	$0.673$	$0.882$	$\mathbf{0}.\mathbf{915}$
$\lambda =0.1$	AR	$0.701$	$0.461$	$0.586$	$0.719$	$0.594$	$0.025$	$0.533$	$0.363$	$0.634$	$\mathbf{0}.\mathbf{753}$
	F-score	$0.704$	$0.467$	$0.589$	$0.722$	$0.616$	$0.038$	$0.217$	$0.403$	$0.638$	$\mathbf{0}.\mathbf{755}$

, where the best performance is in bold font. The evaluation metrics results reported in Table 4 illustrate that the proposed method achieves the best performance across the different networks. For instance, Mx-CRTSA is the only method that detects the correct community structure, with a score of 1 in the different quality metrics in both CKM and Cora networks. In particular, the proposed Mx-CRTSA enhances the performance by $3.7\%$, $98.3\%$, $17.9\%$, $4.2\%$, $17.9\%$, $96.8\%$, $5.2\%$, $52.8\%$ and $22.2\%$ in the CKM network in terms of NMI over the CSNMTF, CPNMF, CSNMF, SCML, RMSC, PM, CentroidCoreg, DiMMA and 2CMV, respectively, whereas in Cora, the performance is boosted by $49.2\%$, $55.8\%$, $39.7\%$, $2.1\%$, $49.7\%$, $2.1\%$, $1.8\%$, $54.5\%$ and $49.3\%$ in terms of ACC over the CSNMTF,, CPNMF, CSNMF, SCML, RMSC, PM, CentroidCoreg, DiMMA and 2CMV, respectively. Another example is the NRC network, where Mx-CRTSA improves the performance by $26.9\%$, $39.6\%$, $21.1\%$, $15.7\%$, $17.3\%$, $17.6\%$, $19.2\%$, $24.7\%$ and $20.8\%$ in terms of the NMI over CSNMTF, CPNMF, CSNMF, SCML, RMSC, PM, CentroidCoreg, DiMMA and 2CMV, respectively. In the C. Elegans network, the Mx-CRTSA algorithm improves the quality of the detected structure by approximately $4.8\%$, $12.7\%$, $6.7\%$, $6.1\%$, $20.6\%$, $8.2\%$, $7.7\%$, $16.8\%$ and $17.5\%$ in terms of the NMI over the CSNMTF, CPNMF, CSNMF, SCML, RMSC, PM, CentroidCoreg, DiMMA and 2CMV method, respectively. Furthermore, Mx-CRTSA performs better than CSNMTF, CPNMF, CSNMF, SCML, RMSC, PM, CentroidCoreg, DiMMA and 2CMV by $70.4\%$, $75.3\%$, $71.7\%$, $25.4\%$, $9.4\%$, $46.9\%$, $22.8\%$, $53.7\%$ and $27.5\%$ in the BBCSport network in terms of ACC, respectively. For the 100leaves network, the proposed Mx-CRTSA improves the performance by $53.8\%$, $71.7\%$, $13.9\%$, $3.3\%$, $16.6\%$, $28.8\%$, $5.1\%$, $35.2\%$ and $11.7\%$ in terms of the F-score over the CSNMTF, CPNMF, CSNMF, SCML, RMSC, PM, CentroidCoreg, DiMMA and 2CMV, respectively. The improvement in performance of the proposed approach with respect to the other algorithms and the different metrics is presented in Table

Table 5. Percent of improvement achieved by the proposed method over the other methods.

Network	Metric	CentroidCoreg	PM	RMSC	SCML	CSNMF	CPNMF	CSNMTF	DiMMA	2CMV
NRC	ACC	$+11.5\%$	$+12.2\%$	$+8.6\%$	$+10.0\%$	$+12.5\%$	$+10.2\%$	$+15.7\%$	$+16.5\%$	$+12.9\%$
	NMI	$+19.5\%$	$+17.6\%$	$+17.3\%$	$+15.7\%$	$+21.1\%$	$+39.6\%$	$+26.9\%$	$+24.7\%$	$+20.8\%$
	AR	$+12.3\%$	$+17.8\%$	$+11.5\%$	$+11.3\%$	$+17.8\%$	$+30.4\%$	$+21.0\%$	$+18.4\%$	$+13.7\%$
	F-score	$+11.1\%$	$+2.9\%$	$+9.0\%$	$+9.1\%$	$+13.6\%$	$+14.6\%$	$+15.2\%$	$+16.2\%$	$+13.4\%$
CKM	ACC	$+1.7\%$	$+52.1\%$	$+5.3\%$	$+1.3\%$	$+0.9\%$	$+68.3\%$	$+1.3\%$	$+39.5\%$	$+58.8\%$
	NMI	$+5.2\%$	$+96.8\%$	$+17.9\%$	$+4.2\%$	$+17.9\%$	$+98.3\%$	$+3.7\%$	$+52.8\%$	$+22.2\%$
	AR	$+2.5\%$	$+99.0\%$	$+17.5\%$	$+2.0\%$	$+14.9\%$	$+99.1\%$	$+2.1\%$	$+78.5\%$	$+20.6\%$
	F-score	$+1.7\%$	$+51.7\%$	$+13.4\%$	$+1.4\%$	$+1.0\%$	$+72.0\%$	$+1.5\%$	$+50.9\%$	$+18.8\%$
C. Elegans	ACC	$+1.9\%$	$+6.9\%$	$+13.7\%$	$+3.7\%$	$+11.6\%$	$+6.9\%$	$+10.5\%$	$+14.8\%$	$+15.2\%$
	NMI	$+7.7\%$	$+8.2\%$	$+20.6\%$	$+6.1\%$	$+6.7\%$	$+12.7\%$	$+4.8\%$	$+16.8\%$	$+17.5\%$
	AR	$+8.7\%$	$+5.3\%$	$+14.6\%$	$+8.6\%$	$+5.1\%$	$+11.3\%$	$+11.3\%$	$+17.5\%$	$+19.6\%$
	F-score	$-7.1\%$	$+3.0\%$	$+9.6\%$	$-4.0\%$	$+8.7\%$	$+4.9\%$	$+6.9\%$	$+15.5\%$	$+16.1\%$
Cora	ACC	$+1.8\%$	$+2.1\%$	$+49.7\%$	$+2.1\%$	$+39.7\%$	$+55.8\%$	$+49.2\%$	$+54.5\%$	$+49.3\%$
	NMI	$+1.9\%$	$+9.4\%$	$+47.2\%$	$+9.4\%$	$+52.4\%$	$+87.7\%$	$+55.3\%$	$+64.6\%$	$+55.9\%$
	AR	$+5.1\%$	$+6.1\%$	$+61.1\%$	$+6.1\%$	$+64.4\%$	$+96.5\%$	$+64.7\%$	$+82.4\%$	$+66.3\%$
	F-score	$+3.4\%$	$+4.1\%$	$+40.8\%$	$+4.1\%$	$+38.0\%$	$+52.7\%$	$+42.1\%$	$+46.7\%$	$+43.1\%$
BBCSport	ACC	$+22.8\%$	$+46.9\%$	$+9.4\%$	$+25.4\%$	$+71.7\%$	$+75.9\%$	$+70.4\%$	$+53.7\%$	$+27.5\%$
	NMI	$+38.5\%$	$+65.2\%$	$+15.4\%$	$+42.6\%$	$+95.9\%$	$+98.3\%$	$+96.4\%$	$+79.6\%$	$+33.3\%$
	AR	$+44.4\%$	$+81.7\%$	$+14.3\%$	$+50.2\%$	$+99.2\%$	$+99.5\%$	$+98.5\%$	$+90.8\%$	$+39.7\%$
	F-score	$+33.8\%$	$+52.3\%$	$+10.8\%$	$+38.1\%$	$+76.2\%$	$+78.1\%$	$+76.3\%$	$+59.1\%$	$+29.8\%$
Wikipedia	ACC	$+1.8\%$	$+3.2\%$	$+3.7\%$	$+1.5\%$	$+0.9\%$	$+36.7\%$	$+1.1\%$	$+24.2\%$	$+21.5\%$
	NMI	$+9.6\%$	$+8.5\%$	$+22.7\%$	$+8.0\%$	$+4.6\%$	$+54.3\%$	$+4.8\%$	$+26.4\%$	$+14.8\%$
	AR	$+3.8\%$	$+3.2\%$	$+21.4\%$	$+2.5\%$	$+3.3\%$	$+43.4\%$	$-1.3\%$	$+37.9\%$	$+22.2\%$
	F-score	$+1.3\%$	$+8.6\%$	$+2.3\%$	$+1.3\%$	$+6.8\%$	$+36.4\%$	$+5.6\%$	$+28.9\%$	$+18.1\%$
COIL20	ACC	$+4.6\%$	$+19.3\%$	$+1.2\%$	$+6.3\%$	$+9.6\%$	$+64.8\%$	$+8.2\%$	$+8.7\%$	$+11.1\%$
	NMI	$+3.5\%$	$+9.9\%$	$+1.9\%$	$+2.5\%$	$+4.5\%$	$+79.0\%$	$+8.1\%$	$+7.9\%$	$+6.2\%$
	AR	$+3.5\%$	$+18.5\%$	$+1.3\%$	$+4.9\%$	$+8.2\%$	$+66.8\%$	$+14.3\%$	$+12.0\%$	$+9.9\%$
	F-score	$+3.2\%$	$+17.0\%$	$+1.1\%$	$+4.5\%$	$+7.6\%$	$+64.3\%$	$+13.4\%$	$+11.2\%$	$+9.2\%$
100leaves	ACC	$+5.1\%$	$+23.5\%$	$+13.1\%$	$+2.7\%$	$+16.8\%$	$+72.0\%$	$+15.4\%$	$+30.3\%$	$+10.8\%$
	NMI	$+1.3\%$	$+10.2\%$	$+5.9\%$	$+1.0\%$	$+12.9\%$	$+87.4\%$	$+16.5\%$	$+24.2\%$	$+3.3\%$
	AR	$+5.2\%$	$+29.2\%$	$+16.7\%$	$+3.4\%$	$+15.9\%$	$+72.8\%$	$+22.0\%$	$+39.0\%$	$+11.9\%$
	F-score	$+5.1\%$	$+28.8\%$	$+16.6\%$	$+3.3\%$	$+13.9\%$	$+71.7\%$	$+53.8\%$	$+35.2\%$	$+11.7\%$

. These results demonstrate the enhancement achieved by the proposed algorithm with respect to the other methods, where estimating a clean adjacency tensor and employing it to discover the communities by optimizing the contribution of the individual subspaces to the common subspace boosts the accuracy of the detected community structure.

4.2.2. Convergence Discussion

The solution of the proposed Mx-CRTSA is conducted through an alternating minimization scheme. We start with estimating the low-rank component through ADMM. The individual and centroid spectral embeddings are then estimated through spectral clustering. More precisely, the individual subspaces, ${\left\{{\mathbf{U}}^{\left(m\right)}\right\}}_{m=1}^M$, and the common subspace across all layers, ${\mathbf{U}}^{*}$, along with the weights vector, w, are estimated efficiently using eigenvalue decomposition and HOOI, respectively.

4.2.3. Parameters Selection

In the proposed objective function there are two regularization parameters that need to be tuned, $(\gamma ,\lambda )$, where $\gamma$ is used to control the error component, E, whereas $\lambda$ controls the individual and the common normalized cut terms across the layers. To investigate the performance sensitivity of the proposed Mx-CRTSA to these parameters, both $\gamma$ and $\lambda$ were varied from $0.01$ to 10 and the effect of tuning one of the parameters is explored while keeping the other one fixed. According to Figure 7, the optimal range for $\lambda$ and $\gamma$ to achieve the best results is found to be $\lambda \in [0.1,0.7]$ and $\gamma \in [0.04,0.09]$.

4.2.4. Run Time Comparison

The average run time of Mx-CRTSA was compared to the run times of the other methods, including CentroidCoreg, PM, SCML, RMSC, CPNMF, CSNMF and CSNMTF. The reported results refer to the average run times over 10 realizations and are displayed in the bar graph shown in Figure 8. As the bar graph shows, Mx-CRTSA is slower than all the other methods except CSNMTF, and has a comparable speed to PM and CPNMF. Nevertheless, the proposed algorithm surpasses the rest of the algorithms regarding detecting the community structure, as demonstrated by the experiments.

5. Conclusions

In this paper, a co-regularized tensor-based community detection approach for multiplex networks is proposed. In particular, in the proposed algorithm, a tensor low-rank representation, a normalized cut and a centroid-based co-regularization are combined to reveal the clusters in multiplex networks. The proposed objective function decomposes the multiplex network’s adjacency tensor into a clean low-rank representation and error components. Moreover, the proposed Mx-CRTSA reveals a common centroid set of eigenvectors to uncover the communities in the network by optimizing the contribution of the individual subspaces to the common subspace. The performance of Mx-CRTSA is tested and compared in various synthetic and real multiplex networks with other community detection algorithms.

The proposed approach exhibits robustness to noise as a consequence of estimating the low-rank representation of the original tensor. Furthermore, the distance co-regularization term considered in the proposed objective optimizes the contribution of the individual subspaces that correspond to the separate layers to the common subspace by minimizing the projection distance between them. The contributions of the individual subspaces are optimized along with the consensus subspace through HOOI efficiently. Finally, the experimental results demonstrate that the Mx-CRTSA approach is capable of discovering the communities in multiplex networks with higher accuracy compared to the other algorithms.

For future work, we will consider reducing the high computational complexity of the proposed approach. In fact, the high complexity comes from the computation of the individual subspaces through EVD and the common subspace through high order SVD. Consequently, truncated SVD and partial EVD will be considered to solve this problem.