Mixture Degree-Corrected Stochastic Block Model for Multi-Group Community Detection in Multiplex Graphs
Authors: Noureddine Henka
Mohamad Assaad
Sami Tazi
Venue: Accepted by TMLR ()

Abstract: Multiplex graphs have emerged as a powerful tool for modeling complex data structures due to their ability to handle multiple relational layers. Clustering within a multiplex graph can involve merging vertices into communities that are consistent across all layers, grouping similar layers into clusters, or creating overlapping clusters among vertices and layers. However, a multiplex graph may exhibit distinct vertex communities based on the specific layers to which a vertex is connected. This scenario, termed multi-group community detection, significantly enhances the accuracy of clustering processes and aids in the interpretation of partitions. To date, the current literature on state-of-the-art community detection has not extensively addressed this modeling approach. In this paper, we introduce a novel methodology referred to as the "Mixture Degree-Corrected Stochastic Block Model." This generative model, an extension of the widely utilized Degree-Corrected Stochastic Block Model (DCSBM), is designed to cluster similar layers by their community structures while simultaneously identifying communities within each layer's group. We provide a rigorous definition of the model and utilize an iterative technique to perform inference computations. Furthermore, we assess the identifiability of our proposed model and demonstrate the consistency of the maximum likelihood function through analytical analysis. The effectiveness of our method is evaluated using both real-word data sets and synthetic graphs.

Published in Transactions on Machine Learning Research (02/2025)
Mixture Regree-Correcioo Stochastic Block Model for Multi- Group Community Detection in Multiplex Graphs
Noureddine Henka
R&D Deportement
RTE Prance
Poris France
Mohamad Assaad
L2S, CentroleSupélec
Gif-sur- Yvette, France
Sami Tazi
sami, com.ttuss@rrte-uuneecoom
RØD Deportement
RTE Prance
Poris France
Reviewed on OpenReview: https: // opt.reveee.r...e...em net/ forum? pKKK9KL
Abstract
Multiplex graphs have emerged as powerful tool for modeling complex data structures due to their ability to handle multiple relational layers. Clustering within multiplex graph can involve merging vertices into communitics that consistent across all layers, grouping similar layers into clusters, or creating overlapping clusters amoug vertices and layers. However, multiplex graph may exhibit distinct vertex communities based on the specific layers to which a vertex is connected. This sconario, termed multi-group community detection, significantly enhances the accuracy of clustering processes and aids in the interpretation of partitions. To date, the current literature on state-of-the-art community detection has not extensively addressed this modeling approach. In this paper, we introduce a novel methodology referred Stochastic Block Model." This generative model,
to the 'Mixture an extension of the widely utilized Pegree-Correctte Stochastic Block Model (DCSBM), is designed to cluster similar layers by their community structures while simultaneously identifying communities within each layer's group. We provide a rigorous definition of the model and utilize an iterative technique to perform inference computations. Furthermore, we assess the identifiaiility of our proposod model and demonstrate the consistency of the maximum likelihood function through analytical analysis. The effectiveness of method is evaluated using both real-wort word data sets and synthetic graphs.
1 Introduction
Recent technological advancements have precipitated an exponential increase in data accumulation, conducting to the of big data. This new ecosystem presents novel challenges in exploring and analyzing extensive datasets, highlighted in recent literature Elgendy & Elragal (2014). Often, data comes from multiple sources, presenting multiple perspectives from diverse sources and features. Such datascts require sophisticated models that capture intricate relationships and interdependenccies across different data types and sources Devagiri et al. b: Niu et. al. (2016). To address the increasing complexity, multiplex graphs have become a relevant asset Hammoud Kramer 2020); Zweig 2016
A multiplex graph consists set of vertices connected across multiple layers Han et al. 2023 Magnani et al.
2021 Each layer in multiplex graph uniquely represents set of edges that models specific similarities

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Group 1
L2
Multiplex Gruph Community
Group 2 L3.L4
Detection
Multiplex Graph
United
ayer Communitees
c) Mulb-Group Communites
Communities
Figure 1: Presentation a multiplex graph with layers. The results community detection algorithms on a multiplex graph are categorized into three distinct types of groupings. Each grouping is represented by color-coded boxes, with each color siguifying a different community. In figure (a), unified communities are depicted, showing single community structure shared across all layers. Figure (b) presents visualization of overlapping communitics, where communitics extend over several but not all layers. Finally, figure (c) displays multi-group communities, in which similar layers clustered together, and each cluster maintains uniform community partition.
between vertices, thereby serving as an effective model for multi-relational data. Additionally, multiplex graphs are adept at modeling time-varying data, offering robust framework for dynamic data analysis Xia et al. 2220)).
Clustering has long been recognized effective means for understanding and exploring data across various domains by identifying groups of individuals with strong similarities Fortunato (2010a); Bedi & Sharma (2016). In multiplex graphs, the clustering process poses a significant challenge, yet offers promising solutions for analyzing complex data structures Wang et al. 2019 Community detection in multiplex graphs aims identify groups characterized by high intra- connectivity and low cnter-connettivity |Fotunnto| (2010b).
To this end, numerous algorithms employing diverse approaches such as optimization De Meo et al. 2011
Que et al. (2015), spectral computation Li al. (2018), consensus clustering Mandaglio et al. (2018 and inference Shuo & Chai 2016)) have been developed.
The results of these methods can generally be categorized into two groups. The first group optimizes commuiticcs to be identical and unified across all layers, which may overlook the inherent diversity within each layer, as presented (a) from Figure [ The second approach of multilayer communities allows for overlapping communities across layers and vertices, where vertices may belong to different communities in different layers, as presented in (b) from Figure [ This model, while computationally intensive, also poses challenges in interpretation and constrains the ro-computation of partitions when new layer arrives, especially as the number of layers increases.
Here, we concentrate on a new outcome of multiplex graph partitioning, named as multi-group clustering. This framework clusters layers into groups, with vertices within each group organized into communities that remain consistent across all layers, as shown the third clustering outcome in part (c) of Figure 1 This multi-group perspective not only captures the intrinsic structure of each layer but also simplifies the interpretation of the clustering.
Moreover, clustering multiplex layers into groups of similar networks has become active research area, treating each layer as an individual within population of networks Mantzion et al. (2023) van der Laan et al. (002)). However, previous approaches have often treated the clustering of layers and the partitioning of vertices independently, neglecting the structural differences in community configurations when layers are clustered. Our dual clustering approach thus aims to cnhance both layer clustering and community detection within multiplex graphs, providing comprehensive solution to the challenges posed by complex network structures.
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Although multi-group community detection has not been widely addressed in the literature, its relevance is apparent in real-world datasets. For example, individuals sharing similar musical tastes may not necessarily align in their political or sports affiliations, that proving the structural diversity of layers. In the financial sector, multiplex graphs represent various transaction types within banking systems such as online, wire, and ATM transactions. Grouping these transactions by similar dynamics and identifying communities within these groups are crucial for detecting anomalous behaviors and preventing fraud. Additionally, as we present in experiment section in the field of neuroscience, the study of brain connectivity through magnetic resonance imaging offers profound insights. Our experiments demonstrate that this approach can effectively group subjects with similar neurological diagnoses, illustrating the practical applications of multi-group community detection in medical research.
We introduce a new approach for Multi-group community detection based on the Stochastic Block Model (SBM); a generative model extensively developed for community detection, which recognizes the presence of communitics within a graph by defining the probability of edges between vertices based on the communitics to which they belong Lee & Wilkinson| (2019). SBM has been applied to various graph types including single-layer [bbb (2017), multiplex [arbillon et al. (2017), and dynamic networks [Crneli (2017). The Degreo- Corrected Stochastic Block Model (DCSBM), an advancement of SBM, addresscs the heterogencity of vertex degrees within communities, relaxing the assumption of uniform degree distribution Karrer & Newman (2011). The previous work in Stanley et al. (2015) proposed a model for SBM with multiple strata, which bears some resemblance to multi-group community detection. However, their implementation faces significant limitations when applied to real-world scenarios. These limitations arise due to the strong assumption within the SBM model of uniform distribution within communities, challenges in initialization, and difficulties in determining the optimal number of communitics when used in practical applications.
To overcome the limitations of the SBM models mentioned above, we introduce the Mixture Degree- CorreeCo Stochastic Block Model (MDCSBM), an innovative framework designed for the multi-group partitioning of multiplex graphs. Our method strategically groups similar layers and allocates the vertices within these layers into distinct blocks, as illustrated part (c) of Figure [ We posit that layers grouped together adhere to a uniform DCSBM distribution. The degree-corrected feature of our model allows us to relax the constraints of uniform distribution within communities, thus extending the scope of application of our model. To implement this dual clustering process, utilize the Expectation Maximization (EM) algorithm (in particular we consider Classifier FM (CEM) in this case Celeux & Govaert (1922) to determine the layer groupings, and the Variational Expectation Maximization (VEM) technique for assigning vertices to specific blocks within each group. We also present a thcoretical proof confirming the identifiability of the model and establish the consistency of the maximum likelihood estimates derived from this model. To achieve rapid and accurate result, we propose a novel and efficient initialization method based on spectral representation. The experimental section of the paper evaluates the model's performance and scalability across multiple synthetic graphs with multiple layers. We further validate our model's efficacy using two real-world datasets, providing interpretations of the results and highlighting the advantages of our approach over existing models. Throughout the paper, we define "partitioning" as the process of forming communities of vertices, and "clustering" as the method of grouping layers.
2 Related Works
Community detection in multiplex graphs can yield diverse types of partitions. For instance, some approaches employ spectral representation-basel algorithms and tensor-based algorithms that fuse layers into a centroid graph. This graph is constrained to have K connected components, each representing a community, where K is the number of clusters aimed at capturing highly consistent communities Kang al. (2019;; Wang et al. (2019): Papalexakis et al. (2016). Other methods focus on consensus clustering, which involves computing communities for each layer individually and then identifying the most consistent community across all layers for multiplex community detection [Berlingerio et al.] (2013); Tagarelli et al. (2017); Tang et al. (2012). Additionally, algorithms originally designed for mono- mono-layer community detection have been adapted for multiplex graphs, often allowing for overlapping between vertioes and layers to form multilayer communitics, such as Generalized Modularity and Multilayer Infomap Mucha et al. (2010): Afsarmanesh & Magnani (016);;
De Domenico et al. (2015); Wilson et al. (2017).
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Recent research has also focused on identifying clusters of similar layers that may exhibit the same structure van der Laan et al. (2022). In this approach, each graph is treated as an individual within a population of networks. Various algorithms aim to identify a set of similar layers using methods such as mixture parametric models Mantziou et al. (2023); Kemp et al. (2006), nonparametric methods derived from minimum code length descriptions Kirkley et al. (2023); Kirkley & Newman (2022), models based on graph distance measurements La. Rosa et al. (2015), and latent space models Young et al. (2022).
However, these models, which aim to partition the graph into communities, often overlook the potential for multiple clusters within the multiplex graph and do not consider the similarity of communities between different layers. This dual consideration is crucial for enhancing the accuracy and comprehensibility of results, especially as the number of layers increases. In the filed of tabular data, the dual clustering of both features and samples- referred to as co-clustering has been extensively explored and has proven beneficial for in-depth data analysis Ailem et al. (2015): Nadif & Govaert (2010). Thus, bridging this gap in graph-type data is crucially relevant.
The Stochastic Block Model (SBM) is generative probabilistic model widely used for community detection in networks. Originally developed for Linglelayer networks, SBM has been extended to accommodate multi-layer clustering. A notable example is the introduction of the Multi-Layer Stochastic Block Model (MLSBM), which incorporates various types of layer aggregation Vallè s-Català et al. (2016). In contrast, a directed graph model with layers generated independently from the same distribution was explored in [De Bacco et al. (2017). Furthermore, Paul & Chen (2016) adapted SBM by assigning each layer its own affinity matrix, yet constrained these matrices to yield consistent communities across layers. Additional developments include the work of Amini et al. (2024); ; Roy et al. (2006), which focuses
hierarchical generalization of SBM. This approach seeks to uncover underlying community structures by considering the hierarchical nature of real-world networks. Other adaptations of MLSBM utilize techniques such as Variational Expectation Maximization to infer multi-layer graph structures Barbillon et al. (2017); Corneli et al. (2016): Han et al. (2015). However, these models often grapple with the challenge of exponential parameter scaling, which complicates their application to real-world graphs. Moreover, they generally lack mechanisms for constraining the formation of multi structures within the multiplex graph, a critical aspect for capturing the complexity of such networks.
3 Mixture DCSBM
This work aims to achieve dual clustering process by grouping similar layers into group of layers, and vertices within each group are further clustered into shared block of vertices. The estimation process of MDCSBM involves determining layer-to-ge to-group variables to identify the group of each layer. It is assumed that edges within each group follow a unique DCSBM distribution, treating layers within the same group as independently sampled from the same distribution. Therefore, vertex-to-block variables are estimated for each group to identify the block of cach vertex within the same group.
3.1 Model Definition
Consider a multiplex graph denoted as G= {G', ,...Gr} comprising L layer, G G²= (V, represents a single layer, where s.t I E [1, L] indicates the layer index, V indicates the set of vertices with V N, E' the set be the corresponding adjacency matrix of multiplex graph G,
of edges within layer I. Let A = where A' stands for the adjacency matrix of the graph G¹. The underlying graph model in this study is an unweighted and undirected multiplex graph, where the edge distributions follow a Bernoulli distribution. The generalization this model to a directed graph is straightforward. Finally, an edge A', is defined by a dyad i.j, representing its extremity.
Let us consider partitioning multiplex's graph layers into K groups and assume that the vertices of group k are divided into Q blocks, where & E [1, K]. The probability of having an edge Ab, in layer I, within group k, giving the block assigned to each vertex, under MDCSBM model is expressed as follows:
P(A Zz II' @²) = 8 07
(1)
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where Z [Z] . is the set of vertex-to-block assigmments in the group k and Z fl.... ..... Q")
+ = 10 .... 0% is the set of degree heterogeneity parameter for vertices in group k s.t. 0 > and
Eizt- 0 Ivi E V, E [1, Q". The matrix II has Q x Q elements "q... Vq, w E {L, O+3² Each element represents the probability of an edge existing within the group k depending the block of its dyad {i,5).
Let consider a given layer Go with known vertex-lo-block variable assignments for each group Z = =Z= ..... z* The probability of existing edge A between dyad (i.j) from the MDCSBM model, conditioned on Z, can be described as a mixture distribution of independent DCSBMs, expressed as:
P(A = 1 3,
k=1
Σ g =1,
(2)
Σ 0 = V (1, [1
1,2/=g
where B {p¹ ..... 3K} denotes the set of probabilities of layer I to be generated from group k, representing the mixture weights of MDCSBM, II (IT', _I,,..,,IIT is the set of parameters of each group, and @ 1 6 ..... @K} is the set of degree heterogeneity for each group. This model incorporates K distributions from which layers can be generated.
To address the challenge of maximizing the log- likelikiooood function due to the sum in the mixture model, we introduce a set of latent variables that represents the layer-to-group assignment. Specifically, yik, s.t E [1, LJ and & E [1, K]. takes the value of one when layer / is generated from group k and zero otherwise. The updated formulation for the probability of an existing edge as follows:
P(A 1| (1,
II
(0) (0 T 22 )
(3)
P(yu 1; (atjmm
Additionally, for any group k we identify the probability of vertex i to be assigned to block as follows:
p(Z = qi a = a
(4)
8.1
k
=1
q=1
such that {al,a?,.. a and ax {at, ab .... o }-
Let us consider A ,..,, A K J,, such that A (II* @# a+) be the aggregation group parameters,
the log-likelihood of the proposod model is written as follows:
ΣΣ Yik In3 +
Z²: Z'A A A+)
(5)
where C(A', Z²: A is the complete ligeliill of layer ( in group & with parameters A formulated follow:
C(A =²: A = in(((A' R 2*)) +m(('')a))''
Σ 1, + 11, x²2,22) Σ In(oto0) + Σ In(a
(6)
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In the context of inferring information from a given multiplex graph, the primary objectives involve assigning each layer to specific group k using variable yik: then assigning each vertex i within group & to particular block using variable z, and optimizing the parameters & and A.
The verification of the paramcter's identifiability and the assessment of the maximum likelihood consistency are performed as follows.
3.1.1 Identifiability
The identifiability of the parameters for uni layer Bernoulli SBM has been proved in Celisse et al. (2012) The proof has been extended to a multiplex graph for pillar division Barbillon et al. (0017). We extend this analysis for multiplex DCSBM with multi groups.
Theorem 3.1. Let assume that there K groups and every group has the some number of blocks Q* =Q"= Q Vk, k (I, Q}² Assume for any q E [I,..., Q}, k E {l,.,,y og O, 3 0. Let EJO, 1 1KKQQXKQ diagonal block that contains matrices 1Ifh at diagonal as follow:
I
0
IT
Let also a be a K * Q K Q matriz, which is the diagonálization of ab.....) K vector, and B be a K Q x K Q matrix, which is the diagonilization of [8 33. vector, where 3 is repeated Q times, E f1...., K}. Assume that the elements of r II..3 are distinct. Then the MDCSBM parameters are identifiable under equivalent solutions.
The proof is presented in the supplementary material in A
3.1.2 Consistency of the Maximum Likelihood
Let us assume that the following assumptions hold:
Assumption 3.2. For every 9 + of ,there erists E (1....,Q") such that vqu * nht, or awy * x Fing
Assumption 3.3. There erists
such that V(q, w) E {1,.... Q*}, T <]0, gw [s. 1 - C]
Assumption 3.4. There exists E 1/Q* such that Vi E (1,..., a E|0, 10 0g E 17, - 7]
Assumption 3.5. There exists E 1/K such that vk E K), 8 E/0, 1[ 3 E 1-11-6 E]
Theorem 3.6. Let (O, d) and d') denote metric spaces and let Mn O v R be random function and M θ + R deterministic function such that for every 0
supa(0.6)) (0) ((0) Mexo M()) I|M, M M 0
(7)
(8)
and (0, 0) = argmaz M, (0, v), then
0.
d(ô, 0))
(9)
The proof is presented in the supplementary material in B
4 Optimization of Log Likelihood Function
As explained previously, the MDCSBM depends on layer-to-rouup and vortex-to-block assignment variables. We set an ilerative approach to address this joint assignment clustering challenge. To elaborate, we estimate layer-to-group assignment variables by utilizing the Expectation Maximization (EM) technique. Then, to infer the DCSBM model within cach group, we use the Variational EM (VEM) technique. This technique proves performance in maximizing DCSBM distribution parameters while estimating the latent vertex-to-block variables.

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4.1 Estimation of Layer-to- Group Variables
The computation of layer-to-grouu latent variables estimation is derived from equation 5 which defines the complete log-likelihodd. The estimation process involves calculating the expectation of the log-likelihood based on the posterior distribution of layer-to-group latent variables, and it can be expressed as follows:
Σ Σ Elyu) In8 + LAA, A*)
(10)
i=k==
where E(y(k) is the posterior expectation probability of layer I to be generated from group k, defined as P(uulA' )² Using Bayes theorem, the estimation of layer-to-group computed as follows:
PLA A*) E(Ur) BOPP((A Z A
(11)
where P(A', Zh; A*) is written follows:
P(A = II @*) P*)
(12)
The estimation of layer-to-grom, prioritizes the layer that maximizes the likelihood distribution for specific group, which is presented as the classification step in CEM algorithm. order to assign each layer to a single group, the selection of the group is based on the following:
VIA = argmax Vij
(13)
4.2
Maximization of Likelihood Parameters and Vertex-to-Bloce Variables
Once the layer-to-group assignment is identified, the MDCSBM parameters can be maximized, and the vertex-to-block variable be estimated too.
4.2.1
Maximization of
Considering equation
5 the optimization of B involves expressing the complete log-likelihood as follows:
L*
C(A, Y: =
Σ
Ing* + C(g*)
1=1
(14)
s.t
gh
where ===
- (1 [1, L), Vık = 1}, set of layer of group k, and C(3*) defined as constant regarding 8h By employing the Lagrange multiplier approach, the solution that satisfies the Karush-Kulbs Kuhn-Tucker (KKT) conditions can be expressed as follows:
N 8
(15)
N
where N₁ = -[A|| is the number of layers in the groups k.
4.2.2 Estimation of vertex-to-blee block and Maximization of Parameter A
Our mathematical models assume that the layers within the same group are generated independently from the same DCSBM distribution specific that group. Let a the multiplex graph that contains the layers of group k based equation D the log-likelihood of group & is expressed as follows:
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L(AR, Zt, A
Σ
Σ
Σ in(a2,)) E E Alym((2212)+ (1 Aly)ln(1 =b(2,)
IELK Ciits
IELK (JifT
Qr s.t
(16)
1,
0 =1
1,2"=9
In order optimize the parameters that maximize the previous equation, it essential to first estimate the latent assignment variables. This task is addressed using the Expectation Maximization (EM) algorithm, which requires computing the posterior probability of the latent variable Z with respect to the observed layers, denoted as P(Z*) ) However, for single-layer graphs, it has been demonstrated that calculating this ional probability is computationally intractable Celisse et al. (2012). Various approaches have been proposed in the literature to tackle this challenge Li et al. (2015): ; Lee & Wilkinson (2019), but they tend to suffer from the curse of dimensionality, mainly when dcaling with large-scale datascts.
The Variational EM technique has been adopted to address this issue an allernative technique for handling DCSBM estimation challenges. Previous studies have established the VEM technique's convergence for single-layer SBM and DCSBM models and multiplex SBM graphs Celisse et al.
(2012) Barbillon et al. (0117) 7). by another distribution RA*
The VEM approach involves approximating the posterior distribution P(Z* (A*)
over zh. By leveraging this approximation, the marginal log-likelihoon over Z can be expressed as follows:
C(A*; A = RA
RA (Z C(A*
Σ z
R (Z*K)n (RA
Z
(17)
KL [RAi (Zk)
P(Z A }
where KL is the Kullback-Leible divergence. Therefore, instead of maximizing L(A"; 8h) for the observed data, the VEM technique optimizes lower bound of L(A"; denoted as To Az))). This lower bound is known the evidence lower bound, and it can be defined as follows:
(R L(A": - KL R (Zk)
P(Z 1 A+)
Σ (Z 2
Σ
RA Nog
(18)
$ Z<A" A
The equality between the evidence lower bound and the log-likelihood holds when R (Zk) is equal to the true posterior distribution P(Zz IA:; A²). Maximizing the lower bound To (A)) ) equivalent to minimizing
the Kullback-Leible divergence KL [ (Zk), P(Z/|At A+)]. Regarding to integer nature of vertex-to-block variables, to approximate the postcrior distribution, we select RA (Zk) as follows:
( II 7Z
(19)
where h(;7/)) is multinomial distribution with parameters T = T The entity Th approximates the probability that vertex belongs to the community q in group k. The Io (RGx) can be written follows:
Io(
1 ΣΣΣ -
k Jhe fw
Al Alylnn(t + (1 Aby) ] + ΣΣ
IEL* jt
(20)
ΣΣ
ΣΣ
Thy tty tnl (
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Algorithm 1 Inference of Likelihood of MDCSBM
Input: 9, K, Q - IQ',..,
Output: Y, Y. II, 0,
Initialize Y. with Algorithm
while Iteration < Iteralion max A Not Converge do Estimate Vik with
Compute Sik with
13
Compute ag with 22
Compute Teu with 23
Compute of with 21
Compute The with
24
end while
The parameters that maximize (+AA))) are derived directly from the previously presented formula. To ensure that the vector a and matrix II satisfy the constraints E, a = 1 and I Tew S Vg w 1h....'''' Lagrange multipliers are employed as follows:
0: = EiErh Eg Alj
(21)
EiEr Liseo Alg
4 Σ
(22)
N
EIeLo Dity T Ti
(23)
Eiel Eits Tig jwu
*k by
x 6
II IEL C t0
7
A + (1 -
(24)
where a² Qt f 4* are the best current parameters for the group k. Due to the interdependence between f and gh an effective way to determine the best estimation to alternate between updating i and şk iteratively until convergence. The optimized parameters define the distribution of DCSBM and vertex-to- block assignments for a group k. The same computation is executed for each group independently. The overall method is summarized in the algorithm
4.3 Initialization model
The initialization process of MDCSBM involves setting up values to laycr-to-group variables Y and vertex-to- block variables Z. Effective initialization of these assignment variables contributes to faster convergence and higher chance of recovering accurate ground truth values. In the context of mixture models, the K-means algorithm is commonly employed for initializing assignment variables due to its simplicity and quick response. In this paper, we introduce novel spectral technique that computes hayer-to-group and vertex-to-blost blockk variables, such that clustering results are used as an initialization for inferring the MDCSBM model.
Consider U 122, (r,,,,,,K,KK set of centroid graphs, with each graph 4h being the centroid that represents the group f We aim find layer-to-group group variables by optimizing centroids that best represent each group. Then, each centroid t gets clustered into Qs community to initialize the vertex-to- to-block assignment variable. One way to find the communitics of centroid i is to ensure that it is composed of Q* disconnected components, where each component corresponds to a community in the graph. In network theory, a graph with a Q* component exhibits a multiplicity of Q* mull eigenvalues in its corresponding Laplacian matrix. These mull eigenvalues are the smallest cigenvalues of the Laplacian matrix. Thus, minimizing the Q* smallest eigenvalue of the Laplacian matrix associated with 24 facilitates the formation of Q* disconnected components within the centroid. Therefore, the model aiming to optimize these representations can be formulated follows:

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K 1 A
K
min
2A
Σ
Tr Tr LynF
u'... F.Y i=1 k=1
R=1
s.t. Vi, uig 2 0, 1d u 1, vk:
(25)
K
(F*)TK* = 3m E {0, 1}.
Σ Uik =
where I1-113- denote the Frobenius norm, and w/j represents element of the centroid 4t where Vi,j E V. The Laplacian representation of centroid 4t is denoted by Lu=, and F represents an embedding vector. The Laplacian matrix is computed in its unnormalized version follows:
Lu = Dup Ut
(26)
where Dy= denotes the degree matrix, diagonal matrix. F't E R NxQ* helps to get the number of connected components in the graph. The embedding vector was included in the cost function to relax the non-linear constraint constructing a centroid with Q* disconnected components that effectively represent the communities, leveraging the number of null eigenvalues corresponding to the number of components the graph. The optimization process for this model described in the next subsection. Experimentally, this initialization helps the MDCSBM to converge faster than random initialization, up to more than 30 times faster, which generally depends the data structure.
4.4 Optimization of initialization model
The initialization model described in Section 4.3 in Equation 25 involves multiple variables, making it challenging to optimize them simultaneously. Therefore, we adopt an ileralive technique where each variable will be optimized while the others are held fixed.
4.4.1 Optimizing Y, while U and F are fixed The model can be represented as follows:
L K
min
Me 1u - 1
k=1
(27)
s.t
Vik E 1),
Σ yik =1
By relaxing the constraint yı E {0, 1} to [0, the model becomes linear, facilitating the application of analytical solutions that satisfy the Karush-Kuhn Tucker (KKT) conditions using the Lagrange technique. The analytical solution expressed as follows:
1121 - 1111
VIk
(28)
EV=1 1 - A'1
The determination of the group to which the layer will be assigned is carried out follows:
Yik = argmaax VI
(29)
4.4.2 Optimizing u, while Y and F are fixed
Firstly, the optimization of centroids 4 performed independently, and according to Wang et al. 2020). the objective function Tr(Pk" Lyss Fk) be expressed Eijilff fj fi Therefore, the optimization
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for each centroid can be formulated follows:
min
ΣΣ t
Mua
AigllE - - |C: - fylluus
LEL
(30)
8.1 úij 2 1, 17 11
LA represent the set of layers for which yik equals one. We denote ||f, fill as dij. Due to the independence of optimization each vertex vector u, from the others, the optimization 2t can be formulated as follows:
min Σ l|u
d,1
2
(31)
IET
s.t Vi.j Uij 2 1, 1,17 uu =1
The model mentioned above is quadratic with linear constraints, indicating that is convex. It resolved using the augmented Lagrange multipliers. Otherwise, any quadratic solver can efficiently resolve this problem.
4.4.3 Optimizing F* while U and Y are fixed
The optimization of each F for every group is performed independently of the remaining groups. For a given group; the model can be formulated as follows:
min Tr(F"*
LuFF
(32)
Fk F =1
s.t
The optimal F* can be obtained by extracting Q eigenvectors of the Laplacian matrix Les, which are associated with the smallest Q* eigenvalues. important to note that Q denotes the number of communities within group k.
Algorithm Multi Centroids algorithın initialization
Input G, K, Q = 1Q ,..., QK1 Output: U, Y
Initialize U = 122 ,K,,
while Iteration < Iteration max Not Converge do
optimize Vι with 29
optimize u, with 31
compute F* from the eigen vector associated to Qh smallest eigen value of Lyg.
end while
5 Model Selection
To determine the optimal number of groups K and the number of blocks Qh in each group k; we propose using the Bayesian Information Criterion (BIC). The BIC is formulated follows:
K BIC =2 =
_ ( Log((L 1)) (N (ogg)(()*))))) + (Q* 1)gg(((+*))))
(33)
+2(K 1) log( L(9, Y, Z; A,
In this expression, ||*|| represents the number of layers associated with group k. The BIC helps determining the number of group and blocks by balancing the fit of the model to data penalized by its complexity. The selected model the with the lowest BIC.
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6 Experiments
To assess the properties of the Mixture Degree- Corrected Stochastic Block Model (MDCSBM), we evaluated its performance across various datasets. This analysis includes real-world data on brain connectivity from cerebral imaging and reality mining data concerning proximity interactions among students within a university. Additionally, we utilized multiple synthetic data partitions to explore the limits of MDCSBM and assess its scalability in handling large datasets.
We conducted comparative analyses of MDCSBM against several established algorithms, including the multiplex DCSBM, Generalized Louvain Mucha et al. (2010), and Graph Fusion Spectral Clustering (GFSC) Kang et al. 2019). These algorithus are designed to optimize partitions within multilayer graphs. They are constrained to return single partition for all multilayer graph, which will be used as the partition for each layer. To quantitatively evaluate the performance of each algorithm, we employed metrics such as Average Normalized Mutual Information (NMI) and Average Adjusted Mutual Information (AMI). These metrics were calculated by computing the NMT and AMT scores for each layer independently, then averaging these scores across all layers. Since the algorithms used for comparison return unified community structures across all layers, the same set of communitics is applicd in the assessment for each layer. The results presented here reflect only the mean values of these metrics.
6.1 Real Data Set
In our real-world application, we examined two datasets: diffusion Magnetic Resonance Imaging (dMRT) and the Reality Mining dataset. To the best of our knowledge, neither dataset includes expert annotations or ground truth labels, meaning there is no reference standard for comparison. Consequently, this lack of ground truth prevents fair comparison with other algorithms. Instead, we conducted in-depth analysis of these datasets using our MDCSBM algorithm, offering interpretations and insights based on the results obtained.
6.1.1 Brain Connectivity from Multiplex Graph Representation
To assess the relevance of the MDCSBM in real applications, we conducted a study focusing on brain connectivity using diffusion Magnetic Resonance Imaging (dMRI) We utilized the HNU1 dataset, which includes 300 undirected graphs derived from 10 brain-scanning sessions across 30 individuals Zuo al. (2015) Each graph contains 200 vertices representing different regions of the brain, with edges indicating the observed neural connections between these regions. The dataset treats each graph as an independent observation, aligning with methodologies from prior studies Mantziou et al. 2023 Arroyo et al. (2020).
Our primary goal was to detect groups of subjects exhibiting similar brain connectivity patterns and to identify communitics within these regions. Considering the brain's division into two hemispheres, each graph inherently comprises two blocks. However, inter-subject scan variations reflect the unique neural states of the individuals.
The MDCSBM was tasked with recognizing subjects with closely related brain scans, accounting for each hemisphere. The model was initialized to distinguish 30 groups, corresponding to the mumber of subjects, with two blocks per group reflecting the hemispheres. Using the centroid method for initialization, the model consistently identified two blocks for each snapshot across 100 runs, which supports neuroscientific evidence of significant hemispherical independence. However, also tended to consolidate subjects into 27 distinct groups from the 300 graphs.
Remarkably, the model effectively grouped each subject's graphs within the identified clusters. Specifically, it clustered subjects 8 and 23, 11 and 14, and 10 and 28 together, as detailed in Table T The grouping of subjects 11 and 14 aligns with findings from a semi-supervised study Arroyo et al. (2020). Notably, while that study employed a semi-supervised method to achiove these results, our MDCSBM model operates in a completely unsupervised manner. Despite the numerous hyper-parameters that need to be carefully adjusted, our approach still provides more comprehensive results. Additionally, in the study by Mantziou et al. (203)), the experiment limited clustering to only two groups across all layers, which is not optimal for this dataset.
hhtpp://neeroodatiio/mn//
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containing 30 different subjects. Even with this simplified task, they were unable to consistently group all layers from the same subject within the same cluster. In contrast, our results demonstrate the robust of the MDCSBM algorithm, as it successfully grouped all layers of each subject into same cluster in every instance.
8mbject
Image
Clustor
Nubjonn
Image
Cinster
Subjoct
Image
Clunter
Nubjoct
Image
Chuster
Nubject
Image
Cluster
Subjert
Image
Clunser
C
82
85
S6
813
834
S11
316
1117
8918
819
820
524
10
10
C26
Table 1: The table shows the cluster obtained for cach image of cach subject by the MDCSBM algorithm. Each of the 30 existing subjects has ten images, which are presented in 10 layers.
6.1.2 Reality Mining Study
In this second study, we analyze a physical proximity graph derived from the reality mining study conducted by Eagle and Pentland Eagle & (Sandy), featuring a group of college students and faculty. This dataset comprises undirected graphs representing interactions among 96 students from the Massachusetts Institute of Technology, collected over a nine-month period. Participants were equipped with mobile phones installed with special software that recorded close proximity encounters using Bluetooth technology. Therefore, each layer of this multiplex graph represents a day, with edges indicating at least one obscrved proximity encounter between individuals on that day. The datasel thus includes L = 234 layers, each with [V] = 96 verlices.
Figure illustrates the results from the MDCSBM model applied to this dataset. Sub-figure (d) displays the BIC values relative to the number of groups. We identified five distinct groups, with two communities in groups O and 2, and one community in groups 1, 3, and 4. The single community structure in groups 1,3, and 4 is attributable to the high sparsity of layers within these groups. In contrast, groups like 0 exhibit dual communities as depicted in sub-figure (c), which shows a layer from group O.
Sub-figures (a) and (b) reveal discernible patterns within these groups. Notably, group (colored red) encompasses layers from weekends and holidays, particularly the Christmas period in December. The other groups capture monthly variations; for instance, group (colored blue) corresponds to the start of the semester, characterized by denser layers. Group 1 (colored orange) represents mid-year, aligning with academic examinations and displaying sparser community interactions. Groups 2 and 4 (colored green and purple, respectively) reflect the end-of-year activities, showing varied layer densities. These interpretations are consistent with typical university schedules, where students primarily interact with classmates during academic sessions but engage with a broader range of acquaintances outside of class times.
Sub-figures (a) and (b) reveal discernible patterns within these groups, particularly for groups and 2, which each contain two communities. From both sub-figures, it is evident. that group 0, represented in blue, consists
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Frequency by
Freas anay Layer Goou I' Mayy
((a))
((b))
305
175
Number Cluster
(fc))
((d))
Figure The figure illustrates the outcomes of the Multi-Group clustering applicd the reality mining distance proximity datasct. Panel (a) depicts the total number of days represented in each group. Panel (b) details the monthly distribution of days for each group. Panel (c) presents the community structures within layer from the group 0. Finally, panel (d) displays the BIC values relative to the number of groups.
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of days of the week when courses are likely to oceur in the first part of the year. This group is characterized by denser layers, indicating that students are taking two different courses. Conversely, group 2, represented in green, functions similarly to group but pertains to the second part of the year, again involving students taking two different courses. This explains the presence two communitics within these groups.
Group 3, colored red, encompasses layers from weekends and holidays, with a notable focus on the Christmas period in December. Other groups capture monthly variations; for instance, group 1, colored orange, represents the mid-year period, which aligns with academic examinations and displays sparser community interactions. Group 4, colored purple, reflects end-of-year activities, showing varied layer densities. These interpretations align with typical university schedulos, where students primarily interact with classmates during academic sessions but engage with a broader range of acquaintances outside of class times.
This experiment demonstrates the effectiveness the DCMSBM in real-world applications, adeptly handling complex data structures and offering insightful interpretations of dynamic social interactions.
6.2 Synthetic Data
6.2.1 Variability in Block Size
In this experiment, we aim to assess each group's sensitivity to block size. Therefore, the dataset composed of groups. Each group of layers consists of graphs, each with 100 vertices organized into four blocks. Vertices inside a block are randomly linked each other with a probability Tintra = 0.5, and vertices from randomly connected with probability Tinter 0.3. What distinguishes the groups of layers is the number of vertex in each block, G = {25, 25, 25, 25), G = {20, 25, 30) and G = {30, 30, 20, 20), where G is the 8th group, and for each group, the first number indicates the number of vertex of the first block, the second number the one of the second block, and so forth, as shown in figure
different blocks
20
Group 1
Group 2
Group 3
Figure 3: The adjacency matrices correspond to single layer for each group. G is presented the right, G2 in the middle, and G3 on the left. The intra-block density is set at Tintra 0.5. For better visualization, the inter-bloct connectivity probability is set at Tinter = 0.1, instead of Tinter = 0.3 as an experiment.
The actor-based clustering algorithm multiplex graphs yiclds an average clustering of vertices. In contrast to MDCSBM, there is no differentiation between blocks of vertices for each group of layers, contributing lo the superior NMI/AM AMI results of MDCSBM over other multiplex algorithms, as shown in Table 2 The mean errors between the predicted and generated parameters of MDCSBM are meangrro DDSBM 0.04, affirming the improved parameter recovery of MDCSBM compared the multiplex DCSBM, which results in error of mean PrraParramMuuttDDDSSS. 0.67.
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MDCSBM
DCSBM
GLouvain
GFSC
NMI
100
61.40
77.44
73.80
AMT
100
58.74
75.74
72.65
Table 2:
The NMI and AMI performances on MDCSBM, multiplex DCSBM, Glouvain and GFSC in variability on block size synthetic datasets.
6.2.2 Variability in Block Distribution
In this experiment, we aim to assess the model's performance regarding varialion in block distribution. Therefore, the dataset for this experiment comprises three groups, each containing ten layers. Each graph consists of 100 vertices distributed across four equally sized blocks ((25, 25, 25, 25)) with the same intra- community probability (Fintra = 0.5). The distinction between layer groups lies in the probability of having an edge between blocks, denoted as Tinter = 0.1, Tinter = 0.3, and G = 0.5. This variability tests the algorithm's capability to identify groups with different distributions. Notably, the third group has Winter = Fintra; resembling a random graph without communities.
MDCSBM accurately identifies vertex blocks for each group with an estimated parameter error of 0.04. Additionally, the algorithm recognizes "1 community' for the layers without communities, consistent with random graph. This result demonstrates the MDCSBM's ability to identify the noisy layer. In contrast, multiplex algorithms exhibit an average effect in the vertex-to- block assignment, especially multiplex DCSBM estimates the vertex-to-block with a higher estimation error PreerrgrrrPPrraMMatttt.ooo = 0.71), as observed previously.
mean FrroParam.MDCCSE
MDCSBM
DCSBM
GLouvain
GFSC
NMT
100
55.54
66.66
53.55
AMI
100
55.03
66.66
53.02
Table 3: The NMI and AMI performance for MDCSBM, multiplex DCSBM, Glouvain and GFSC in variability on block size synthetic datasets.
6.2.3 Variability in Number of Layer
In this experiment, test the scalability of the method regarding the number of layers. Therefore, we keep the mumber of vertices, groups, and the block distribution constant while we vary the number of layers for cach group. Similar to Experiment 6.2.1 we define three groups with an equal number of layers but distinct block divisions for each group. Specifically, we set Tittra 0.5 and Tinter = 0.3. The block distribution within each group consists four blocks, as follows: G = [25, 25, 25, 25), G (20, 25, 30), and G (30, 30, 20, 20), where G represents the 5th group.
The result of the following experiments is shown in the figure We can that the performance of the MDCSBM in retrieving the optimal blocks for each layer augments when the number of layers augments, compared to the other methods. It can be explained by the law of large numbers, which delincates the convergence in probability to the expected value as the number of samples Saellesthhheee layers in our case increases. As the layers in the multiplex graph augment, the computation time increases linearly due to the number of parameters scaling lincarly with the number of layers, given a fixed number of groups and blocks. This contrasts with other tested algorithms, which do scale well with large datasets.
6.2.4 Variability in Number of Vertices
In this experiment, we assess the scalability the method regarding the size each graph. Therefore, we fix the number of layers, the number of groups, and the block distribution of each group, and then we vary the number of vertices of the multiplex graph. We set three groups with an equilibrium of 10 layers for each and different block division such that G [25%N, 25%N, 25%N, 25%N), @ (20%N, 25%N, 25%N, 30%%),
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100
NDCSOM AMI
NDCSBM-AMI
DCSBK NNI
DCSBMAMI
GLovain
GFSC
Metrics
15
Layers
Figure The performance of MDCSBM, DCSBM, Glouvain, and GFSC to find the clusters of vertices regarding the number of layers for each group. The NMI and AMT were used as metrics of performance.
and GO = N30% N,,,, 20%N, 20%N]. The x%, means percent from a number of vertices of the layer. The intra-block distribution is Tintra = 0.5 and the inter-distribution is Winter 0.3.
In this experiment, we maintain a constant number of layers, groups, and block distribution for cach group while varying the graph size by raising its number of vertices. Similar to the previous experiments in 6.2.1 define three groups with equal number of layers and distinct block divisions for each.
NDCSBM NM
100
MDCSBM-AMI
NHI
DCSBMAMI
NMI GFSC
Metrics
100
200
300
420
500
600
700
500
Vertices
The performance of MDCSBM, DCSBM, Glouvain, and GFSC to find the clusters of vertices regarding the number of vertices in a multiplex graph. The NMI and AMI were used as metries of performance.
Figure 5:
The obtained result is represented in the figure 5 with thousands of nodes. The time complexity varies regarding the graph's 's size and the block's structure. We study the complexity time of the MDCSBM in the supplementary materials.
We can notice that the MDCSBM scales to large dalaset
6.2.5 Variability in the Number of Groups
The objective this experiment is assess the method's sensitivity with respect to the number of groups within the multiplex graph. Consequently, keep the number of layers in the multiplex graph fixed at 20, then vary the number of groups from 2 to ensuring that each group has distribution distinct from the others.
Figure shows the performance of the algorithins in this experiment. The MDCSBM performs better than the others in finding multigroup community detection. However, we can see that for a high number of groups, the performance of MDCSBM decreases; such result may be explainable by the reduced number of layers for the high number of groups. We can see that this result will be enhanced when each group's 's layer number increases.
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NDCSOM AMI
NDCSBM-AMI
DCSBK NNI
DCSBMAMI
GLovvin
GFSC
Metrics
Number of Groups
Figure 6: Performance of MDCSBM, DCSBM, GLouvain and GFSC when the number of groups varies in fixed number of layer.
6.2.6 Sensitivity to Block Size
In this experiment, we aim to assess the ability of the method to perform in an unbalanced block size. Therefore, construct multiplex graph with 20 layers and groups, such that each group has 10 layers of 100 vertices for each, and each group has two blocks with the same distribution. We fix the size of blocks of one group to be 50 vertex per block, and for the other group, we change the block size from 10 to 50 vertex.
100
NDCSBM-AMI
DCSBM NHI
DCSBMAM
GLovvnn NMI
GLouvain-AN
GPSC XMI
Metrics
Ration between the block sizes %
Figure Performance of MDCSBM, DCSBM, GLouvain and GFSC when the size of block variate. The variate group has two blocks; the sensitivity computed by the fraction between the size of the two blocks.
The performance of the algorithms in this experiment is presented in Figure 6 It is observed that MDCSBM outperforms the others in multi-group community detection. However, with a high number of groups, the performance MDCSBM decreases, potentially due to the reduced number of layers for the high nunber of groups. This result is expected to improve when the number of layers in each group increases.
6.2.7 Sensitivity to Group Size
This experiment aims assess the model's performance when the groups have an unbalanced distribution of layers. We consider multiplex graph with two groups featuring distinct block distributions to achieve this. The number of layers in the first group is fixed at 10, while the number of layers in the second group varies from I to 100. Each layer consists of 100 vertices.
Figure g illustrates various algorithms' performance in this application. Compared to the others, the MDCSBM algorithm demonstrates notable stable performance, even sccnarios with high imbalance in the number of layers. This result is attributed the effectiveness of the joint clustering approach, which we believe contributes to achieving accurate results.
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NDCSOM AMI
NDCSBM-AMI
DCSBK NNI
DCSBMAMI
GLovain
Metrics
Number of layers
Figure 8: Performance of MDCSBM, DCSBM, GLouvain, and GFSC for different numbers of layers in the group. It varies from layers.
6.2.8 Time Complexity
To illustrate the model's time complexity and scalability to large datasets, present the time complexity of the model. To achieve this, we conduct two experiments where the time complexity is evaluated for varying numbers of layers and vertices.
Comoleuit Time (Secen))
Complexty Time (Second)
Layers
Mertices
(())
((b)
Figure 9: Complexity time regarding a number layers and vertices. At the right, the time is counted when the number of layers augments. At the left, the time counted when the number of vertioes augments, The time is counted in seconds.
Figure illustrates the time complexity for each case and for each algorithm. Firstly, MDCSBM models scale better all cases than all other algorithms. Evidently, the time complexity scales linearly with the number of layers. This behavior is attributed to the increase in variables that linearly correlate with the number of layers. Additionally, the time complexity rises more rapidly with an increase in the number of vertices. This outcome results from the significant growth of the combinatorial solution as the number of vertices increases. Despite this, the modeling approach maintains a reasonable time complexity in scenarios with high combinatorial solutions, owing to our efficient initialization process, which facilitates faster convergence compared random initialization. Furthermore, our initialization accelerates convergence a favorable local minimum, achieving speeds over 50 times faster than random initialization.
6.2.9 Comparison Between MDCSBM and the Initialization Model
The proposed initialization model, based spectral representation, be regarded comprehensive approach capable of performing multi-group community detection. To evaluate its performance against the MDCSBM model, we used the test set detailed in 6.2.1 This dataset consists of 3 groups, each containing 10 layers, with each layer comprising 100 vertices organized into four blocks. Vertices within the same block are connected randomly with a probability of Fnntra = 0.5, while connections between vertices different. blocks occur with a probability of Tinter 0.3. The distinguishing characteristic of each group lies in the distribution of vertices within the blocks: G 25, 25, 25, 25, G² 20, 25, 25, 30, and G 30, 30, 20, 20,
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where G" denotes the 1t group. Within each group, these values represent the number of vertices in the first, second, third, and fourth blocks, respectively, as depicted in Figure 3
For this evaluation, we used the MDCSBM algorithm initialized with the proposed spectral model. The experiment was repeated 100 times, and the performance results are summarized in Table
Model Initialization (Spetrale Representation)
MDCSBM
NMI
75.48+-22.80 74.60+-23.88
95.60 +- 17.33
AMI
95.36+-17.98
Table 4: The NMI and AMI performance for MDCSBM and model initialization on block size synthetic datasets.
The superior performance of the MDCSBM model compared to the spectral model can be altributed to the way the MDCSBM utilizes the results produced by the spectral model. The spectral model provides a partition that corresponds to a fixed point of its own cost function, serving as an effective initialization. Although this partition does not necessarily represent fixed point for the MDCSBM, enables the MDCSBM to further refine and enhance the clustering results, ultimately achieving better overall performance.
6.2.10 Comparison Between Random and Proposed Initialization
The MDCSBM model is highly non-convex, and the resulting outputs are significantly influenced by the initialization of layer-to-group and vertex-to-block values. To address this, we proposed an efficient ini- tialization method based on spectral representation. To illustrate the advantages of this technique, we present a comparison of performance and computation time between random initialization and our proposed spectral-based initialization model.
The same test set described in 6.2.1 was used for this comparison. This dataset compriscs 3 groups, cach containing 10 layers, with each layer composed of 100 vertices organized into four blocks. Vertices within the same block are randomly connected with probability of Tintra = 0.5, while vertices between different blocks are connected with a probability of Tinter = 0.3. The distinguishing factor among the groups is the number of vertices in each block: G = 25, 25, 25, 25, G = 20, 25, 25, 30, and G = 30, 30, 20, 20, where G represents the jth group. Within each group, the numbers correspond the vertex count of the first, second, third, and fourth blocks, respectively, as illustrated in Figure
We repeated the experiment 100 times and present the performance results and computation times in Table 5
Model Initialization
Random Initialization
NMI
95.60 +- 17.33
23.734-26.40
AMT
95.36+1-77.
22.70+-27.00
Time Complexity (Second)
3.65+-11.77
130.84+-74.79
Table 5: The NMI and AMI performance, and time computation for MDCSBM by using the proposed model initialization and random initialization on block size synthetic datasets.
We observed that both the computation time and performance were notably improved when using our proposed initialization method compared to random initialization. This improvement can be attributed to the non-convex nature of the MDCSBM model and its sensitivily to initialization. A robust and well-structured initialization method, such as the one we proposed, is essential for achieving better performance and faster convergence.
6.2.11 Model Selection
The defined MSBM model ncods prior knowledge of number of cluster. Ones can think about formulation that allows to optimize the number of cluster in the inference presented in Roy et al. (2006); Amini et al. 2024).. Such models has been based Chinese Restaurant Process (CRP) to determine the number of
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cluster. The CRP modeling in out of the scope of this paper, further enhancement will be reserved for future work. Here in this experiment, we show the results from using he BIC criteria to determine the number of cluster as defined in equation 33 It worth noting to remember that the BIC criteria determine the optimal number of blocks and groups by balancing the fit of the model to the data with a penalization of the model complexity.
In this experiments, we create a multiplex graph of 30 layers. Within this multiplex graph, we set groups of 10 layers for each. We set Tintre 0.5 and Titter 0.3 for all layers. The block distribution within each group consists of four blocks, as follows: G {25, 25, 25, 25), G2 {20, 25, 25, 30), and G {30, 30, 20, 20), where G' represents the tth group.
Number of Cluster
Number of Cluster
((a))
((b))
Figure 10: The BIC metric for finding the optimal number of groups and block. At the left, the number of group is fixed to 3, then the BIC is computed for variate number of blocks. At the right, the number of block is fixed to 4, then the BIC is computed for variale number of group.
Figure 10 shows the variation of the Bic regarding the number of blocks when the number of groups is fixed to 3 at the left, and the variation of BIC regarding number of layers when the number of block is fixed to 4 at the right. We can clearly notice that the minimal value of BIC for both experiences indicates the optimal value of cluster as what is attending, showcasing its performance in toy example to define the number of clustering.
7 Conclusion
Throughout this paper, we have introduced the Mixture Degree-Corrected Stochastic Block Model (MDCSBM) for multi-group community detection in multiplex graphs. The MDCSBM defines the existing groups that share similar community structure. Therefore, for each identified group, a distinct DCSBM is derived to (EM)
ascertain the community structure of each vertex. We have devised an framework for estimating layer-to-group assignment, variables, followed by a Variational EM technique for estimating vertex-ted to-bokkk assignments, A novel centroid methodology has been proposed to initialize layer-to-group to-uu and vertex-to-block variables, enhancing the model's convergence.
This model has been formulated to refine the estimation of the generating model underlying multiplex graphs. It significantly contributes to better comprehension of community structures within multiplex graphs characterized by multigroups of community memberships. While the current presentation exclusively addresses unweighted graphs, potential extensions encompassing the incorporation of weights through alternative probability distributions such as Gaussian or Poisson distributions exist. Such extensions would undoubtedly enrich the model's applicability in capturing the intricacies of diverse real-world scenarios.
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A Identifiability
Proof. We consider the degree node beterogeneity parameter constant and extend the proof from Celisse et al. (2012) to the MDCSBM model as follows. For any group k, rg.k the probability for giving member from block in group h to have conncction with another in the same group Tq.k= = B Let R be Q K square matrix such that Rig. (rg,a) for i E 0, Q x k 1. R is Vandermonde matrix that is invertible by assumptions.
Therefore, for any i = O,... (2Q - 1) K, let set
14 Σ (a q.k
(34)
and M is a k(Q + 1) x KQ matrix such that
Mij Pitj
(35)
For any = 0, ..., Q k, define the Q*K square matrix M by removing line i from the matrix. In hence,
M = RRRR
(36)
where a is Q * K matrix as defined previously, where all og #0. Because R being invertible, then del(M) > 0.
Let us define
QxK Σ del(M
B(X,0)
(37)
1=0
B is of degree Q x K. For vi(0) = (1.r (T. then
B(r((0), 0) = det(M (M), Vi(0))
(38)
The column are linearly combinations of V,, then (rr((0), 0) = 0 for any It means that B can factorized as follow:
KQ-1
B(x, 0) = QXK
II (x rr)
(39)
Let assume the 0 (II, B) and 0 (II', a', 88) are two sets of parameters such that for any multiplex G graph with multi-group model, O) L(g; 0¹). Therefore, get, (x(0) (a(0"), that means that
25

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Published in Transactions on Machine Learning Research (02/2025)
(0) (0") for any i. The B(:0) -((80) because it dependents on the determinant of M, which leads to (0) r0)). Ths R(0)- R(0"), and
( (((()))->M' R(0) a())
(40)
Therefore a= ='. The same steps be applied to proof the idcntifiability of f where the matrix diagonal a is replaced by diagonal matrix of B where every BR, wk E {I,..., K} will be repeated Q times before set B641. It leads to a matrix with the same dimension Q x K. For II, let's define
U. R(0)3(0)) (1033((000((000((0))000000(0000)
From previously, R(0) R(0), (0)) (x())) and ((0) ((0)) then
U (())) II II'
(41)
B Consistency of Maximum likelihood
Proof. The asymptotic consistency of the maximum likelihood estimator of Bernoulli uni layer SBM has been studied in Celisse et al. (2012); Bickel et al. (2013). The proof of the consistency of MDCSBM is straightforward from the proof of MSBM model, which can derived from the proof of uni-layer SBM. The proof can be performed by the same steps by taking
Mn(T, 0,3) = N(N - 1)L
(42)
log
ΣΣ>
3
Bernoulli
2 II
h 1(m
ijj
M(T)
max a.jEAA
ΣΣ
grth ao att
9.w
(43)
Σ *k *k Ogga Owwe logg (T
( ar) ) (1 TTtT Top..o)))
g
where Bernoulli(T) is the Bernoulli distribution with parameter and 8r, a and x² denote the true parameters respectively, with
A= A= =(att) a 2 0, Σ dg. 1}
(44)
26


Reviews:
[{"review_id": "Fir119eP2R", "reviewer": "Reviewer_sYid", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Given the assumptions in the paper the extension is natural and well-formulated. The approach addresses a different niche -- multi-group communities -- compared to existing approaches for multiplex graph. The presentation is clear and the initialisation strategy is reasonable. The experimental evaluation on synthetic data confirms that when the data is indeed generated from the proposed model the algorithm can infer the ground-truth groups and communities. In contrast, the evaluation on realistic data is less convincing. For example the brain connectivity dataset seems relatively easy and is likely closer to a \"toy\" dataset. The data in the reality mining study seems more suited for temporal approaches (the authors assume each layer is one day). Overall, it is not very convincing that there is a \"need\" for such a model in the real world, or that real-world graphs are likely to follow a mixture of DCSBMs.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose an extension of the Degree-Corrected Stochastic Block Model (DCSBM) for multiplex graphs. The graphs in different layers are partitioned into K groups, and the vertices of a given group are partitioned into a group-specific number of communities. The model is essentially a mixture distribution of K independent DCSBMs. This kinds of models are known to be sensitive to initialisation, thus, the authors propose an alternating optimisation procedure to find a good initialisation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- By \"Estimation Maximization (EM)\" and Variational EM do you mean \"Expectation Maximization\"? If yes, please use the established terminology.\n- What is the performance after initialisation (Alg. 2) before running Alg. 1? In other words, the initialisation strategy can serve as a very good baseline for comparison. This can be added to all synthetic experiments. \n- Please provide more details on the experimental evaluation. How is NMI/AMI computed? Is it the average of the NMI/AMI on groups + NMI/AMI on communities? \n- Please provide more details on the baselines. Since based on the exposition none of the baselines seem to be directly designed for multi-group communities how are the used in this setting?\n- Can you provide the code to reproduce the experiments?\n- A lower priority but nonetheless interesting addition would be to compare the proposed initialisation to random initialisation.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "QfKlLT22rZ", "reviewer": "Reviewer_FFDE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The method and the experiments are well described.\n- The considered problem has a wide range of applications\n- The proposed solution seems to identify meaningful layer groups and vertex blocks on real graphs, and its computation requirements seem low enough to make it practically useful.\n\nWeaknesses:\n\n- There are some typos and an incomplete section (see changes) so the paper needs another pass of proofreading\n- The experiments on real data are somewhat limited. For the first experiment, identifying individuals from diffusion MRI scans is of limited usefulness and the approach is not compared to other methods. In the second method, the validation is only a qualitative assessment of the identified clusters and the approach is not compared to other methods.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors consider the problem of community detection within multiplex graphs.\nA multiplex graph is made of a set of nodes, and several layers, each layer being a set of edges between the nodes.\nThe considered task is to detect communities, ie define partitions of the nodes.\n\nPrevious approaches are divided by the authors into \"unified communities\", in which a single partition is found and shared across all layers, and \"multilayer communities\", in which a community can span several layers and a node belongs to a single community for a given layer, but can belong to a different community in another layer.\n\nThe authors propose a third kind of community detection, where layers are clustered into groups, then unified communities are defined for each group.\nThis approach is named \"multi-group communities\".\nIt thus requires a dual clustering to (i) cluster layers into groups and (ii) for a given cluster of layers, cluster the nodes into blocks (ie find a partition).\nThe authors formalize this model, describe an algorithm for identifying the groups and the blocks, and perform experiments on real and synthetic data.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall the method and the experiments are well described so I have mostly minor comments.\n\nIn Eq.3 why does Θ not appear on the left hand-side?\n\nAbove Eq. 6 bold minuscule θk seems to be used to refer to the degree heterogeneity parameter of group k which is elsewhere denoted with a majuscule Θk?\n\nIn section 4.2, should \"vertex-to-group\" be \"vertex-to-block\" instead?\n\nIn 4.2.2: \"writhed\" → \"written\"\n\nSection 4.3 states that the optimization algorithm for the initialization is described in the supplementary material but it is actually in section 4.4\n\nIn section 5.1.1 the proposed algorithm (almost) correctly groups multiple dMRI scans from each of 30 subjects (only collapsing 3 pairs of subjects and never placing scans from a single subject in different groups). While this is a good sanity check, being a structural imaging technique we expect dMRI to be somewhat stable for a given individual. It would be useful to also apply other methods for multiplex graph partitioning to this dataset to see if they succeed or fail to identify individuals. The following points are suggestions for future work rather than requests for this paper: for a more challenging task, it could be worth considering a dataset that has several populations (eg patients for a certain disease and controls) and seeing if the algorithm can group individuals who share some characteristic such as a diagnosis. Finally, the proposed method focuses not only on clustering layers into groups but also vertices into blocks. The experiment is done with a fixed number of 2 blocks (one per hemisphere). It could be interesting to increase the number of blocks and explore the identified partitions, to see if the algorithm identifies densely connected networks.\n\nIn section 5.2.8 it would be useful to compare the time complexity to that of other methods from the literature.\n\nIn section 5.2.9 a reference to an equation is broken (??)\n\nI did not understand section 5.3 \"real world data\". It seems to be truncated and starts to describe experiments that are not mentioned elsewhere in the paper.", "broader_impact_concerns": "none", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "cGzeu2Ihi3", "reviewer": "Reviewer_tnnY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\nThe theoretical analysis of the model definition and variable estimation in this paper is logically clear, and the article includes both theoretical and experimental content, making it relatively comprehensive.\n\n\nWeaknesses:\n1. The main contribution of grouping layers and partitioning vertices is not novel, as it was previously analyzed in \"Clustering Network Layers with the Strata Multilayer Stochastic Block Model\" by Natalie Stanley et al. (IEEE Transactions on Network Science and Engineering, 2016).\n2. While the paper introduces a new statistical model and employs algorithms with strong theoretical foundations, it lacks an analysis of the theoretical performance of these algorithms.\n3. The main theoretical results are provided in the appendix, but they seem to be direct extensions of previous work without demonstrating technical difficulties or innovative points in the analysis.\n4. As a theoretical model with strong assumptions, the experiments in this paper mainly focus on analyzing the proposed algorithm, with comparisons to other methods conducted only on simulated datasets, lacking substantial practicality analysis. And the analysis of experimental results is confusing. \n\nOverall, the paper lacks sufficient theoretical depth, and the proposed algorithm does not possess significant practical value.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "1. This paper proposes the \"Mixture Degree-Corrected Stochastic Block Model\" (MDCSBM), which aims to cluster similar layers by their community structures while simultaneously identifying communities within each layer's group. \n2. This paper provides a rigorous definition of the model and uses an iterative technique for inference computations.\n3. The effectiveness of the method is assessed using both real-world datasets and synthetic graphs.\n4. This paper evaluates the identifiability of the proposed model and demonstrates the consistency of the maximum likelihood function through analytical analysis in supplementary materials", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Please review the format of the paper, paying special attention to the misuse of mathematical symbols, unclear mathematical representations, and the layout of figures and tables.\n2. There are also unclear analysis in the experimental section. Explanations and descriptions of the figures and tables are unclear. Specifically, real-world data is mentioned again in Section 5.3 at the end of the experiments; how does this differ from Section 5.1? What are the experimental results, figures, and tables in this part?\n3. Does the analysis of the algorithm's consistency in the appendix appear to be a simple extension of prior work? If not, please explain what technical innovations are present.\n4. Please provide clear definitions or explanations of the metrics used in the experiments, such as BIC, and offer a more detailed introduction of the datasets used in the experimental section.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: All of reviewers feel that the paper provides a contribution to the problem of community detection. The problem is well formulated and the approach addresses a different niche (multi-group communities). Some of concerns raised by reviewers were well resolved. Although there are still some concerns about experiments on real-world data, I believe the paper is deserved to be published.
Decision: Accept (poster)