Rafika Boutalbi
email: rafika.boutalbi@ipvs.uni-stuttgart.de

Lazhar Labiod
email: lazhar.labiod@u-paris.fr

Mohamed Nadif
email: mohamed.nadif@u-paris.fr

Psbm Plbm

Splbm Tsplbm

Implicit Consensus Clustering from Multiple Graphs

Keywords: Three-way data, Multiple graphs, Co-clustering, Consensus Accuracy | NMI Metrics Accuracy NMI Accuracy | NMI Metrics Accuracy NMI PSBM PLBM SPLBM TSPLBM Nus-Wide-8 Dataset Simimarity CH Similarity CM55 Similarity CORR Similarity EDH Similarity Accuracy | NMI Metrics Accuracy NMI Accuracy | NMI Metrics Accuracy NMI

Dealing with relational learning generally relies on tools modeling relational data. An undirected graph can represent these data with vertices depicting entities and edges describing the relationships between the entities. These relationships can be well represented by multiple undirected graphs over the same set of vertices with edges arising from different graphs catching heterogeneous relations. The vertices of those networks are often structured in unknown clusters with varying properties of connectivity. These multiple graphs can be structured as a three-way tensor, where each slice of tensor depicts a graph which is represented by a count data matrix. To extract relevant clusters, we propose an appropriate model-based co-clustering capable of dealing with multiple graphs. The proposed model can be seen as a suitable tensor extension of mixture models of graphs, while the obtained co-clustering can be treated as a consensus clustering of nodes from multiple graphs. Applications on real datasets show the interest of our contribution.

Introduction

Relational data are ubiquitous in various fields (web, biology, neurology, sociology, communication, economics, etc.), and their accessibility has kept increasing in recent years. These data, as a whole, form a network formalized by a graph, where each node is an entity, and each edge is a connection between a pair of nodes; this graph can be directed or not. We find this situation in various scientific publications; the relationships between documents can often be described as multiple graphs with different types of links. In fact, several relationships, such as co-terms, co-authors, co-keywords, and co-references between documents can be used. The objective of this work is to address the clustering of multiple graphs. This is a graph mining task of clustering vertices into several groups in the presence of multiple types of proximity relations. We could hypothesize that the combination of different information that arises from multiple graphs may improve the clustering results. For instance, two documents which share a number of words and/or have one or more authors in common and/or quote each other, are likely to deal with the same topic. Incorporating this additional information leads us to consider a tensor representation of the data.

To deal with multiple graphs, various models and methods under different approaches are proposed to analyze these networks. In [START_REF] Banerjee | Multi-way clustering on relation graphs[END_REF][START_REF] Tang | Clustering with multiple graphs[END_REF], the authors proposed a multiway clustering framework for relational data, where different types of entities are simultaneously clustered, based not only on their intrinsic attribute values, but also on the multiple relations between the entities. Other works use a spectral decomposition-based approach relying on the combination of adjacency matrices [START_REF] Tang | Clustering with multiple graphs[END_REF][START_REF] Chen | Block spectral clustering methods for multiple graphs[END_REF][START_REF] Nie | Self-weighted multiview clustering with multiple graphs[END_REF]. In these works, the clustering is not the main objective of the proposed approaches, nevertheless it can be deduced from decomposition results.

On the other hand, one of the most used methods in this context is the Stochastic Block Model (SBM) [START_REF] Nowicki | Estimation and prediction for stochastic blockstructures[END_REF] which is a probabilistic approach. SBM is commonly used for network modeling and discovering the latent community structures from a graph. It provides a statistical approach able to model data matrix, symmetric or not, into homogeneous blocks. This leads to consider SBM [START_REF] Daudin | A mixture model for random graphs[END_REF] as a particular case of the Latent Block Model (LBM) proposed by Govaert andNadif (2003, 2005) and extended in [START_REF] Shan | Bayesian co-clustering[END_REF][START_REF] Govaert | Mutual information, phi-squared and model-based co-clustering for contingency tables[END_REF], which models any kind of data matrices not necessarily square or symmetric. In other words, the clustering of the graph directed or not, is in fact, a particular case of co-clustering. In this work, we consider graphs represented by adjacency matrices assimilated to contingency tables. Thus, considering the previous example of document clustering, the relations between documents (co-terms, co-authors, etc.) are count data and can be represented by particularly sparse contingency tables. Many works in the literature show the interest of Poisson distribution for graph theory and clustering of random graphs [START_REF] Janson | Poisson convergence and poisson processes with applications to random graphs[END_REF][START_REF] Daudin | A mixture model for random graphs[END_REF].

To the best of our knowledge, this is the first attempt to formulate a model-based co-clustering for sparse three-way data. To this end, we rely on the latent block model [START_REF] Govaert | Mutual information, phi-squared and model-based co-clustering for contingency tables[END_REF] for its flexibility to consider any data matrices. Figure 1 presents a binary three-way dataset constructed from multiple graphs, and the expected results in terms of co-clustering. The key contributions of this work are: -Finally, using the ensemble method, we prove that the proposed algorithm, which can be viewed as an implicit consensus clustering for multiple graphs, is more effective than explicit clustering obtained by traditional consensus clustering methods.

v v v v
The remainder of this paper is organized as follows. In Section 2, we present related work and show the strong points of our approach. Section 3 reviews Poisson LBM, shows the limits of traditional PSBM and describes Sparse PLBM (SPLBM). Section 4 discusses the extension of SPLBM to consider multiple graphs. In Section 5, we present a variational Expectation-Maximization algorithm. Section 6 is devoted to evaluating our approach. Finally, section 7 concludes the paper and gives some directions for future research.

Related Work

Although SBM is popular in social networks analysis, dealing with the count data and due to the degree of heterogeneity, the traditional SBM fail to detect relevant clusters of edges to adress community detection problem (Qiao et al., 2017). Thereby, several authors have developed a degree-corrected SBM. In [START_REF] Karrer | Stochastic blockmodels and community structure in networks[END_REF], using a Poisson SBM, they introduced a parameter θ i controlling the degree of expected degrees of vertices i. They consider that each x ij with i = j is distributed according to Poisson(θ i θ j δ k ), where δ k is the expected value of the adjacency matrix for the vertices i and j lying in block (k, ) while x ii is distributed according to Poisson(1 2 θ 2 i δ kk ). Doing so and under some constraints on the θ i 's, they proposed the DC-SBM (Degree-Corrected SBM) clustering algorithm (DC-SBM 1 ) from an undirected graph on n vertices, possibly including self-edges. Furthermore, they established the equivalence between the maximization of the log-likelihood and the maximization of mutual information used as an objective function for clustering bipartite graphs [START_REF] Dhillon | Information-theoretic co-clustering[END_REF]. It is important to emphasize that the model proposed in [START_REF] Karrer | Stochastic blockmodels and community structure in networks[END_REF] is similar to that proposed by [START_REF] Nadif | Block clustering of contingency table and mixture model[END_REF], where the authors also showed this connection with the maximization of mutual information; they proposed the Croinfo algorithm as illustrated in Figure 2. In fact, the objective function maximized by DC-SBM, which can also be used for the co-clustering of an undirected graph, is associated with a constrained Poisson LBM commonly used in the co-clustering context; see e.g.; (Ailem et al., 2017a,b). To sum up, considering DC-SBM which implies that the data are generated according to a Poisson LBM with P(x ij , x i. x .j γ k ) where P(x ij ; λ) = e -λ λ x ij

x ij ! , the proportions of the classes of the nodes are assumed to be equal. In addition, although both algorithms DC-SBM or Croinfo are different, the objective is the same, and the clustering considered is based on an approach similar to that of the traditional hard clustering algorithms; for more detail, the reader can refer to recent works (Govaert and[START_REF] Govaert | Mutual information, phi-squared and model-based co-clustering for contingency tables[END_REF].

In our contribution, we structured graphs as three-way data where the clustering is the principal objective. We propose an extension of LBM to tackle the co-clustering of multiple undirected/directed graphs where each cell of the diagonal is not necessarily equal to an even number as conventionally considered in community detection. To do this, we adopt an EM-type approach to refer to the Expectation-Maximization algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF][START_REF] Mclachlan | Finite Mixture Models[END_REF]) and not Classification EM [START_REF] Celeux | A classification EM algorithm for clustering and two stochastic versions[END_REF]. Furthermore, we will show that this purpose can be viewed as an implicit consensus clustering from Multiple Graphs.

Poisson Latent and Stochastic Block Models

Given an n × d data matrix X = (x ij , i ∈ I = {1, . . . , n}; j ∈ J = {1, . . . , d}), it is assumed that there exists a partition on I and a partition on J. A pair of partitions (Z, W) will represent a partition of I × J into g × m blocks. The partition Z for rows can be represented by a label vector (z 1 , . . . , z n ) where z i ∈ {1, . . . , g} or a binary matrix in {0, 1} g satisfying g k=1 z ik = 1. In the same manner the partition W for columns can be represented by a label vector (w 1 , . . . , w d ) where w j ∈ {1, . . . , m} or a binary matrix in {0, 1} m satisfying m =1 w j = 1.

Poisson Latent Block Model (PLBM)

Denoting Z and W the sets of possible labels Z for I and W for J, the marginal density function f (X; Ω) of the Poisson Latent Block Model (PLBM) [START_REF] Govaert | Mutual information, phi-squared and model-based co-clustering for contingency tables[END_REF]) can be written

f (X, Ω) = (z,w)∈Z×W i,k π z ik k j, ρ w j i,j,k P(x ij ; x i. x .j γ k ) z ik w j (1)
where Ω = (π, ρ, γ), with π = (π 1 , . . . , π g ) and ρ = (ρ 1 , . . . , ρ m ) where (π k = P (z ik = 1), k = 1, . . . , g), (ρ = P (w j = 1), = 1, . . . , m) are the mixing proportions of row and column clusters respectively, and γ = (γ k ; k = 1, . . . g, = 1, . . . , m). For this model, the complete data are taken to be the vector (X, Z, W) where unobservable Z and W lead to the labels, the resulting complete data log-likelihood can be written as follows:

L C (Z, W, Ω) = log f (X, Z, W; Ω) = i,k z ik log π k + j, w j log ρ + i,j,k,
z ik w j log P(x ij ; x i. x .j γ k ).

To estimate Ω, we consider the EM algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. However, the E-step using the log-likelihood of (1) directly is intractable due to the dependence structure among the rows and columns. [START_REF] Govaert | An EM algorithm for the block mixture model[END_REF] suggest a variational approximation in relying on the interpretation of EM due to [START_REF] Neal | A view of the em algorithm that justifies incremental, sparse, and other variants[END_REF]. This leads to consider the following criterion

L C ( Z, W, Ω) + H( Z) + H( W) (2)
where L C ( Z, W, Ω) is the fuzzy complete-data log-likelihood. H( Z) =i,k z ik log z ik with P (z ik = 1|X) = z ik , and H( W) =j, w j log w j with P (w j = 1|X) = w j are the entropies.

Poisson Stochastic Block Model

As we mentioned earlier, Poisson SBM, even DC-SBM, are particular cases of Poisson LBM insofar as the latter can model matrices, symmetric or not, oriented or non-oriented graphs, numbers of row clusters and columns clusters not necessarily equal (g = m) and finally with proportions of clusters equal or not. Therefore the transition from LBM to SBM is easy to show. Thereby, for undirected graph, the maximization of (2) leads to maximizing

L C ( Z, Ω) + 2H( Z)
which is proportional to

i,k z ik log π k + 1 2 i =j,k = z ik w j log P(x ij ; x i. x .j γ k ) + 1 2 i,k z ik log P(x ii ; x i. x i. γ kk ) - i,k z ik log z ik .
The main differences between them are a) considering the Poisson SBM, the last term, which concerns the diagonal of X, is skipped and it does not take into account the degree of nodes, unlike LBM which considers the diagonal elements. b) with Poisson LBM, x ij |z ik w j = 1 ∼ P(x i. x .j γ k ), while with SBM x ij |z ik w j = 1 ∼ P(γ k ). Notice that γ k depends only on the block k and not on the margins. Thereby, starting from PLBM, next we will see how to take into account the sparsity often present in the graphs.

3.3 PLBM for sparse data: Sparse PLBM (SPLBM)

Recently, in (Ailem et al., 2017b), the authors proposed a generative mixture model for co-clustering document-term matrices referred to as SPLBM. With this model, they assume that for each diagonal block kk the values x ij ∼ Poisson(λ ij ) where

λ ij = x i. x .j k [z ik w jk ]γ kk or x ij |z ik w jk = 1 ∼ P(x i. x .j γ kk )
and for each block k with k = , x ij ∼ Poisson(λ ij ) where the parameter λ ij takes the following form:

λ ij = x i. x .j k, =k [z ik w j ]γ or x ij |z ik w j = 1 ∼ P(x i. x .j γ).
Assuming ∀ = k, γ k = γ leads to suppose that all blocks outside the diagonal share the same parameter. SPLBM has been designed from the ground up to deal with data sparsity problems. As a consequence, in addition to seeking homogeneous blocks, it also filters out homogeneous but noisy ones due to the sparsity of the data. The pdf of SPLBM can be written as follows:

f (X, Ω) = (z,w)∈Z×W i,k π z ik k j,k ρ w jk k i,j,k (P(x ij ; λ kk )) z ik w jk i,j,k, =k (P(x ij ; λ)) z ik w j .
Assuming that the complete data are (X, Z, W), the complete data log-likelihood L C (Z, W, Ω) takes the following form : To estimate the parameters Ω, Z and W. To this end, a variationnel EM has been proposed (Ailem et al., 2017b) to maximize (2) where L C ( Z, W, Ω) is the new fuzzy complete-data log-likelihood.

log   i,k π z ik k j, ρ w jk k i,j,k e -x i. x .j γ kk (x i. x .j γ kk ) x ij x ij ! z ik w jk i,j,k, =k e -x i. x .j γ (x i. x .j γ) x ij x ij ! z ik w j   . (a) (b) (c)
Note that although SPLBM is a co-clustering model, we can derive a graph clustering algorithm from an adjacency matrix (symmetric or not). Thereby, when we are dealing with undirected graphs; strating with the same initialization of z and w (z (0) = w (0) ), we obtain the same row and column clusters, that is essential for the undirected graph clustering problem.

PSBM, PLBM and SPLBM for graphs

Although PLBM can deal with sparse matrices, SPLBM can be more suitable for sparse matrices (Figure 3). It is designed to seek a diagonal block structure and capture the most reliable associations between the rows and columns object clusters. SPLBM assumes that each diagonal block (or cocluster) is generated according to the Poisson distribution with some specific parameters, and each non-diagonal co-cluster representing noise data is generated according to Poisson distribution with identical parameters. In Figure 4 we report the graphical models of Poisson models discussed in the paper.

xij zi π λk PSBM x ij z i θ i θ j δ k DC-PSBM/CroInfo xij zi π wj ρ xi.x.jγ k PLBM xij zi π wj ρ xi.x.jγ kk xi.x.jγ SPLBM Fig. 4: Graphical models: z i is the label of row i, w j is the label of column j.
To clarify expectations and the impact of this parameterization, on the political blogs dataset 2 , we applied the clustering algorithms derived from SBM, PLBM, and SPLBM, using 30 random ini-tializations and measured the clustering accuracy. Figure 5 shows the interest of SPLBM, which takes into account the sparsity often present in a graph network.

The properties of this parameterization prompt us to adopt it for co-clustering with multiple graphs, as illustrated in Figure 1. Next, to avoid confusion between all the rows and columns that are identical in our case, we still keep the notations using the z ik 's and w j 's. The presented models PSBM, PLBM, and SPLBM deal with adjacency matrices (2D data matrix) to tackle the problem of graph clustering. In the sequel, we deal with multiple graphs organised as 3D data matrix; each matrix depicts a graph.

SPLBM with multiple graphs

Three-way tensor characteristics

A tensor is a multidimensional array, which is also known as the N -way, Nth-order tensor. A tensor can be viewed as an element product of N vector spaces [START_REF] Kolda | Tensor decompositions and applications[END_REF]. This notion of tensors should not be confused with tensors in physics and mathematics fields such as stress and strain tensors [START_REF] Frankel | The Geometry of Physics: An Introduction[END_REF].

A three-way tensor or third-order tensor has three dimensions and then has three indices, as shown in Figure 6. A first-order tensor is a vector, a second-order tensor is a matrix, and tensors of order three or higher are called higher-order tensors. The notation used here is very close to that introduced by [START_REF] Kiers | Towards a standardized notation and terminology in multiway analysis[END_REF] for third-order tensor. Notice that scalars are represented by lowercase letters e.g. x, and vectors are expressed by a bold lowercase letter e.g. x. The matrices are denoted by bold capital letters e.g. X. And finally, tensors are indicated by bold capital Euler letters e.g. X . The ith element of vector x is denoted as x i , the element (i, j) of a matrix is expressed by x ij , and x b ij represents the element (i, j, b) of a tensor. The order of tensor is referred to as the number of dimensions, also called ways or modes. Onemode tensor is a vector, second-order tensor is a matrix, and third-order tensor is a cuboid. In the case of matrix X, a row and column can be denoted by x i: and x :j , respectively. In the case of three-way tensor x ij: , x i:b , and x :jb represents the vector of the three different modes respectively. As we focus on frontal slices, the tensor can be represented by {X b , b = 1 . . . , v} (Figure 7(c)). For convenience, in the following, we will denote the tensor entry x ij: by

v d n b = 1 . . . v j = 1...d i = 1...n
x ij = (x 1 ij , . . . , x b ij , . . . , x v ij ) (Figure 8); then x b i. = j x b ij and x b .j = i x b ij .
In this sequel, we aim to extract homogeneous sub-tensors from three-way data.

Definition of the proposed model

We extend SPLBM to Three-way tensor data leading to Tensor SPLBM (or TSPLBM). The proposed model seeks not only to discover homogeneous tube co-clusters (a three dimensional co-clusters) but also discover important blocks and ignore noisy ones. Thereby, TSPLBM allows to discover a diagonal co-clusters structure, which are tubes (through all slices) from the three-way tensor. It makes it more useful for sparse tensor with high sparsity close to 90%, as shown in the experiments. TSPLM provides a better partitioning than the classical co-clustering algorithm applied on each slice of tensor separately or a consensus clustering used on these independent results.

Our proposal Tensor SPLBM considers 3D data matrix X = [x ij ] ∈ R n×n×v where n is the number of nodes, and v the number of graphs (slices). Figure 1 presents a tensor data with v graphs. Assuming the independence per graph, the conditional Poisson pdf is given by

n i,j=1 g k=1 { v b=1 P(x b ij ; x b i. x b .j γ b k )} z ik w jk g k, =k { v b=1 P(x b ij ; x b i. x b .j γ b )} z ik w j .
As X is symmetric per slice b, when i = j we have z ik = w jk and for k = 1, . . . , g we have

π k = ρ k .
Then to optimize the lower bound of log-likelihood criterion noted F C ( Z, Ω) leads to optimize the following criterion (Eq. 3) (Appendix A for more details).

1 2 F C ( Z, Ω) = 1 2 L C ( Z, Ω) + H( Z) (3)
where H( Z) =i,k z ik log z ik is the entropy, and L C ( Z, Ω) is the fuzzy complete log-likelihood function expressed by:

L C ( Z, Ω) = 2 i,k z ik log π k + i,j,k z ik z jk v b=1 log P(x b ij ; x b i. x b .j γ b kk ) + i,j,k, =k z ik z j v b=1 log P(x b ij ; x b i. x b .j γ b ).
After some algebraic calculations, we can simplify the criterion (up a constant) that takes the following form (for more details, please see Appendix B)

i,k z ik log π k + 1 2 b k x b kk log( γ b kk γ b ) -x b k. x b .k (γ b kk -γ b ) + N b (log(γ b ) -N 2 b γ b ) + H( Z). (4)
where

x b k. = i z ik x b i. = j z jk x b .j = x b .k , x b kk = i,j z ik z jk x b ij , and N b = i,j x b ij ,

Variational Inference

To estimate the parameters of the model, we rely on the Variational EM algorithm [START_REF] Govaert | An EM algorithm for the block mixture model[END_REF], and we extend it to multiple graphs. In the sequel, the proposed algorithm is referred to as TSPLBM.

E-step. It consists in computing, for all i, j, k the posterior probabilities z ik and w jk given the estimated parameters Ω. As k z ik = k z jk = 1, using the corresponding Lagrangians, up to terms which are not function of z ik , leads to log z

(t+1) ik ∝ log π k + 1 2   j,k z (t) jk v b=1 P ijb kk + j =i,k = z (t) j v b=1 P ijb k   , (5)
where

P ijb kk = log P(x b ij ; x b i. x b .j γ b kk ) and with k = , P ijb k = log P(x b ij ; x b i. x b .j γ b ). The update of z (t+1) ik
is described in Appendix C, and z

(t)
ik represents the value of z ik in the previous iteration (t).

M-step. Given the previously computed posterior probabilities Z, the M-step consists in updating, ∀k, the parameters π k , γ b kk and γ b . The estimated parameters are defined as follows. First, taking into account the constraints k π k = 1, it is easy to show that π k = i z ik n . Secondly, it is easy to obtain for all b, k

γ b kk = i,j z ik z jk x b ij i z ik x b i. j z jk x b .j = x b kk [x b k.
] 2 and,

γ b = N b -i,j,k z ik z jk x b ij N 2 b -k i z ik x b i. j z jk x b .j = N b -k x b kk N 2 b -k [x b k. ] 2 . ( 6
)
The TSPLBM algorithm (Algorithm 1) for multiple graphs (MG), alternates the two previously described steps Expectation-Maximization. At the convergence, a hard co-clustering is deduced from z ik 's using the maximum a posteriori principle.

Algorithm 1: TSPLBM Input: X , g.

Initialization: Z (0) randomly and compute

Ω (0) , t = 0 repeat E-Step: Compute z(t+1) ik z (t+1) ik ∝ π k exp 1 2 j z (t) jk v b=1 x b ij log( γ b kk γ b ) M-Step: Update Ω (t+1) = (π (t+1) k , (γ b kk ) (t+1) , (γ b ) (t+1) ) given by π k = i z (t+1) ik n , γ b kk = x b kk [x b k. ] 2 , and γ b = N b -k x b kk N 2 b -k [x b k. ] 2
until the objective function value change is small or there is no change; return Z, Ω 6 Experiments

In our experiments, we aim to discuss three important questions about (i) The importance of considering multiple graphs simultaneously on clustering results through TSPLBM and comparison with baselines considering one graph each time. (ii) The second point shows how the proposed model can help with the interpretation of the obtained results. (iii) And finally, we made a parallel between the proposed approach and clustering ensemble, and we compare implicit consensus obtained by TSPLBM and the explicit consensus achieved by the clustering ensemble method.

Datasets and evaluation

We use four datasets with a different number of graphs (slices) and clusters. Table 1 shows the characteristics of datasets in terms of the type of instances (image or image+text), the number of graphs/slices (#Graphs), the number of instances (#Nodes), and the number of clusters (#Clusters).

Hereafter, we give in detail, the description of each dataset.. Co-terms Title, and Co-terms Abstract, are adjacency matrices representing the co-terms between documents on the title and abstract, respectively. The Co-terms T matrix is computed using BB , where B is a binarized documents-terms matrix, then ∀i, T ii > 0. We also have Co-authors graph denoting the number of joint authors for two documents.

Nus-Wide-8 dataset: It is a part of the Nus-Wide images dataset4 extracted using Flickr API. This dataset is composed of eight topics, namely Animals, Persons, Plants, Snow, Street, Temple, Town, and Wedding. We constructed six graphs -the Co-tags graph, which is an adjacency matrix of common tags between images. As described in the previous paragraph for Co-terms matrix, we used a binary matrix images-tags M to compute Co-tags matrix H by MM . Other graphs are also created based on extracted features from images The computed similarity matrices are converted to adjacency matrices by putting one if the similarity is higher than ninety-seven percent quantile and zero otherwise.

Amazon-products-10 dataset: It is a part of the Amazon-products dataset5 , composed of product images. We consider ten product categories, namely Beauty, Digital music, Home and kitchen, Office products, Cell phones, Sports and outdoors, Health and personal care, Clothing-Shoes-Jewelry, Patio-garden, and Baby. We constructed seven graphs. The three first one Similarity LBP, Similarity Haralick and Similarity Gabor are constructed based on Low Rank Representation (LRR) method (Liu et al., 2013a) ). The computed similarity matrices are converted to adjacency matrices by putting one if the similarity is higher than ninety-seven percent quantile and zero otherwise.

Co-terms Title and Co-terms Description are adjacency matrices representing the co-terms between the title and description of products, respectively. Finally, Co-viewed and Co-purchased are adjacency matrices Y, where Y ij = 1 means that these two products are viewed (respectively purchased) simultaneously when users make a query.

Figure 9 shows all graphs (slices) for the Amazon-products-10 dataset. The dataset is composed of seven graphs. We notice that each slice has different structures and different degrees of complexity. Our TSPLBM input is a tensor (Node × Node × Graph) for each dataset DBLP1, DBLP3, Nus-Wide-8, and Amazon-products-10 with different sparsity 0.96, 0.99, 0.83, and 0.98 respectively. Fig. 9: Amazon-products-10 dataset.

What is the impact of considering multiple graphs on clustering results? We first compare TSPLBM applied on all graphs simultaneously with PSBM, PLBM, SPLBM used on each graph. The goal is to evaluate TSPLBM in terms of clustering with a comparison with the baselines. On the other hand, we aim to measure how the combination of different information through graphs, impacts, and improves results. Note that TSPLBM can be viewed as an ensemble method.

We perform 30 random initializations and compute Accuracy and Normalized Mutual Information (NMI) [START_REF] Strehl | Cluster ensembles -a knowledge reuse framework for combining multiple partitions[END_REF] metrics by averaging all runs. The clustering accuracy noted (ACC) discovers the one-to-one relationship between two partitions and measures the extent to which each cluster contains data points from the corresponding class. However, NMI is based on Mutual Information (MI) and measures the amount of retrieved information considering our knowledge about the clusters and the obtained results by a clustering method while respecting the proportions of clusters.

In Figure 10, the performances of the four algorithms PSBM, PLBM, SPLBM, and TSPLBM on the four datasets, are reported. PSBM, PLBM, and SPLBM are applied on each slice (graph) separately. TSPLBM is applied to the tensor considering all graphs simultaneously. We notice that, in most cases, TSPLBM is better than other algorithms applied to each graph and allows us to achieve the best trade-off. TSPLBM includes all graphs and also the graphs with a very complex structure. DBLP3 obtains the lowest results due to the complex structure of dataset composed of 12K papers with very close or complementary topics on computer science. We observe that PLBM and SPLBM do a better job than PSBM for all datasets on the more informative slices. It is also worth noting that PLBM does good performances in terms of Accuracy on DBLP1 and in terms of NMI on DBLP3. TSPLBM performs a natural consensus when considering all slices and allows us to obtain a unique partition at the end with good clustering results.

How does the TSPLBM differ from multiview methods in terms of clustering performance?

The Multi-view clustering (MvC), [START_REF] Bickel | Multi-view clustering[END_REF] aims to perform clustering from diverse sources or domains, where each object is described by several sets of features (or views). The MvC methods are used in several applications such as image clustering, where we can have different kinds of features. They allow to take into account the information arsising of each view.

Because of the diversity of feature sets, each view can be converted to a symmetric instancesxinstances similarity/dissimilarity matrix. This brings us back to a tensor representation of these views where each of them is a graph where the edges are continuous. Thereby, even though each view is not a count matrix, we compared TSPLBM-after binarisation-with two recent and effective algorithms SwMV Liu et al. (2013b). To evaluate the three algorithms, we selected six multi-view datasets UC-digits, 3sources, BBC, 100leaves, DBLP1, and Nus-Wide-8. These datasets are available at https://github.com/KunyuLin/Multi-view-Datasets.

Nie et al. (2017) and MultiNMF

We perfomed the same experimentation procedure as TSPLBM with 30 runs, and we compute the average of ACC, NMI, and Purity [START_REF] Sripada | Comparison of purity and entropy of k-means clustering and fuzzy c means clustering[END_REF]. For the MultiNMF, we pricked up the results in terms of ACC and NMI that are available in these two papers [START_REF] Wang | Gmc: Graph-based multi-view clustering[END_REF][START_REF] Wang | Feature extraction via multi-view non-negative matrix factorization with local graph regularization[END_REF].

In Table 2 are reported the obtained results on the six multi-view datasets. Thereby SwMV does a better job than MultiNMF ; it achieves good results on UC-digits and 100Leaves. However, SwMV could not give the clustering for DBLP1. On the other hand, TSPLBM achieves highly better results than SwMV on the four datasets.

Overall, from these experiments, even with binary edges, we observe that TSPLBM gives encouraging results compared with SwMV and MultiNMF applied on graphs with continuous edges.

Comparison between TSPLBM and tensor decomposition approaches Undoubtedly and for a long time, to deal with tensor data X ∈ R n×n×v , the tensor decomposition methods are the most popular [START_REF] Kolda | Tensor decompositions and applications[END_REF]. Even if they are not devoted to clustering, they allow ton contribute to this task. Actually, these methods return a factor matrix ∈ R n×r (r is a given rank) for each mode that can be used for clustering. In the following, we focus on only one mode. Thus, we used a list of suitable algorithms for the clustering: Kmeans++ [START_REF] Arthur | K-means++: The advantages of careful seeding[END_REF], Spectral clustering (SC) [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF], and the EM algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] derived from diagonal Gaussian Mixture Model (GMM) available in the Scikit-Learn package. Thereby, we compared the sparse tensor co-clustering algorithm TSPLBM with PARAFAC [START_REF] Harshman | Parafac : parallel factor analysis[END_REF] and Tucker decomposition [START_REF] Tucker | Some mathematical notes on three-mode factor analysis[END_REF] on the six datasets presented in the previous section.

We use different ranks (10, 20 and 50). We performed 30 runs with random initialization. Then we computed ACC, NMI, and purity by averaging all runs. In figure 11 clustering results for the six datasets according to the different tensor-based algorithms (PARAFAC, Tucker decomposition, and TSPLBM) and the clustering algorithms applied on the obtained tensor decomposition. The results concern tensor decomposition approaches with rank number equal to 10 (The results for rank 20 and 50 are similar to those using rank equal to 10). We observe that in major cases for the six datasets the TSPLBM does a better job than PARAFAC and Tucker decomposition. For the 3sources and Caltech-7 datasets, PARAFAC and Tucker decomposition with GMM obtained close results in terms of Purity and Accuracy but TSPLBM achieves higher performances in terms of NMI.

Fig. 12: Time complexity analysis.

To evaluate the computing time of TSPLBM comparing to tensor decomposition approaches, we represent in figure 12 the time execution in seconds of the compared methods for each datastet. We notice that for the four datasets 3sources, BBC, DBLP1, and UCI-digits, TSPLBM is close to all other approaches in terms of time execution. However, with Nus-wide-8 and 100Leaves, the time execution is more important, this is due to the dataset size and the number of clusters for Nus-wide-8 and 100Leaves. In figure 11, we observe however, that TSPLBM outperforms tensor decomposition approaches with approximately 25 points of ACC for these two datasets.

How can the proposed model help us in the interpretation of the obtained results? The objective of this part is to analyze the obtained topics and demonstrate how the proposed model can help and then improve the interpretation of the obtained clusters.

The second analysis that we made is dimensionality reduction of topics-tags matrix using the correspondence analysis method (CA) [START_REF] Benzecri | L'analyse des données, tome 2 : l'analyse des correspondances[END_REF][START_REF] Nenadic | Correspondence analysis in R, with two-and three-dimensional graphics: The CA package[END_REF]. The choice of CA is due to the connection between mutual information and chi-square, which is based in CA, see, e.g., [START_REF] Govaert | Mutual information, phi-squared and model-based co-clustering for contingency tables[END_REF]. The matrix topic-tags Z T M is constructed from image-tags M based on obtained topics (or partition) Z obtained by TSPLBM. In Figure 13, are projected the tags and topics on the two first dimensions of CA including the top tags in terms of contribution6 on the CA results.

We can notice that there are some close topics and other very different one. For instance, topic 3 about weddings is opposed to topics 8 and 6 about snow and temple considering the first and the second dimension respectively. On the other hand, we can see that topics 1 and 2 about plants and animals are close.

Figure 14 presents the tags whose contribution is important. We show the frequencies of each term for each topic. For topics 2 and 5 (pink and purple color respectively), we can see that the four top tags are Nature, Green, Macro, and Flower related to Plants topic and Street, City, Night and Architect related to Town topic.

Based on the Co-tags graph and the obtained topics, we construct a graph of image clusters linked by edges representing the intensity of joint tags between all topics, this can be computed by Z HZ where Z is obtained by TSPLBM, and H is the co-tags matrix. We can notice that there are some topics with a strong relationship like plants-snow and town-persons. On the other hand, some topics with a weak link like animals-town and animals-temple. This representation highlights that there are Fig. 13: CA applied on topic-tags matrix. Fig. 14: Frequency matrix of subject tags whose contribution is important. some tags used with confused meaning. In this context, it is possible to use tensor models for tags completion and tags correction [START_REF] Tang | Tri-clustered tensor completion for social-aware image tag refinement[END_REF][START_REF] Veit | Separating self-expression and visual content in hashtag supervision[END_REF].

Implicit consensus vs. explicit consensus

In the first part of our experiments, we have observed that TSPLBM applied on all slices simultaneously is, in most cases, better than other algorithms. As we are in an unsupervised context, we have found it helpful to run the calculation with several different random initial conditions and take the best result in terms of maximum log-likelihood, overall runs.

Figure 15 shows the 30 performed runs sorted according to Normalized log-likelihood (NL), which is the objective function of TSPLBM. We also draw the ACC and NMI curve according to the 30 runs. We observe that for DBLP1, the best runs leading to maximal NL are the best runs in terms of clustering (ACC and NMI). However, this observation is not noticed in all datasets; for instance, some best runs can achieve less good results in terms of ACC and NMI. This problem is recurrent with all unsupervised methods where the best runs in terms of the objective function are not necessarily the best ones in terms of clustering. On the other hand, we may see the proposed model as an implicit consensus model for graphs clustering, and it is tempting to compare the proposed model to ensemble-based clustering methods. The first works about consensus or ensemble classification have emerged in the context of supervised learning; see for instance [START_REF] Maclin | An empirical evaluation of bagging and boosting[END_REF][START_REF] Schapire | The boosting approach to machine learning: An overview[END_REF][START_REF] Dietterich | Ensemble methods in machine learning[END_REF]. However, only the majority voting type algorithms work on the model output level, and the most well-known classification ensembles approaches are based on different variants of voting [START_REF] Bauer | An empirical comparison of voting classification algorithms: Bagging, boosting, and variants[END_REF][START_REF] Crammer | Learning from multiple sources[END_REF][START_REF] Gao | Graph-based consensus maximization among multiple supervised and unsupervised models[END_REF]. This approach has been extended to unsupervised learning [START_REF] Strehl | Cluster ensembles -a knowledge reuse framework for combining multiple partitions[END_REF][START_REF] Vega-Pons | A survey of clustering ensemble algorithms[END_REF]. A clustering ensemble, also known as a consensus clustering or clustering aggregation, is defined in the same manner as for classification [START_REF] Hanczar | Ensemble methods for biclustering tasks[END_REF][START_REF] Alqurashi | Clustering ensemble method[END_REF][START_REF] Yu | Co-clustering ensembles based on multiple relevance measures[END_REF]. It consists in combining multiple clustering models (partitions) into a single consolidated partition that we refer to as explicit consensus clustering. In other words, from r partitions {Z 1 , Z 2 , Z 3 ,. . . , Z r }, a consensus clustering leads to a unique partition Z * . Based on consensus functions, many approaches exist; see for instance [START_REF] Strehl | Cluster ensembles -a knowledge reuse framework for combining multiple partitions[END_REF][START_REF] Hanczar | Ensemble methods for biclustering tasks[END_REF]Affeldt et al., 2020a,b).

In [START_REF] Strehl | Cluster ensembles -a knowledge reuse framework for combining multiple partitions[END_REF], the authors introduced three ensemble clustering methods that can produce a consensus partition. All of them consider the consensus problem on a hypergraph representation of the set of partitions. More specifically, each partition is a binary classification matrix (with objects in rows and clusters in columns) where the concatenation of all the set defines the hypergraph. Figure 16 presents this matrix and different steps to construct a combination of these different graphs of clusters, emerged from different partitions, to obtain a unique graph. To this end, we rely on the three hypergraph clustering-based approaches proposed by [START_REF] Strehl | Cluster ensembles -a knowledge reuse framework for combining multiple partitions[END_REF], namely CSPA (Cluster-based Similarity Partitioning Algorithm), HGPA (HyperGraph Partitioning Algorithm), and MCLA (Meta-CLustering Algorithm). To improve clustering results of TSPLBM we will adopt the ensemble approach. We explore in the next part, how implicit consensus clustering through TSPLBM behaves compared to explicit consensus through cluster ensembles of multiple graphs. In Figure 17, we report the proposed approach to compare TSPLBM with the clustering ensemble methods proposed by Strehl [START_REF] Strehl | Cluster ensembles -a knowledge reuse framework for combining multiple partitions[END_REF]. To do this, we used the implementation of python package Cluster Ensembles7 . It relies on CSPA, HGPA, and MCLA and returns the best results in terms of the mean of NMI between the obtained consensus clustering Z * and the different clustering solutions {Z 1 , Z 2 , Z 3 ,. . . , Z r }. Therebey, with TSPLBM, we select the top ten runs maximizing log-likelihood then we carry out the consensus by using the cluster-ensembles methods. With SPLBM, PLBM, and PSBM, we consider two steps. The first step is the same as that used with TSPLBM to select the top ten runs and apply the cluster-ensembles methods. The second one consists in applying another clustering consensus between graphs to obtain a unique partition. Note that the slice consensus is implicitly provided by the TSPLBM algorithm. In Figure 18 are reported the obtained results in terms of NMI using the comparison approach described above. We can notice that TSPLBM achieves the highest NMI for all datasets. SPLBM does a better or similar job than PLBM on three datasets, while PSBM obtains the lowest NMI measures on all datasets. These results can be explained by the fact that the implicit consensus achieved by TSPLBM is optimized within the objective function of the algorithm, unlike the explicit consensus, where the partitions are obtained separately.

Conclusion

It is well known that the traditional Poisson SBM fails to detect relevant clusters of edges, this requires a degree-corrected SBM (DC-SBM). Drawing on this, we first established some connections between Poisson SBM and the corrected version DC-SBM with Poisson LBM commonly used for the co-clustering of contingency tables. We justified the extension of the latter to deal with multiple graphs clustering. To take into account the sparsity of the tensor, we modified the parametrization of the model and proposed a Tensor SPLBM (TSPLBM). We derived, thereby, an EM-like learning algorithm called TSPLBM capable of performing clustering from a tensor data. On real datasets of text and image graphs, we have shown that TSPLBM, is better than the cited baselines algorithms in terms of clustering.

On the other hand, we can note that the proposed clustering algorithm TSPLBM can be seen as an implicit consensus clustering between multiple graphs. To reinforce our idea that TSPLBM can be used in this sense, a comparative study with explicit consensus through ensemble clustering methods was realized. Experiments on several real graphs datasets highlight the effectiveness of TSPLBM. Thereby, this work gives an extra dimension to LBM as an ensemble method.

Our approach has made it possible to propose a like-EM learning algorithm. It is possible to develop a like-Classification EM version. To do this, all that is needed is to insert a classification step between E and M steps. This could lead to propose an extension of DC-SBM for multiple graphs. In this paper, the number of clusters has been assumed to be known, it would be interesting to propose an extension of some criteria, such as ICL (Integrated Completed Likelihood) criterion, already used with SBM [START_REF] Daudin | A mixture model for random graphs[END_REF]. C Appendix: Update z ik ∀i, k

To obtain the expression of z ik , we maximize (3) with respect to z ik , subject to the constraint k z ik = 1. The corresponding Lagrangian, up to terms which are not a function of z ik , is given by : L( z, β) = Setting this derivative to zero yields:

z ik = π k exp 1 2 j,k

Fig. 1 :

 1 Fig. 1: Goal of co-clustering of multiple graphs.

Fig. 2 :

 2 Fig. 2: Political blogs dataset: Clustering with PSBM and DC-SBM/Croinfo.

Fig. 3 :

 3 Fig. 3: (a): Original data -(b): co-clustering according PLBM -(c): co-clustering according SPLBM.

Fig. 5 :

 5 Fig. 5: Political blogs dataset: Comparison of PSBM, PLBM, and SPLBM in terms of accuracy.

Fig. 6 :

 6 Fig. 6: Third-way tensor data representation.

  (a) Horizontal slices. (b) Lateral slices. (c) Frontal slices.

Fig. 7 :

 7 Fig. 7: Slices representations of the three-way tensor.

Fig. 8 :

 8 Fig. 8: The three-way tensor structure.

  . The followed process to build graph similarity based on six extracted features form images including 64-D Color Histogram (CH), 144-D Color Correlogram (CORR), 73-D Edge direction histogram (EDH), 128-D Wavelet texture (WT), 225-D block-wise color moments (CW55).

  for three different features namely 256-D Local Binary Patterns (LBP), 216-D Haralick features (Haralick et al., 1973) (considering distances d = 1 . . . 9, orientations θ = [0°, 45°, 90°, 135°]) and 192-D Gabor features (Chengjun Liu and Wechsler, 2001) (considering scales σ = 1 . . . 4, orientations θ = [0°, 45°, 90°, 135°]

Fig. 10 :

 10 Fig. 10: Comparison in terms of Accuracy and NMI for all datasets with PSBM, PLBM, SPLBM and TSPLBM.

Fig. 11 :

 11 Fig. 11: Comparison between TSPLBM and tensor decomposition approaches based on clustering performances.

Fig. 15 :

 15 Fig. 15: Normalized Log-likelihood vs NMI and ACC for all runs.

Fig. 16 :

 16 Fig. 16: Process of the transition from clustering to consensus clustering.

Fig. 17 :

 17 Fig. 17: Ensemble methods with PSBM, PLBM, SPSBM and TSPBLM. Description of the assessment process of all algorithms in terms of ACC and NMI.

Fig. 18 :

 18 Fig. 18: Consensus clustering based NMI comparison.

  log(x b i. x b .j ) -log(x b ij !) + H( Z) Note that the term i,j,b x b ij log(x b i. x b .j )-log(x b ij !) is a scalar which does not depend on z, w, and Ω and therefore can be ignored for optimization purpose.To keep formulas uncluttered we therefore discard this term in the subsequent development. Thus, we obtain: kk -γ b ) + N b (log(γ b ) -N b γ b ) + H( Z).

z-

  ik log( z ik ) + β(1k z ik ).Taking derivatives with respect to z ik , we obtain: log z ik -1 -β.

.

  Summing both sides over all k yields exp(β+1) = k π k exp 1 2 j,k z jk v b=1 P ijb k k + j =i,k = z j v b=1 P ijb kPlugging exp(β + 1) in z ik leads to:z ik ∝ π k exp 1 -x b k. x b .k (γ b kk -γ b ) + N b (log(γ b ) --[x b k. ] 2 γ b kk -γ b (N 2 b -[x b k. ] 2 ) + N b (log(γ b )Thus, plugging the estimation of γ b kk 's and γ b (equation 6) yields to :k [x b k. ] 2 (N 2 b -[x b k. ] 2 ) + N b (log(γ b )) + N b (log(γ b ) -1)The term N b (log(γ b ) -1) does not depend on z ik we can simplify the expression to: a particular k, we can re-write the equation C as:

Table 1 :

 1 Characteristics of datasets. The two datasets DBLP1 and DBLP3 are document datasets constructed from the global DBLP 3 dataset. The clusters are represented by journals/conferences where the papers are published. We selected three journals ((and conferences) for DBLP1, namely Discrete Applied Mathematics, IEEE software, and SIGIR. For DBLP3, we selected ten journals (and conferences), which are ICC, IJCAI', SIGMOD, Discrete Applied Mathematics, Electr. Notes Theor. Comput. Sci., DAC, GECCO, ICIP, ICCV, and Journal of Systems and Software. We constructed three graphs.

	Datasets	Type	#Graphs #Nodes #Clusters
	DBLP1	Text	3	2223	3
	DBLP3	Text	3	12550	10
	Nus-Wide-8	Text+Images	6	2738	8
	Amazon-products-10	Text+Images	7	9897	10
	DBLP1 and DBLP3:

Table 2 :

 2 Mutiview Clustering performance comparison. -symbol means that we could not retrieve the results for MultiNMF for these datasets.2 NA symbol means that the SwMV algorithm could not find clustering solution.

	Datasets	MultiNMF 1 ACC NMI	ACC	SwMV 2 NMI	Purity	ACC	TSPLBM NMI	Purity
	UC-digits	0.88	0.80	0.94	0.91	0.95	0.74	0.80	0.76
	3sources	0.48	0.46	0.35	0.10	0.36	0.66	0.54	0.66
	BBC	0.48	0.33	0.33	0.05	0.33	0.66	0.66	0.66
	100leaves	0.67	0.86	0.59	0.87	0.61	0.46	0.81	0.46
	DBLP1	-	-	NA	NA	NA	0.83	0.57	0.85
	Nus-Wide-8	-	-	0.28	0.004	0.28	0.56	0.41	0.56
	1

In the paper, to distinguish between a model and its derived algorithm we use typewriter font for an algorithm, thereby DC-SBM is the model and DC-SBM its derived algorithm.

https://dl.acm.org/citation.cfm?id=1134277

https://aminer.org/citation

https://dl.acm.org/citation.cfm?id=1646452

http://jmcauley.ucsd.edu/data/amazon/links.html

With CA each tag contributes to the inertia of each axis. The contribution of a tag to axis α is expressed as a percent of the inertia for axis α.

https://pypi.org/project/Cluster_Ensembles/

Funding Our work is funded by the German Federal Ministry for Economic Affairs and Energy (BMWi) under grant agreement number 01MK20008F (Service-Meister).

A Appendix: Proof of Equation 3The marginal density function f (X; Ω) of TSPLBM can be written as:

Thus, the complete-data log-likelihood function is given by:

Hence, the aim is to maximize the following lower bound of the log-likelihood criterion:

where L C ( Z, W, Ω) is the fuzzy complete-data log-likelihood function. As X is symmetric per slice b, when i = j we have z ik = w jk and for k = 1, . . . , g we have π k = ρ k and H( Z) = H( W). The F C ( Z, Ω) takes the following form:

Then, we can simplify the model by optimizing

B Appendix: Proof of Equation 4The simplified optimization criterion can be written as:


Geoffroy Godet
email: geoffroy.godet@easii-ic.com

Member, IEEE Emmanuel Augendre
email: emmanuel.augendre@cea.fr

Jose Lugo-Alvarez

Hélène Jacquinot
email: helene.jacquinot@cea.fr

Frédéric X Gailla

Thomas Lorne
email: thomas.lorne@cea.fr

Emmanuel Rolland
email: emmanuel.rolland@cea.fr

Member, IEEE Thierry Taris
email: thierry.taris@ims-bordeaux.fr

Florence Servant
email: florence.servant@cea.fr

Frédéric X Gaillard
email: frederic-x.gaillard@cea.fr

Influence of Substrate Resistivity on Porous Silicon Small-Signal RF Properties

Keywords: Crosstalk, linear attenuation coefficient, porous silicon (PS), S-parameters, substrate resistivity

This article provides guidelines to design porous silicon (PS) layers regarding optimization of smallsignal properties in passive structures for radio frequency (RF): insertion loss and crosstalk. Results are based on high-frequency measurements on 200-mm wafers and electromagnetic simulations up to 40 GHz. Using substrate resistivity below 1 cm allows the best tradeoff with minimized PS thickness.

I. INTRODUCTION

W ITH the advent of pervasive mobile and wireless devices, porous silicon (PS), a long studied material [START_REF] Uhlir | Electrolytic shaping of germanium and silicon[END_REF]- [START_REF] Zhang | Porous silicon formation and electropolishing of silicon by anodic polarization in HF solution[END_REF], attracts a renewed interest as blanket or local substrate material for radio frequency (RF) applications [START_REF] Rack | Small-and large-signal performance up to 175 • C of low-cost porous silicon substrate for RF applications[END_REF]- [START_REF] Gautier | Porous silicon for electrical isolation in radio frequency devices: A review[END_REF]: it is compatible with Si technology and offers lower permittivity and higher resistivity than Si.

The latter helps to reduce signal propagation loss (attenuation). Using PS benefits a wide range of devices including coplanar waveguides (CPWs), inductors, and filters [START_REF] Gautier | Porous silicon for electrical isolation in radio frequency devices: A review[END_REF]. There are reports indicating that for a given device geometry, increased PS thickness improves device performance [START_REF] Welty | Porous silicon technology for RF integrated circuit applications[END_REF], [START_REF] Billoué | Integration of RF inductors and filters on mesoporous silicon isolation layers[END_REF]. However, the systematic effect of starting bulk silicon resistivity on RF small-signal properties has not yet been described with PS.

In this article, we report for the first time on the impact of silicon resistivity on substrate loss of CPW made on PS. First, we fabricated CPW on PS variants (made with different thicknesses on silicon with variable resistivity). We then measured the S-parameters of CPW fabricated on these substrates up to 40 GHz. A simple high frequency structure simulator (HFSS) [START_REF]Description of the Ansys HFSS Simulation Software[END_REF] model reproduces these measurement results. This model enables predicting the impact of substrate resistivity on CPW substrate loss and crosstalk, allowing to draw guidelines for the design of future RF PS layers.

II. EXPERIMENTAL

A. PS Fabrication

We prepared blanked PS layers starting from 200-mm p-type doped silicon wafers with resistivity in the 1-2-• cm and 5-10-• cm ranges, respectively. For each resistivity, Si substrates were electrochemically anodized in hydrofluoric acid (35%) with isopropanol as surfactant and a current density of 14 mA/cm 2 . The thickness and porosity of these layers were determined using differential weighting and cross section scanning electron microscopy. Fig. 1 illustrates the morphology of these layers. For each resistivity, layers of 12-and 20-μm thickness were fabricated. In all samples, porosity was estimated close to 50%.

B. CPW Fabrication

After PS formation, a thin (400 nm) silicon dioxide layer was deposited by plasma-enhanced chemical vapor deposition at 260 • C. The wafers were then coated with 100 nm of titanium and 1 μm of gold. Photolithography and wet etching lead to the patterning of this metal stack into CPW transmission lines (see Fig. 2). RF probing pads with 100-μm pitch allow CPW connection.

C. S-Parameter Measurements and Parameter Extraction

Scattering parameters were measured between 100 MHz and 40 GHz using an Anritsu 37369A Vectorial Network Analyzer and ground-signal-ground (GSG) probes from Cascade. Measurements were carried out on different CPW pairs differing only by their lengths (500 and 1500 μm). RF pad parasitic contributions were removed using Mangan's method proposed in [START_REF] Mangan | De-embedding transmission line measurements for accurate modeling of IC designs[END_REF]. Losses were calculated as follows:

α C = R/2Z 0 (1) α G = GZ 0 /2 (2) α total = α G + α C ( 3
)
where α C and α G are the respective contributions of metallic conductor and substrate material to α total , the total loss of the CPW. R and G are the linear resistance and conductance values of the resistance-inductance-capacitance-conductance (RLCG) equivalent model of the transmission line and Z 0 relates to its characteristic impedance.

III. RESULTS AND DISCUSSION A. Choice of CPW Dimensions

Losses were extracted on two contrasting CPW geometries: a narrow signal line with small signal-to-ground gap (S/G = 10 μm/4.6 μm) and a wide signal line with a large signal-to-ground gap (S/G = 70 μm/42 μm). Both CPW lines have a 50-characteristic impedance for a 3-k • cm Si substrate without PS. Fig. 3 presents the results obtained on two different Si substrate resistivities.

For the compact CPW, it is noticeable that conductor contribution dominates losses. This originates both from the higher linear resistance of the signal line and from reduced substrate losses, as electromagnetic fields are mainly confined within the low-loss PS layer. In contrast, in the wide signal-line and large gap configuration, the substrate component (including bulk Si contribution) determines CPW linear loss. For this reason, the investigation of substrate resistivity impact focuses on the larger layout.

B. HFSS Electromagnetic Simulations

Different configurations were implemented in the HFSS software [START_REF]Description of the Ansys HFSS Simulation Software[END_REF], using parameter values listed in Tables I andII. Experimental difference between conductor and substrate contribution to propagation linear loss versus frequency, on 20-µm-thick PS layers, as a function of CPW geometry and substrate resistivity.

TABLE I STACK DESCRIPTION TABLE II ELECTRICAL PROPERTIES

In these structures, the substrate resistivity was chosen to be 1.5 and 7.5 • cm for hardware in the respective 1-2-• cm and 5-10-• cm resistivity ranges. PS layer resistivity value was set in agreement with [START_REF] Ménard | P type porous silicon resistivity and carrier transport[END_REF]. The relative permittivity value of 4.5 originates from [START_REF] Sarafis | Dielectric properties of porous silicon for use as a substrate for the on-chip integration of millimeterwave devices in the frequency range 140 to 210 GHz[END_REF] and best describes the permittivity around 50% porosity.

Fig. 4 compares measured and simulated losses as a function of frequency, on CPW structures with the same dimensions: 70-μm-wide signal line, 308-μm-wide surrounding ground lines separated by a 42-μm-wide gap. Two variants of silicon resistivity are combined with two variants of PS layer thickness (12 and 20 μm).

The loss values in this work are of the same order of magnitude as literature data with similar stacks: in [START_REF] Welty | Porous silicon technology for RF integrated circuit applications[END_REF], losses of 1.5 dB/mm at 40 GHz have been reported on 15-μm-thick PS on 1-3-• cm Si and with a gap of 2.5 μm. In this work, this value is obtained with a 20-μm PS layer. The difference is due to CPW geometry: in our case, this was selected in order to maximize substrate impact, not to optimize performance (which would require narrower line-to-ground gaps as in [START_REF] Welty | Porous silicon technology for RF integrated circuit applications[END_REF]).

The obtained trend agrees with literature data showing that increasing PS thickness yields attenuated losses [START_REF] Welty | Porous silicon technology for RF integrated circuit applications[END_REF]. Comparing at 16 GHz, increasing PS thickness from 12 to 20 μm decreases attenuation coefficients from 0.8 to 0.4 dB/mm.

The change in substrate doping has a strong impact on the attenuation dependence on frequency. Below 16 GHz, for a given PS thickness, it is more favorable to use 1.5 than 7.5 • cm. This trend reverses at higher frequencies, especially with thinner PS.

In all cases, the simulations capture the observed dependence on PS thickness and Si substrate resistivity. In order to have a better insight of these trends, HFSS simulations were performed to investigate wider parameter ranges. For those thickness and resistivity ranges, the variation of the characteristic impedance of the chosen coplanar line is between 30 and 80 . Both PS thickness and substrate resistivity appear to influence the observed trends.

C. Simulated Dependence of Loss on Resistivity

For each PS layer thickness, attenuation reaches a peak value in the 10-20-• cm resistivity range.

1) For higher resistivity values, eddy currents fade, reducing losses. 2) For lower resistivity values, the electromagnetic field is mainly confined into the PS layer, which also dims eddy currents. The 10-20-• cm resistivity range leads to a worst case regime where both mechanisms are most ineffective.

The effect of substrate resistivity is amplified in the case of thin PS layers. Conversely, substrate resistivity influence on losses decreases as soon as PS layer thickness reaches a sufficiently high value (which is around the same order of magnitude as CPW signal-to-ground spacing).

The exercise was repeated at 40 GHz. Fig. 6 summarizes the corresponding main results.

Worst case attenuation values appear in the 1-2-• cm resistivity range, which is one order of magnitude lower than the 4 GHz case.

In that resistivity range, the peak attenuation value increases with frequency (about 10-fold higher than at 4 GHz). The dependence of the peak attenuation value on PS layer thickness seems relatively unchanged.

Minimum attenuation appears for the thickest PS layers and is weakly dependent on substrate doping. This minimum level is higher than in the case of 4 GHz operation. Both observations can be explained by the fact that metallic conductor contribution determines the minimum value and skin effect sets its frequency dependence.

D. Simulated Dependence of Crosstalk on Resistivity

E. Guidelines for PS Layer Design

At a given frequency and for a given PS layer thickness, Figs. 5 and6 suggest that a given loss target value could be achieved with different substrate resistivities in the 10 -2 -50-• cm range.

However, it might be preferable to use low-resistivity silicon as a default starting material.

1) It requires relatively thin PS layers for low-frequency operation.

2) It contributes in minimizing crosstalk over the whole frequency range.

3) It allows keeping the same PS features, as changing Si resistivity can affect PS morphology [START_REF] Korotcenkov | Silicon porosification: State of the art[END_REF].

IV. CONCLUSION

This article reports the first systematic evaluation of substrate resistivity influence on the attenuation coefficient of CPWs fabricated on various thicknesses of PS.

First, we fabricated CPW on PS variants (with different thicknesses and made on silicon substrates with variable resistivity). We then measured the S-parameters of CPW fabricated on these substrates up to 40 GHz. HFSS simulation reproduces these measurement results. This enables the prediction of the substrate resistivity impact on CPW substrate loss and crosstalk.

At each frequency, there is a substrate resistivity range in which attenuation is maximal, suggesting that either low or high resistivity could provide well-controlled attenuation values with minimum PS thickness. At the same time, simulations suggest that low substrate resistivity is necessary to minimize crosstalk up to high frequencies.

While 1-• cm substrate resistivity is suitable for PS to operate in the sub-6-GHz domain, we recommend using lower resistivity silicon if the PS layers are meant to be compatible with a wider range of operating frequencies.

Fig. 1 .

 1 Fig.1. Cross-sectional imaging of the Si/PS interface for two distinct substrate resistivities, using scanning electron microscopy.

Fig. 2 .

 2 Fig. 2. Material stack used in the fabrication of CPW on PS.

Fig. 3 .

 3 Fig.3.Experimental difference between conductor and substrate contribution to propagation linear loss versus frequency, on 20-µm-thick PS layers, as a function of CPW geometry and substrate resistivity.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Total linear propagation loss as a function of frequency, on 12-and 20-µm PS layers, for two different substrate resistivities. Symbols denote measurement results and dashed lines denote HFSS simulation output.

Fig. 5

 5 Fig.5presents simulated attenuation coefficients for PS layers between 5-and 200-μm thickness, on Si substrates in the 10-50-• cm resistivity range, at a fixed frequency of 4 GHz.For those thickness and resistivity ranges, the variation of the characteristic impedance of the chosen coplanar line is between 30 and 80 .

Fig. 6 .

 6 Fig.6. HFSS simulated total attenuation as a function of substrate resistivity, for various PS thickness values, at 40 GHz.

Fig. 7

 7 Fig.7compares the simulated magnitude of S 21 versus frequency between 100 MHz and 40 GHz, for three different substrate resistivities, with 20-μm-thick PS layer, on a coplanar test structure with 25-μm coupling distance and 150-μm coupling width, as described in the inset of Fig.7.While S 21 bears a weak dependence on substrate resistivity below 1 GHz, only the 0.15-• cm resistivity case shows

Fig. 7 .

 7 Fig.7. HFSS simulated S 21 magnitude as a function of frequency for three different substrate resistivities, using a 20-µm PS layer, extracted on a coplanar structure.

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This work was supported in part by European Commission, French State and Auvergne-Rhône Alpes Region through the funding of Electronic Components and Systems for European Leadership (ECSEL) Project BEYOND5 part of Important Project of Common European Interest (IPCEI) Microelectronics and in part by the French Nano2022 Program. The review of this article was arranged by Editor H. Wong.