Salma Ben Dhaou
email: salma.bendhaou@hotmail.fr

Mouloud Kharoune
email: mouloud.kharoune@univ-rennes1.fr

Arnaud Martin
email: arnaud.martin@univ-rennes1.fr

Ben Yaghlane

An Evidential Method for Correcting Noisy Information in Social Network

Keywords: Theory of Belief Functions, Communities, Social Networks, Attributed Networks

Nowadays, social networks have become an important part of our daily lives.

Hence, several researchers have been interested in the study and analysis of the interactions between the entities composing this type of networks. By modeling a social network, we can assign attributes to nodes and links based on network and community structure. These attributes which may be uncertain, imprecise or even noisy, involve obtaining a non-coherent network. In order to remedy this problem, we propose, in this paper, a method that corrects the noise in the network using the theory of belief functions.

Introduction

Nowadays, the use of computer technology and Internet has become essential. As a result, social networks became an important part of our daily lives. Therefore, it is interesting to study and analyze the types of relationships that exist in these networks. To do so, the study of the community structure as well as the nodes and links attributes represent main characteristics that must be taken into account to analyze these networks.

In social network analysis [START_REF] Wasserman | Social network analysis: Methods and applications[END_REF][START_REF] Prell | Social network analysis: History, theory and methodology[END_REF], the observed attributes of social actors are understood in terms of patterns or structures of ties among the units. These ties may be any existing relationship between units; for example friendship, material transactions, etc.

Currently, if we observe any social network, we will soon realize that the entities composing this network are grouped, for example, according to a center of interest, a category of age, a preference, etc.

In his work, Santo Fortunato [START_REF] Fortunato | Community detection in graphs[END_REF] explained that communities, also called clusters or modules, represent groups of vertices which probably share common properties and/or play similar roles within the graph. He argues also that the word community itself refers to a social context. In fact, people naturally tend to form groups, within their work environment, family or friends.

In a social network, we can deal with missing or modified information. In addition, the information exchanged can be often imperfect, due to the heterogeneous nature of the sources. Therefore, it would be interesting to use a vector of values which represent the nodes and links attributes.

In the same context, many studies focus on modeling the uncertain social network. In fact, they represent an uncertain network by weighting the nodes or links with values in [0, 1] to model uncertainties. Hence, it will be easier to monitor the behavior of the social network [START_REF] Adar | Managing uncertainty in social networks[END_REF][START_REF] Ben Dhaou | Belief approach for social networks[END_REF]. In addition, as shown in [START_REF] Ben Dhaou | The advantage of evidential attributes in social networks[END_REF], the use of evidential attributes, from the theory of belief functions, gives better results compared to the probabilistic ones.

The theory of belief functions offers a mathematical framework for modeling uncertain and imprecise information [START_REF] Shafer | A mathematical theory of evidence[END_REF]. It has been employed in different fields, such as data classification [START_REF] Denoeux | A k-nearest neighbor classification rule based on dempstershafer theory[END_REF][START_REF] Liu | A new incomplete pattern classification method based on evidential reasoning[END_REF] and social network analysis [START_REF] Wei | Identifying influential nodes in weighted networks based on evidence theory[END_REF].

Furthermore, the theory of belief functions provides a flexible way of combining information collected from different sources. In the majority of cases, this combination is followed by decision-making. It also allows conflict management.

The aim of this paper is to show that even with noise in the network, our algorithm is able to classify the nodes in their initial clusters. In the case of a large noise, the algorithm guarantees the coherence of the information of any network even when it is a network whose nodes and links attributes have been strongly modified.

In this paper, we focused on the use of a limited number of communities. In terms of scaling up, there are several strategies that can reduce complexity like the one presented in [START_REF] Martin | Implementing general belief function framework with a practical codification for low complexity[END_REF]. This will be the subject of future work. This paper is structured as follows. In section 2, we remind some basic concepts of the theory of belief functions and review some community detection methods as well as some other related works. Section 3 is dedicated to our contribution. Section 4 is devoted to the experimentations and finally section 5 concludes the paper.

Background

In this section, we start by recalling some basis of the theory of belief functions, we use it in this paper in order to model uncertainties. Then we present some community detection methods that use both the structure and the attributes of the network.

Theory of Belief Functions

The theory of belief functions allows explicitly to consider the uncertainty of knowledge using mathematical tools [START_REF] Shafer | A mathematical theory of evidence[END_REF][START_REF] Shafer | Perspectives on the theory and practice of belief functions[END_REF]. It is a useful and effective way in many fields such as classification, decision making, representation of uncertain and inaccurate information, etc.

In fact, it is a suitable theory for the representation and management of imperfect knowledge. It allows to handle the uncertainty and imprecision of the data sets, to combine mass functions and make decisions.

The principle of the theory of belief functions consists on the manipulation of functions defined on subsets. However, it does not represent uncertainty using sets of probability measures. These functions are called mass functions and range from 0 to 1.

Let Ω be a finite and exhaustive set whose elements are mutually exclusive, Ω is called a frame of discernment. A mass function is a mapping

m : 2 Ω → [0, 1] such that X∈2 Ω m Ω (X) = 1 and m Ω (∅) = 0 (1)
The mass m Ω (X) expresses the amount of belief that is allocated to the subset

X. We call X a focal element if m Ω (X) > 0.
A categorical mass function is a mass function with an unique focal element such that m Ω (A) = 1.

In this work, we used also another interesting concept which is the distance 70 of Jousselme [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF]. This distance represents the degree of similarity between bodies of evidence. It is defined by:

d j (m Ω 1 , m Ω 2 ) = 1 2 (m Ω 1 -m Ω 2 ) T Jac(m Ω 1 -m Ω 2 ) (2)
where the elements Jac(A, B) of Jaccards weighting matrix Jac are defined as:

Jac(A, B) =      1 if A = B = ∅ |A ∩ B| |A ∪ B| , A, B ∈ 2 Ω \ ∅ (3)
We also consider the normalized conjunctive rule called the Dempster rule [START_REF] Shafer | Dempster's rule of combination[END_REF], given for two mass functions m Ω 1 and m Ω 2 for all X ∈ 2 Ω , X = ∅ by:

m ⊕ (X) = 1 1 -k A∩B=X m Ω 1 (A)m Ω 2 (B) (4)
where

k = A∩B=∅ m Ω 1 (A)m Ω 2 (B)
is the global conflict of the combination. The Dempster combination rule reinforces the mass values of the elements on which the sources are agree. This rule is adapted when the combined mass functions are cognitively independent. In the case of dependent mass functions, one can use the mean rule given for two mass functions m Ω 1 and m Ω 2 for all X ∈ 2 Ω , X = ∅ by:

m Ω (X) = 1 2 (m Ω 1 (X) + m Ω 2 (X)) (5)
In order to make decision, we use the pignistic probability introduced by Smets in [START_REF] Smets | Decision making in the tbm: the necessity of the pignistic transformation[END_REF] for normal mass functions by:

BetP (X) = Y ∈2 Ω ,Y =∅ | X ∩ Y | | Y | m Ω (Y ) (6)

Some Community Detection Methods with Graphs Structure and Attributes

In this section, we introduce some community detection methods based on graph structure and attributes.

According to [START_REF] Seong | Incremental clustering of attributed graphs[END_REF], an attributed graph G a = (V a , E a ) can be defined as a set of attributed vertices V a = {v 1 , . . . , v p , . . . , v q , . . . , v n } and a set of attributed edges E a = {. . . , e pq , . . .}. The edge e pq connects vertices v p and v q with an attributed relation.

The presented model in [START_REF] Zhou | Graph clustering based on structural/attribute similarities[END_REF] uses both information. In fact, an unified neighborhood random walk distance measure allows to measure the closeness of vertex on an attributed augmented graph. Then, the authors use a k-Medoids clustering method to partition the network into k clusters.

A second method presented in [START_REF] Leskovec | Learning to discover social circles in ego networks[END_REF] consists on a model dedicated to detect circles that combine network structure and user profile. The authors learn for each circle, its members and the circle-specific user profile similarity metric.

They model the membership of a node to multiple circles in order to detect overlapping and hierarchically nested circles.

A third method presented in [START_REF] Trabelsi | Handling uncertain attribute values in decision tree classifier using the belief function theory[END_REF] Thus, it is important to consider both information structure and attributes in order to detect the network communities. In fact, if one source of information is missing or noisy, the other can solve the problem.

The works cited above [START_REF] Zhou | Graph clustering based on structural/attribute similarities[END_REF][START_REF] Leskovec | Learning to discover social circles in ego networks[END_REF] use only a probabilistic attributes as well as the structure of the graph to do the clustering. In our previous work [START_REF] Ben Dhaou | The advantage of evidential attributes in social networks[END_REF], we show that the use of evidential attributes gives better results than the probabilistic ones in the clustering.

The works cited [START_REF] Zhou | Graph clustering based on structural/attribute similarities[END_REF][START_REF] Leskovec | Learning to discover social circles in ego networks[END_REF][START_REF] Trabelsi | Handling uncertain attribute values in decision tree classifier using the belief function theory[END_REF] are interesting, but they do not assume that network information can be noisy or perturbed. In addition, they do not consider the use of node and link attributes simultaneously to do clustering.

Other Related Works: Homophilic Behaviors in Social Networks

In addition of the presented community detection methods above, there are works that are related to our research such as the reconstruction of an initial network and the propagation of labels.

In [START_REF] Zhou | Selp: Semi-supervised evidential label propagation algorithm for graph data clustering[END_REF], the authors present a new method using the theory of belief functions that aims to detect communities on graphs after the stabilization of the label propagation process. In fact, SELP permits to propagate the labels from the labeled nodes to the unlabeled ones based on a propagation rule. The proposed algorithm computes the dissimilarities between nodes based on the graph structure. The main advantage of the proposed algorithm is that it can effectively use limited supervised information to guide the process of the detection.

Another interesting work presented in [START_REF] Guimerà | Missing and spurious interactions and the reconstruction of complex networks[END_REF] aims to identify missing and spurious interactions (links connecting nodes) and to reconstruct network whose properties are closer to the 'true' underlying network. To do so, the authors focus on the family of stochastic block models. The proposed method can also guide new discoveries. In fact, if a given interaction between 2 nodes exists but with a very low reliability for the interaction, that means that the function of the interaction is very specific.

The method proposed in [START_REF] Vuokko | Reconstructing randomized social networks[END_REF] aims to address the problem of reconstructing the original network and set of features given their randomized counterparts.

The technique of data randomization consists of removing some of the original edges of the network in addition of new ones. Furthermore, the features can be also randomized. In this work, the authors assume that data-randomization method do not completely destroy the original dataset. For the case of features, every node is associated with k binary features. If the node has that feature, it will take 1 otherwise it will take 0.

All the works presented are interesting. However, we can not do a comparison at the experimental level since we do not consider the resolution of the same problem. Indeed, the first work consider a network with few nodes having labels and aim to propagate them to the unlabled ones. In our case, we consider that all nodes and links have a prior lables. In the second research [START_REF] Guimerà | Missing and spurious interactions and the reconstruction of complex networks[END_REF], the authors are interested in predicting links based on observations. In our work, we do not modify the initial structure of the network. Regarding the third work [START_REF] Vuokko | Reconstructing randomized social networks[END_REF], the authors remove links from the graph and add new ones whereas in our case, we do not modify the structure of the graph.

An Evidential Method for Correcting Noisy Information in Social

Networks

In this section, we will introduce our proposed approach. First, we will present the important notions used in this work. Then, we will explain the formalization of our method and finally, we will detail the main steps of the proposed algorithm.

Important Notions

In the networks, noisy or imperfect information can transit. Therefore, if we limit ourselves to the network structure as well as the nodes and links attributes in the classification, the error rate may increase and the network information may become inconsistent.

To solve this problem, we propose a method that allows the classification of nodes in the case of a noisy network, based on the community structure as well as the nodes and links attributes.

In the case of a significant noise introduced, our algorithm corrects inconsistent information. Thus, even if we do not find the initial network, we get a new coherent network. In this context, we present two notions used in this work:

Noise. A noisy element (i.e. a node or a link) is an element whose attribute has been modified.

Consistency.

A network is composed of a set of nodes belonging to communities C i and linked together by links. Two nodes connected by a link represent a triplet. Depending on the community structure of the network, a node belongs to a single community C i while the link may be of different types. If it is inside the community C i , then it will be of the type IC i . However, if it connects two nodes belonging to two different communities, then it will be of type BC.

We use only one type of link representing the link between two communities (BC) in order to minimize the possible hypotheses, since the more the number of communities increases, the more the types of links connecting two communities increase too.

In what follows, we will present the general idea of the proposed method.

Formalization

In this work, we consider a coherent triplet as a triplet (V k1 , L k12 , V k2 ) that satisfies one of the following possibilities: Thus, the triplet is defined as follow:

• V k1 ∈ C i , V k2 ∈ C i , L k12 ∈ IC i with i = 1, . . . N • V k1 ∈ C i , V k2 ∈ C j , L k12 ∈ BC with (i = j),
• V k1 modelized with a mass function m Ω N k1 • V k2 modelized with a mass function m Ω N k2 • L k12 modelized with a mass function m Ω L k12
We remind that a categorical mass function is a mass function with an unique focal element such that m Ω (A) = 1. The representatives below represent the community centers. We calculate the distances between the mass functions of the nodes and links and categorical mass functions of the representatives in order to be able to place these elements in a group.

• For the nodes: the categorical mass functions are defined by m

Ω N ω (ω) = 1 with ω ∈ Ω N , i.e. m Ω N Ci (C i ) = 1, with i = 1, . . . , N .
• For the links: the categorical mass functions are defined by m

Ω L ω (ω) = 1 with ω ∈ Ω L , i.e. m Ω L BC (BC) = 1 or m Ω L ICi (IC i ) = 1 , with i = 1, . . . , N .
The aim of the proposed approach is to correct the noise added to a network by considering each triplet independently of the others. To do this, our algorithm proceeds by calculating the distances between the mass functions of each element of the triplet and the mass functions of the representatives of the communities. Then, it calculates the average distances of the 3 elements of the triplet and compares them with the average distances of the coherent triplets defined initially. The algorithm then keeps the minimum average distance which

gives us an idea about the type of the triplet.

The value of this minimum average distance is considered as a mass function from the current information of the network and is combined thereafter with the initial mass functions. Subsequently, for each node with several links, we will combine with the mean rule all the mass functions that are related to it.

Finally, we will use the BetP to make a decision about the membership of a node to a community and a link to a given type.

We will detail in the following the different steps of the proposed approach.

Main Steps of the Algorithm

The proposed approach is applied in 4 steps detailed below. In order to simplify the notations, we present in the following the equations used in one iteration t of the algorithm.

Step 1:

For each element of a triplet k, we calculate the distances between the latter and the corresponding categorical mass functions.

In the theory of belief functions, a distance is used to describe the difference between two distinct sources of information. We use the distance of Jousselme which takes into account the quantification of the similarity between the focal elements using Jaccard similarity coefficients.

By calculating the distance between the mass function of a node or a link and the corresponding categorical mass functions that are "ideals", we have an idea about its belonging to a community or a kind of link. In fact, we keep the minimum distance and the decision corresponds to the categorical mass functions having the lowest distance with the mass function of the nodes or of the links. Hence, for each triplet (V k1 , L k12 , V k2 ), with k = 1, . . . , M , M the number of triplets (or links) we calculate at iteration t:

C k1 = arg min ω∈Ω N d J (m Ω N k1 , m Ω N ω ) (7)
C k2 = arg min ω∈Ω N d J (m Ω N k2 , m Ω N ω ) (8)
L k12 is determined according to the coherent triplets by:

L k12 =    IC k1 if C k1 = C k2 BC if C k1 = C k2 (9)
Table 1 shows the coherent values of a triplet for the case of a network containing 3 communities. This process of decision is given by [START_REF] Essaid | A Distance-Based Decision in the Credal Level[END_REF].

Step 2:

For each triplet k, at the iteration t we calculate the average distance d k 235 obtained from each possible combination presented previously.

Hence, d k represents a minimal distance between the triplet k and the most possible categorical triplet. This average distance makes it possible to calculate the dissimilarity between any triplet and another coherent one defined initially.

It is defined by:

d k = d J (m Ω N k1 , m Ω N C k 1 ) + d J (m Ω L k12 , m Ω L L k 12 ) + d J (m Ω N k2 , m Ω N C k 2 ) 3 (10)
Step 3: Knowledge Review

In this step, we will use the obtained value of the average distance d k to define a mass function, that will be combined with the initial mass functions of the nodes and links composing each triplet. Therefore, the average distance d k value is assigned to the focal elements that represent the types of the two nodes and the link composing the triplet k and the rest will be assigned to the ignorance.

Hence, we have:

   m Ω N k 1d (C k1 ) = 1 -d k m Ω N k 1d (Ω N ) = d k (11)    m Ω L k 12d (L k12 ) = 1 -d k m Ω L k 12d (Ω L ) = d k (12)    m Ω N k 2d (C k2 ) = 1 -d k m Ω N k 2d (Ω N ) = d k (13)
Once the minimum average distance has been found, we know to which coherent triplet initially defined, the current triplet k is the closest. Therefore, we know what is the nature of each of its elements. Hence, we know if the link which connects the two nodes is of type IC i or BC.

The minimum average distance d k is an information provided by a network whose initial mass functions can be noisy. Therefore, this should be taken into account when reviewing knowledge.

Calculation of final Mass Functions

In this step, we update at the iteration t + 1 the mass functions obtained from the previous step with the initial mass functions given at the iteration t by the following equations:

m t+1,Ω N k1 = m t,Ω N k1 ⊕ m t,Ω N k 1d (14)
m t+1,Ω L k12 = m t,Ω L k12 ⊕ m t,Ω L k 12d (15)
m t+1,Ω N k2 = m t,Ω N k2 ⊕ m t,Ω N k 2d (16)
where m t,Ω N k 1d , m t,Ω L k 12d , m t,Ω N k 2d are given respectively by equations ( 11), ( 12) and [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF].

The combination of the mass functions derived from the minimal average distance calculation and the initial generation by the Dempster rule provides a final idea of nodes and links belonging to their clusters. The Dempster rule affects the generated conflict to the focal elements and therefore we will not have a mass on the empty set.

Step 4:

As we treat each triplet independently of the others, we can have cases where several links start from the same node and so we have several mass functions for the same node. In order to determine an unique mass function for each node (e.g. V k1 ), we combine by the mean rule (given by equation ( 5)), all the mass functions obtained for the given node V k1 in step 3 (equation ( 15)). The choice of the mean is due to the fact that mass functions are dependent. Hence, for a given node V k1 , with M k1 links, we modify the mass functions by:

m Ω N k1 = 1 |T | {k:V k 1 ∈T } m Ω N k ( 17
)
where T = {(V k 1 , L k12 , V k2 )} represents the triplets that contain the node V k1 and m Ω N k is given by the equation ( 14).

Finally, we use the BetP given by equation ( 6) to make decision about the belonging of the triplet (V k1 , L k12 , V k2 ). We have at the iteration t + 1, in the order of the triplet:

C k1 = arg max X∈Ω N Y ∈2 Ω N ,Y =∅ | X ∩ Y | | Y | m Ω N k1 (Y ) (18)
L k12 = arg max X∈Ω L Y ∈2 Ω L ,Y =∅ | X ∩ Y | | Y | m Ω L k12 (Y ) (19)
C k2 = arg max X∈Ω N Y ∈2 Ω N ,Y =∅ | X ∩ Y | | Y | m Ω N k2 (Y ) (20)
Algorithm 1 shows the outline of the process followed for correcting noise in social network using evidential attributes.

The use of the Dempster combination rule makes it possible to reinforce from one iteration to another the mass values of the elements on which the sources agree. Indeed, if we have a mass coming from each source on the same focal element, the combination rule of Dempster allows to increase the belief on the latter. From the fact that we will have an increase in mass functions values, there will be no change in the decision. Hence, we can confirm that the proposed method is still converging to a single element by the decision process given by equations ( 18), ( 19) and (20).

Experimentations

We start the experiments by the generation of mass functions on the nodes and links according to the structure of the network. Indeed, for each node belonging to C i , we generate two focal elements: one on C i and the second one the mass function of the element and the corresponding categorical mass functions using Eqs ( 7), ( 8), (

2. for each triplet k, compute the minimum average distance d k by using Eq (10)

3. Define mass functions from the computed d k using the Eqs (11), ( 12),

4. Update the mass functions using the Eqs ( 14),( 15), ( 16), 5. Combine the mass functions for the same node in order to have a unique mass function by using the Eq (17)

6. Make decision about the belonging of each element of the triplet k using Eqs ( 18), ( 19), (20)

7. t = t + 1
until The results of Eqs ( 18), ( 19) and ( 20) are stable.

on Ω N and we assign the highest generated value to C i . We do the same for the links: depending on the type of the link, we generate two focal elements.

In a second step, we noised this network according to three scenarios:

• Noisy Nodes Only: In this case, we have selected randomly a certain number of nodes of the initial network and we have modified their mass functions by randomly generating two focal elements (ignorance and another element except the empty set).

• Noisy Links Only: In this case, we selected randomly a certain number of links of the initial network and we modified their mass functions by randomly generating two focal elements (ignorance and another element except the empty set).

• Noisy Nodes and Noisy Links: In the latter case, we selected randomly some nodes and links of the networks. Then, we modified their mass functions.

After that, for each triplet we calculate the distances between the attributes of the link and the two nodes and the attributes of the representatives. As we consider different networks with N communities, the coherent triplets are defined on the basis of the community structure of the networks. That is to say, a node can belong to only one community C i . From this hypothesis, the links that we can have will be of type IC i if they are inside the community C i , if not the links will be of type BC (if the nodes belong to two different communities).

Then, we calculate the average of the distances of the elements composing the triplet based on the possibilities defined initially. Table 1 presents the possible triplets for the case of a network of 3 communities.

Thereafter, we keep the minimum average distance that will be combined with the initial mass functions by the Dempster rule. Here, the initial mass functions represent the mass functions before the calculation of our model is applied. For each node V ki belonging to several triplets, we will combine by the mean rule all the mass functions obtained at the end of the calculation of the

V k1 V k2 L k12 C 1 C 1 IC 1 C 1 C 2 BC C 1 C 3 BC C 2 C 2 IC 2 C 2 C 1 BC C 2 C 3 BC C 3 C 3 IC 3 C 3 C 1 BC C 3 C 2 BC

Dempster combination.

The proposed algorithm is iterative since, for several cases of noisy nodes and/or noisy links, the corrections are made only after a certain number of iterations.

The mass functions obtained at the end of each iteration represent the input of the next iteration. For each iteration, we calculate the confusion matrix. We remind that a confusion matrix is a technique for summarizing the performance of a classification algorithm.

In order to know the accuracy value at each iteration for each case to be tested, we compared the result of the pignistic probability applied at the end of each iteration with the initial information of the network before introducing the noise. The accuracy represents the ratio of correct predictions to total predictions made.

In order to show the efficiency of our method, we will compare the obtained results with those of the baseline. All experiments were repeated 10 times. All figures represent the average of the accuracy calculated for 10 runs. In addition, the evidential approach and the probabilistic one are tested under the same conditions: The same elements randomly selected and noisy in the evidential case are noisy during the probabilistic approach test.

In the tables presented in the following, we will present the accuracy averages as well as the confidence intervals obtained from the evidential approach and the baseline for each type of experiment.

Possible Corrections

In the presence of noise, the algorithm corrects the information of the network as a function of the noisy elements and the coherent triplets initially defined. In this section, we will present the possible corrections for the case of a network containing 3 communities:

One noisy node and the link and the other node are corrects. Initially we have the triplet:

V k1 ∈ C 1 , L k12 ∈ IC 1 , V k2 ∈ C 1 .
Suppose that one of the nodes is modified and belongs now to C 2 or C 3 . The algorithm will detect that according to the information given by the link and the other node, the modified one should be corrected. Therefore, the noisy node will be affected to C 1 . It is the same if we have a triplet

V k1 ∈ C 2 , V k2 ∈ C 2 , L k12 ∈ IC 2 or a triplet V k1 ∈ C 3 , V k2 ∈ C 3 , L k12 ∈ IC 3 .
The noisy node will be reassigned to its initial community.

Two noisy nodes and the link is correct. In that case, the algorithm will change the nature of the link to obtain a coherent triplet. If the modified nodes belongs to the same community, the algorithm will change the link in such a way that it will be internal to the same community. If the modified nodes belongs to different communuties, the algorithm will change the nature of the link to "Between Clusters" (BC).

One noisy node, one noisy link and one correct node. Suppose that initially we had,

V k1 ∈ C 1 , L k12 ∈ IC 1 and V k2 ∈ C 1 . V k1 was modified to belong to C 2 or C 3 , L k12 ∈ BC and V k2 ∈ C 1 .
In that case, the algorithm will not change the information of the triplet because it's coherent. However, if we had for example Two noisy nodes and noisy link. In that case, the algorithm will compute the minimal distance between the current triplet and the coherent ones defined initially and then modify the information of the current triplet.

V k1 ∈ C 2 or C 3 , L k12 ∈ IC 2 or IC 3 and V k2 ∈ C 1 ,

Used Networks

In this work, we performed our experiments on the real data Karate Club network and on networks generated by LFR.

• Karate Club Network: The Zachary Karate Club is a well-known real social network studied by Zachary [START_REF]The uci network data repository[END_REF]. The study was carried out over a period of three years from 1970 to 1972.

In this network, we find:

-34 nodes that represent the members of Karate Club.

-78 pairwise links between members who are interacted outside the club.

During the study a conflict arose between the administrator "John A" and instructor "Mr. Hi", which led to the split of the club into two. Half of the members formed a new club around Mr. Hi, members from the other part found a new instructor or gave up karate.

• LFR: The LFR benchmark [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF] is an algorithm that generates artificial networks that simulate real-world networks. The generated network has a prior known communities and it is used to compare different community detection methods.

Convergence

The previous presented algorithm is iterative which allows to obtain better results of the accuracy from one iteration to another. The stop criterion used is the stabilization of the value of the accuracy. In these experiments we will limit ourselves to 5 iterations since beyond this number, the variation of the accuracy becomes negligible.

In order to show the convergence of our evidential approach, we will consider an LFR network composed of 99 nodes, 191 links and 3 communities. We will noise 30 nodes and 50 links and evaluate the behavior of the proposed algorithm.

Figure 2 shows the evolution of the accuracy from an iteration to another.

We took the case of 30 noisy nodes and 50 noisy links (Evidential Attributes).

We can notice that from an iteration to another, the accuracy value increases which means that the algorithm succeeds in correcting the noise.

Baseline

In order to show the efficiency of our method, we have performed an algorithm that uses the same principle in probabilistic version.

Step 1: Generation of Probabilities

In this step, we generate randomly N values in [0, 1] for each node and N + 1 probabilities for each link then we normalize. We generate N + 1 probalities as we have IC i links within communities and BC links that connect communities to each other. Then, we associate the maximum probability generated with the class to which the node/link belongs. The vector of probabilities will be defined as follow:

• (p(C 1 ), p(C 2 ), . . . , p(C N )) for each node.

• (p(IC 1 ), p(IC 2 ), p(IC 3 ), . . . , p(IC N ), p(BC)) for each link.

Step 2: Calculation of Distances

In this step, we will calculate the Euclidean distances between the attributes of each node/link composing a triplet with those of the representatives of each group:

• For the nodes: certain events are defined by p

Ω N ω (ω) = 1 with ω ∈ Ω N i.e. p Ω N Ci (C i ) = 1, with i = {1, . . . , N }.
• For the links: certain events are defined by p

Ω L ω (ω) = 1 with ω ∈ Ω L i.e. p Ω L BC (BC) = 1 or p Ω L ICi (IC i ) = 1 , with i = {1, . . . , N }.
Depending of the number of communities composing the network, every representative will have 1 on the attribute of its class and 0 on the others. For example, if we consider a representative of C 1 and we have 3 communities in the network, its probabilities vector will be R 1 = (1, 0, 0).

Hence, we have:

C k1 = arg min ω∈Ω N d E (p Ω N k1 , p Ω N ω ) (21)
C k2 = arg min ω∈Ω N d E (p Ω N k2 , p Ω N ω ) ( 22
)
L k12 is determined according to the coherent triplets by:

L k12 =    IC k1 if C k1 = C k2 BC if C k1 = C k2 (23)
Step 3: Calculation of Average Distances

In this step, we will calculate the minimal average distance of each triplet k defined by:

d k = d E (p Ω N k1 , p Ω N C k 1 ) + d E (p Ω L k12 , p Ω L L k 12 ) + d E (p Ω N k2 , p Ω N C k 2 ) 3 (24)
Step 4: Assignment of probabilities from distances

In this step, we will assign the probabilities resulting from the computation of the distances between triplets. We use the values of the minimal average distance d k .

Hence, we have:

     p Ω N k 1d (C k1 ) = 1 -d k p Ω N k 1d (C k1 ) = d k N -1 (25)      p Ω L k 12d (L k12 ) = 1 -d k p Ω L k 12d (L k12 ) = d k N (26)
     p Ω N k 2d (C k2 ) = 1 -d k p Ω N k 2d (C t k2 ) = d k N -1 (27)
We precise that C k1 , L k12 , C k2 represent respectively the elements contrary to C k1 , L k12 , C k2 .

Step 5: Calculation of the average between the new probabilities and the initial ones

In order to have a single probability distribution for each node/link, we will calculate the average between the probabilities generated in the first instance and those resulting from the calculation of the distances.

p t+1,Ω N k1 = p t,Ω N k1 + p t,Ω N k1d 2 (28)
p t+1,Ω L k12 = p t,Ω L k12 + p t,Ω L k12d 2 (29)
p t+1,Ω N k2 = p t,Ω N k2 + p t,Ω N k2d 2 (30)
where p t,Ω N k 1d , p t,Ω L k 12d , p t,Ω N k 2d are given respectively by equations ( 25), ( 26) and (27). In order to determine a unique probabilities vector for each node (e.g. V k1 ), we combine by the mean rule (given by equation ( 5)), all the probabilities ob-tained for the given node V k1 . Hence, we have:

p Ω N k1 = 1 |T | {k:V k 1 ∈T } p Ω N k (31)
where

T = {(V k 1 , L k12 , V k2 )} and p Ω N k
is given by the equation ( 28).

435

Step 6: Making Decision

In this step, we will decide on the membership of each node/link. To do this, we decide the singleton having the maximum of probability.

Algorithm 2 A Probabilistic Approach for Correcting Noise Require: Graph G(V, E), The set of labeled nodes, the set of labeled links Ensure: The corrected graph.

t = 0 repeat 1.
for each element of a triplet k, compute the Euclidean distance between the element and the corresponding categorical representative using Eqs

for each triplet k, compute the minimum average distance d k by using Eq (24)

3. Define probabilities from the computed d k using the Eqs ( 25), ( 26), (27)

4. Update the probabilities using the Eqs (28),( 29), (30), 5. Combine the probabilities for the same node in order to have a unique vector of probabilities by using the Eq (31)

6. Make decision about the belonging of each element of the triplet k

7. t = t + 1
until Number of iterations equal to 5.

Algorithm 2 shows the outline of the process followed for correcting noise in social network using probabilistic attributes.

440

In order to test the effectiveness of the baseline, we will add the noise as we did with the evidential approach. To do this, we will add noise to the same nodes and links selected randomly when we tested the evidential approach.

Improvement Rate

Tables 2, 3, 4, 5 show the rate of improvement of the evidential approach compared to the baseline at the fifth iteration. We consider the variation of noise in the LFR network composed of 99 nodes, 191 links and 3 communities.

The rate of improvement is calculated by making the difference between the average values of the accuracy obtained with the evidential approach at the fifth iteration with that given by the baseline.

Noise

Experiments on Real Data: Karate Club

As the karate club network has 2 communities, the frames of discernment of the nodes and links will be defined by:

• Ω N = {C 1 , C 2 } • Ω L = {IC 1 , IC 2 , BC}
In this part, we will show the results obtained in the case of noisy nodes only, noisy links only and noisy nodes and links at the same time.

Noisy Nodes Only

In figure 3 we present the accuracy average values at the fifth iteration when we vary the number of noisy nodes.

We notice that the more the number of noisy nodes increases, the more the accuracy average value decreases for both evidential and probabilistic methods.

However, we remark that we obtain a better accuracy average results with the belief function theory comparing to the probability theory. This can be explained by the fact that the theory of belief functions manages ignorance as well as conflict.

Table6 presents the accuracy averages and the confidence intervals obtained from the evidential approach and the baseline for each level of noise added to the nodes only in the case of the Karate Club. We show in figure 4 the accuracy average results at the fifth iteration after noising 20, 40, 60 and 78 links of the network.

According to the curve, the average accuracy values given by the evidential approach are better than that given by the baseline in each level of noise.

We show in table7 the obtained accuracy averages and the confidence in-475 tervals given by the evidential method and the probabilistic approach when we vary the number of noisy links only in the case of the Karate Club.

Noisy Nodes and Noisy Links

In this third case, we proceed by noising the nodes and the links at the same time. Figure 5 shows the obtained results of accuracy average after noising the We notice that the accuracy average values decreases as the noise level increases for both evidential and probabilistic approaches. However, the proposed 485 method gives better results than the baseline.

Table8 shows the obtained accuracy averages and the confidence intervals given by the evidential method and the probabilistic approach in the case of noisy nodes and noisy links in the case of the Karate Club.

Experiments on LFR

In the second part of our experiments, we used different networks generated with LFR [START_REF]Lancichinetti-fortunato-radicchi benchmark[END_REF]. We present in table 9 the parameters used to generate our networks.

We will perform several experimentations which will be repeated 10 times and show the obtained average of the accuracy. All the figures present the results given by the evidential approach and the baseline.

We will start by varying the noise of the nodes, links and both of the LFR network composed of 99 nodes, 191 links and 3 communities.

For the rest of the experiments, we will vary each time one of the parameters of the LFR network such as the number of communities, the size of the network and the mixing parameter µ and observe their impact on the noise correction rate. For each of these experiments we will noise 60% of the nodes and 50% of the links.

In this work, we used the LFR parameters presented in table 9 for the We vary the number of noisy nodes by 30 at each step and then we add noise on all the nodes of the network. As for the links, we vary the noisy links by 50, then we add noise on all the links of the network.

We chose these values in order to have a better view on the impact of the noise introduced on the network information.

We compare the obtained results with those of the baseline.

Figure 8 shows the results of the accuracy average for every level of noise used in these experiments. We compare the obtained results with those of the baseline after noising 30 nodes and 50 links, 60 nodes and 100 links, 90 nodes and 191 links and finally, 99 nodes and 191 links.

From this figure, we can notice that the accuracy average results are better with the evidential attributes. We remark also that when it is very noisy, it becomes impossible to obtain good results.

It should be noted that in the case of adding a maximum noise, the value of the accuracy average is stable from the beginning. This is due to the fact that when we noise the data, the mass functions are generated randomly and therefore there are two possibilities:

• Either the new mass function makes sure to change the class of the node/link.

LFR: Variation of the Communities number

In this part of experiments, we vary the number of communities. We generate 4 LFR networks:

• a network with 200 nodes, 402 links and 3 communities.

• a network with 200 nodes, 472 links and 4 communities.

• a network with 200 nodes, 477 links and 5 communities.

• a network with 200 nodes, 501 links and 6 communities.

Figure 9 shows the obtained results of the accuracy average for each network.

We can remark that for all the networks, the evidential model gives better results on links and nodes accuracy average than the baseline. We notice also that there is not really a big difference in the values of the accuracy average when we vary the number of communities. We can, therefore, conclude that the proposed approach is stable.

Table13 presents a comparison between the accuracy averages and the confidence intervals given by the evidential approach and the probabilistic one when we vary the number of communities in the case of LFR networks. method is stable since the values of the precision calculated for each network are close to each other.

Table14 shows the obtained accuracy averages and the confidence intervals from the evidential approach and the probabilistic one when we vary the size of the network in the case of LFR.

LFR: Variation of the Mixing Parameter µ

In this section, we will present the obtained results of the accuracy average following the variation of the mixing parameter µ. We consider 5 networks whose mixing parameter was varied and containing 3 communities:

• a network with 200 nodes, 484 links and µ = 0.1.

• a network with 200 nodes, and 402 links and µ = 0.3.

• a network with 200 nodes, and 467 links and µ = 0.5.

• a network with 200 nodes, and 488 links and µ = 0.7.

• a network with 200 nodes, and 502 links and µ = 0.9. We find that the accuracy average of the nodes is greater than the accuracy average of the links when µ < 0.5, while the latter becomes greater than the accuracy average of the nodes when µ > 0.5. This change is explained by the fact that the more the mixing parameter approaches 1, the more we get a network with more links between clusters than within the community.

We present in table15 the obtained accuracy averages and the confidence intervals given by the evidential approach and the baseline when we vary the mixing parameter in the case of LFR.

Comparison of execution time

In this section, we will compare the execution time put by the model's evidential version as well as the probabilistic one. We will present the execution time at the fifth iteration. We will observe the evolution of the execution time in the case of LFR networks with 6, 5, 4 and 3 communities. The execution time will be expressed in seconds.

The table 16 shows that the evidential method takes more time compared to the baseline. We notice also that as the number of communities increases, the execution time increases too.

We remind that in this paper, we focused on the use of a limited number of communities. In terms of scaling up, there are several strategies that can reduce complexity such as representing only the focal elements or grouping them together if their values are negligible. This will be the subject of future work.

Conclusion

Researches that have focused on clustering using the network structure as well as the nodes attributes, ignore the links information. In order to remedy this problem, we propose a method which allows to classify the nodes in their initial clusters even when there is a significant noise added to the network. In the case of a large noise, the algorithm guarantees the information coherence of any network even when it is a network whose nodes and links attributes have been strongly modified.

Throughout this work, we first recalled some basic notions of the theory of belief functions as well as some methods for the communities detection based on graph structure as well as the attributes and some other related works. Then, we presented our method which consists in first generating attributes on the nodes and the links according to the network structure. In a second step, we added noise on the attributes and then reclassified the nodes and/or the links.

We tested our approach on real data: the Karate Club network. Then, we varied the noise on a LFR network composed of 3 communities and we presented the obtained results during the noising of the nodes, links and both. Finally, we studied the behavior of the proposed method according to the variation of the number of communities, the size of the network as well as the mixing parameter.

All the obtained results were compared with those of the baseline. Experiments

have shown that the more we noisy the network, the farther we get away from the initial network, but we are sure to have a coherent network. In addition, our proposed approach is stable when we vary the number of communities and the size of the network and gives better results in all studied cases than the baseline.

As future work, we intend to deal with the case of overlapping communities.

Given the fact that a node can belong to several communities, it has become interesting to analyze the evolution of a social network over time. This study could help to better identify the types of nodes as well as their exchanges on the network. In addition, the theory of belief function offers a very effective way to analyze the evolution during the time of evidential networks composed of overlapping communities.

We also intend to improve the code and the execution time of the proposed method. In fact, although the proposed approach yields better results, it takes much longer time than the baseline.

  consists on dealing with the uncertainty that occurs in the attribute values within the belief function framework in the case of clustering. In this work, the authors present a new version of decision trees with the theory of belief functions to handle the case of uncertainty present only in attribute values for both construction and classification phases.

Figure 1

 1 Figure1shows the notations for a given triplet k. It consists of two nodes (starting node, arrival node and link that connects them) having each one a mass function which shows the belonging possibilities of a node to a community C i . Nodes are connected through a link, that also has a mass function which indicates the possibilities of its label (A link can be of the type IC i if it is inside the community or BC if it connects two nodes belonging to two different communities).

Figure 1 :

 1 Figure 1: Triplet k.

Algorithm 1

 1 An Evidential Approach for Correcting Noise Require: Graph G(V, E), The set of labeled nodes, the set of labeled links Ensure: The corrected graph t = 0 repeat 1. for each element of a triplet k, compute the distance of Jousselme between

  the algorithm will change the link to BC and if one of the nodes (or both) are connected to other nodes, so the algorithm will have another information and can change one of the node based on that.

Figure 2 :

 2 Figure 2: LFR: corrected nodes and links: case of 30 noisy nodes and 50 noisy links.

Figure 3 :

 3 Figure 3: Karate Club: comparison of probabilistic and evidential accuracy: case of noisy nodes.

Figure 4 :

 4 Figure 4: Karate Club: comparison of probabilistic and evidential accuracy: case of noisy links.

480

  attributes at the fifth iteration. The abscissa represents respectively the level of noise 10 nodes and 20 links, 20 nodes and 40 links, 30 nodes and 60 links and finally, 34 nodes and 78 links.

Figure 5 :

 5 Figure 5: Karate Club: comparison of probabilistic and evidential accuracy: case of noisy nodes and links.

Figure 8 :

 8 Figure 8: LFR: comparison of probabilistic and evidential accuracy: case of noisy nodes and links.

Figure 9 :

 9 Figure 9: LFR: comparison of probabilistic and evidential accuracy: case of noisy nodes and links.

Figure 10 :

 10 Figure 10: LFR: comparison of probabilistic and evidential accuracy: case of variation of the size of the network.

Figure 11 :

 11 Figure 11: LFR: comparison of probabilistic and evidential accuracy: case of variation of the mixing parameter.

Figure 11

 11 Figure11shows the results obtained by the evidential method and the baseline after varying the mixing parameter.

Table 1 :

 1 Coherent Triplets For 3 Communities.

Table 2 :

 2 Improvement Rate: Case of Noisy Nodes Only.

	Noise	Rate of improvement
	50 Links	41%
	100 Links	36.7%
	191 Links	36%

Table 3 :

 3 Improvement Rate: Case of Noisy Links Only.

Table 4 :

 4 Improvement Rate for Nodes: Case of Noisy Nodes and Noisy Links.

	Noise	Rate of improvement
	30 Nodes + 50 Links	50%
	60 Nodes + 100 Links	27%
	90 Nodes + 191 Links	6%
	99 Nodes + 191 Links	4%

Table 5 :

 5 Improvement Rate for Links: Case of Noisy Nodes and Noisy Links.

Table 6 :

 6 Accuracy Average and Interval of Confidence: Case of Noisy Nodes Only in the

		Karate Club.
	470	4.6.2. Noisy Links Only

Table 7 :

 7 Accuracy Average and Interval of Confidence: Case of Noisy Links Only in the

	Karate Club.

generation of our networks: n represents the number of nodes, K the average degree, maxK the maximum degree, mu the mixing parameter, t1 the minus exponent for the degree sequence, t2 the minus exponent for the community size distribution, minC the minimum for the community size, maxC the maximum for the community size, on the number of overlapping nodes, om the number of memberships of the overlapping nodes and C the average clustering coefficient.

Since LFR generates the links of the graph in both directions and in this work we consider non-directed graphs, we will use a single link to represent the connection between two nodes. As a result, the number of links we present in the experimental part is half the number of links initially generated.

The first set of experiments consists of varying the noise in an LFR network composed of 99 nodes, 191 links and 3 communities. We will proceed by noising the nodes at first, then the links and finally we will simultaneously noise both.

The frames of discernment of the nodes and links for this network are defined as follows:

• Ω L = {IC 1 , IC 2 , IC 3 , BC} with IC i represents the links inside the community C i and BC represents the links between 3 communities.

Noisy Nodes Only

In this first case of experiments, we will add noise to a number of nodes randomly selected of the network. The noise consists on modifying the mass functions of the selected nodes by randomly generating two focal elements (ignorance and another element except the empty set). We will then compare the obtained results with those given by the baseline. Figure 6 shows the obtained results of the accuracy for every variation of the noise. We vary the number of noisy nodes from 30 to 99.

We notice that the more the number of noisy nodes increases the more the accuracy average decreases. We also note that for each level of noise, we obtained better results with the evidential model. This is because the theory of belief functions offers a very effective way to handle ignorance and conflict.

Table10 shows the obtained accuracy averages and the confidence intervals

given by the evidential method and the probabilistic approach in the case of noisy nodes only in the case of LFR network.

Noisy Links Only

The second part of the experiments consists in keeping the initial generation of the mass functions of the nodes and adding noise only to the mass functions 540 of the links.

Figure 7 shows the obtained results of the accuracy average due to the variation in the number of noisy links. In this figure, we compute the accuracy average for 50, 100 and 191 noisy links. We notice that we obtain better results when we use the evidential attributes. These results can be explained by the fact that the evidential approach better manages ignorance than the probabilistic approach.

We present in table11 the accuracy averages and the confidence intervals obtained from the evidential approach and the baseline in the case of noisy links only in the case of LFR network.

Noisy Nodes and Noisy Links

In this third part of the experiments, we noised simultaneously the nodes and the links of the network.

The aim of simultaneously noising the nodes and the links is to make the network totally incoherent and to evaluate the ability of the algorithms to correct the noise and to find a network comparable to the initial one. • Either the element always retains its initial membership but with a different mass function.

Hence, we will always have elements that are correct even when it's the case of maximal noise. These correct attributes help in the finding of other correct triplets.

We present in table12 a comparison between the accuracy averages and the confidence intervals given by the evidential approach and the baseline in the case of noisy nodes and noisy links in the case of LFR network.

In what follows, we will noise 60% of nodes and 50% of links by varying each time a parameter of the LFR algorithm. The idea is to see the impact of each parameter on the correction rate of noisy information for the same level of noise. To do this, we will first vary the number of communities. Then, we will vary the n which represents the number of nodes composing the network and finally, we will vary the mixing parameter.

Case of

LFR: Variation of the Size of the Network

In this section, we will present the obtained results of the accuracy following the variation of the network size. We consider 5 networks whose number of nodes was varied and containing 3 communities:

• a network with 50 nodes and 115 links.

• a network with 99 nodes and 191 links.

• a network with 200 nodes and 402 links.

• a network with 300 nodes and 721 links.

• a network with 400 nodes and 932 links.

Figure 10 presents the obtained accuracy average results. It shows that the evidential approach was able to correct more information than the baseline whatever the network considered. Moreover, figure 10


A Michrafy
email: michrafy@enstimac.fr

H Diarra

J A Dodds

M Michrafy

Experimental and numerical analyses of homogeneity over strip width in roll compaction

Keywords:

Homogeneity of properties over the width of strips produced by roll compaction of microcrystalline cellulose powder (MCC) has been examined by light transmission through the compact, by measurements of the porosity distributions and by three-dimensional finite element modeling. Light transmission through compacts revealed periodic heterogeneity in the form of alternate dark and light zones. The period seems to be connected to the geometry of the screw and independent on the feed screw velocity which was varied with the roll speed with a constant ratio. Measurements of the porosity of samples cut from the compacted strip show heterogeneity of the density over the width of strips with a higher density in the centre of the strip and a lower density on the sides. These two techniques clearly showed the heterogeneous behavior across the width of the compacted strip of MCC. However, the light zones (respectively dark zones) did not correspond to the lower porosity zones (respectively higher porosity zones). Three-dimensional finite element modeling (FEM) of roll compaction of powders was conducted with two inlet feed conditions: constant feed pressure and constant feed velocity. Results of the simulations using the constant feed pressure show a uniform maximum principal stress and density across the width of the strip. When a constant inlet feed velocity is assumed the maximum principal stress over the strip width was higher at the centre of the strip and decreases to the sides. This profile also corresponds to the density profile over the width of the strip. In this case, the predicted results present a similar tendency to that found by mercury intrusion porosimetery and are in agreement with the measured bulk density of strips produced with different roll speeds.

Introduction

Roll compaction is a powder agglomeration process used in variety of industries including the pharmaceutical, mineral and chemical industries. In the pharmaceutical industry, roll compaction is used as a dry granulation method, well suited to powders which are sensitive to water and other solvents. The process can produce composite granules from mixtures of drug substances and excipient powders thus improving flowability, giving more uniform composition and chemical stability for direct compression. Roll compaction of poor flowability powder mixtures requires screw feed of the powder between two counter-rotating rolls. These then draw the powder into the compaction zone and apply a high pressure forming a strip of compacted powder. Despite the simplicity of the roller compaction, a quantitative understanding of the process has proved difficult to develop because of the complexity of the powder behavior and the large number of operating parameters. The heteroge-neity of properties over the compacted strip width is one of the complex features that have been demonstrated in several experimental investigations. Tundermann et al. [START_REF] Tundermann | Deformation and densification during the rolling of metal powders[END_REF] performed roll compaction of sponge-iron powders MH300 (fine) and MH100, and showed non-uniform density over the strip width with a distribution that depended on the particle size of the powder. Simon and Guigon [START_REF] Simon | Correlation between powder-packing properties and roll press compact heterogeneity[END_REF] showed the role of the feed screw on the heterogeneity observed on strips of compacted lactose monohydrate mixed with 4% coal. In the work of Miguelez-Moran et al. [START_REF] Miguelez-Moran | Characterisation of density distributions in roller compacted ribbons using micro-indentation and Xray micro-computed tomography[END_REF][START_REF] Miguelez-Moran | The effect of lubrication on density distributions of roller compacted ribbons[END_REF] on microcrystalline cellulose, characterizations of the density distribution of ribbons of MCC showed a non-homogeneous behavior. However, in the investigations of Lecompte et al. [START_REF] Th | Dry granulation of organic powders-dependence of pressure 2D-distribution on different process parameters[END_REF] on the roll compaction of organic powder measurements of maximum pressures for several consecutive rotations were practically similar and hence no heterogeneity of the pressure distribution over the strip width was observed. This observation was explained by the fact that the screw was positioned further from the rolls. From the works cited above, it is important to point out the role of the design of roll press, the feeding system, the roll width and the geometry of the screw on the heterogeneity generated over the width of strips of compacted powders. Despite the difficulty in comparing results from the literature due to the type of roll press and to the operating conditions, it is commonly accepted that the two main factors responsible for strip heterogeneity are: the powder feed system and the powder flow behavior in the compaction area between rolls.

This paper presents experimental and numerical investigations to analyze the homogeneity of properties over strip width of microcrystalline cellulose powder compacted with a laboratory roll press. Results from the simulations and from the experiments are discussed in terms of the heterogeneity of properties over the width of the strips.

Experimental

Roll press

Experiments were carried out with a Komarek® B050PH laboratory press with horizontal feed screw. This press was fitted with 100 mm diameter and 38 mm wide smooth rolls. Fig. 1 is a schematic diagram of the press. The roll speed can be varied in the range 0-7.5 rpm, the feed screw can be varied in the range 0-137 rpm and the separation force of rolls (known as "hydraulic pressure") can be varied up to 50 kN (corresponding to the linear force 13.2 kN cm -1 ). Two cheek plates are placed on both sides of the rolls as it is shown in Fig. 1.

The upper roll was instrumented with three strain gauges to measure the normal pressure along the width of the roll. The strain gauges were positioned at 10 mm from the roll surface and are aligned in the direction of the axis (one gauge in the centre and the two others at 5 mm from both sides of the roll). Thus, there is no direct contact of the sensor with the powder. The signals are transmitted to the computer using a slip ring assembly fixed to the press as it was showed in the picture of Fig. 2.

Powder

Microcrystalline cellulose (MCC) known commercially as Avicel PH102, with a mean particle size (d 50 ) of 90 μm, and apparent and true densities of 0.27 and 1.59 g/cm 3 , was used in the experiments.

Compact and characterization

It is well known, for roll press with horizontal feeding that screw and roll speeds are the dominant parameters in the roll process. To avoid the lack of powder and the overfeeding of powder, the correlation between the screw and roll speeds (Vs and Vr) for obtaining a compact with good mechanical properties was first characterized (Fig. 3). Three speed zones were identified: an underfeeding zone where the powder material is not compacted, an overfeeding zone where the machine becomes blocked, and a good compaction zone (giving compact with a good mechanical properties). In this latter zone strips of compacted powder MCC were formed using constant ratio Vs/Vr. The gap was fixed to 2 mm and a high roll separation force of 34 kN was employed to avoid fluctuations in the thickness of the compacted strip (no fluctuations of the gap). With the above operating conditions and for each pair of screw and roll speeds with Vs/Vr = 9.2, plates of length around 15 cm were considered and each one was cut into three rectangular pieces. The thickness of samples was in the range 0.19-0.21 cm, the width in 3.5-3.8 cm and the length in 4-5 cm. The bulk density of the pieces was calculated from their mass and volume. The density of the plate was evaluated by averaging the three measurements. Results of the averaged density were plotted in Fig. 4 according to the roll speeds. The bulk density evolution showed a decrease for formed strips using roll speeds higher than 6.4 rpm. This is generally a consequence of the short residence time of the powder under the pressure.

The analysis of the homogeneity of properties of the strips produced was made in two ways: by light transmission through the compacted strip [START_REF] Simon | Correlation between powder-packing properties and roll press compact heterogeneity[END_REF] and by mercury porosimetry measurements of small samples carefully cut from the compacted strip in the rolling and the across directions. The image of the light transmitted through the compact was plotted in Fig. 5 and reveals periodic heterogeneity represented by dark and light zones. The period seems independent on the screw velocity (the roll speed was varied in the good compaction zone without changes of the period size) and seems to be linked to the geometry of the screw.

To aid understanding of the heterogeneity observed, a grid was superimposed according to the alternate lighted and darker zones (Fig. 6). Three groups in the rolling direction of four samples (across the wide) were cut following the plane of the grid. In Fig. 6, the designation G, MG, MD and D corresponds to the position of samples across the width. The symbols "+S, S, C and +C" correspond respectively to "more dark, dark, clear, and more clear".

With the mercury porosimeter the volume of mercury introduced in each sample at high pressure (up to 30 MPa) was measured and was used as an index of the porosity of the sample: "the greater the volume introduced, the higher is the porosity of the sample and hence the lower is the density ". Results were plotted in Fig. 7a.

To verify that the intrusion of mercury at high pressure does not introduce errors by modifying the microstructure of samples (e.g. local densification or destruction of internal pores) results at low pressure (0.1 MPa) as plotted in Fig. 7b were also examined.

Three-dimensional finite element modeling

The experimental results presented above demonstrate the existence of the heterogeneity of properties over the strip width of compacted MCC powder. This heterogeneity probably results from the non-uniform feeding of powder by the screw, the flow of powders between rolls under the effect of the complex mechanisms of powderroll friction, and the presence of two cheek plates on the roll sides to limit powder loss.

The objective of this section is to use three-dimensional finite element modeling to examine the stress and density distributions over the strip width of compacted MCC powder.

Powder behavior

The analysis was conducted by assuming that the powder behaves as a single phase porous media (without air). As for die compaction [START_REF] Michrafy | Modelling the compaction behaviour of powders: application to pharmaceutical powders[END_REF][START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF][START_REF] Wu | Modelling the mechanical behaviour of pharmaceutical powders during compaction[END_REF][START_REF] Bonnefoy | Guidelines for modeling cold compaction behaviour of various powders[END_REF][START_REF] Frenning | Analysis of pharmaceutical powder compaction using multiplicative hyperelasto-plastic theory[END_REF] or for roll compaction in two dimensions [START_REF] Dec | Comparison of various modeling methods for analysis of powder compaction in roller press[END_REF], the Drucker Prager Cap model implemented in Abaqus® [START_REF]Abaqus/Explicit[END_REF] was used. The elastic behavior was assumed to be linear and to depend on the relative density. Results of such modeling depend on the identification of material parameters and the boundary conditions employed. In the space (p,q) "hydrostatic pressure, equivalent stress", the yield surface is essentially represented by the shear surface (shear failure line) and the cap surface. The constitutive equations of the model are voluntarily not listed here and can be found in the references cited above.

The material parameters required by the simulations, and which have to be identified from experiments such instrumented die compaction, simple compression and diametrical compression [START_REF] Bonnefoy | Guidelines for modeling cold compaction behaviour of various powders[END_REF] are: powder cohesion d, internal friction β eccentricity of the cap R and P b the evolution of the cap surface according to the volumetric inelastic strain. In the elastic plastic representation used here the relative density is used as variable describing the microstructural state, hence, the parameters cited above and elastic properties depend on this.

Calibration of model parameters. Method based on simple tests

Tablets were prepared by compacting MCC powder in an unlubricated cylindrical die of 11.28 mm diameter with pressures in the range 20-168 MPa with an instrumented reciprocating press [START_REF] Michrafy | Compaction behavior of binary tablets[END_REF]. Axial stresses and displacements of the upper and lower punches (σ u , σ l , u u , u l ), and radial stress σ r on the die were recorded. Fig. 8a andb shows typical axial stress versus the axial strain and equivalent stress (q = σ uσ r ) versus hydrostatic pressure p = (σ u + 2σ r ) / 3.

Young modulus and Poisson ratio

A simple axial compression test was used to evaluate the elastic properties using the tablets prepared as above. The upper and lower punches were lubricated with a magnesium stearate to reduce the role of wall friction. The axial pressure σ z and strain ε z =ln (h / h 0 ) and radial (transverse) strain ε r =ln (d / d 0 ) were monitored during loading and unloading, (h 0 and d 0 are the initial height and diameter of the tablet). Then, the Young modulus was evaluated from the initial unloading slope of the simple compression curve and the Poisson ratio was given by ν = -εr εz . The resulting changes in elastic properties according to the relative density are plotted in Fig. 9a andb.

Cohesion and angle of internal friction

For a given relative density corresponding to a given (p max ,q max ) of the die compressive cycle (Fig. 8b), the shear failure line of the yield surface is defined by two parameters: the cohesion d, which is the intersection with the deviatoric axis q and the internal friction angle β, the slope of the line. They can be identified from the failure stress σ d in the diametrical compression test and the failure stress σ u based on the simple compression test. In the space (p,q) the two resulting points (2σ d /3, ffiffiffiffiffiffi 13 p σ d ) and (σ u / 3,σ u ) define the failure line and hence the cohesion and the angle of internal friction. In the range of the relative density studied, the evolution of the cohesion according to the relative density is plotted in Fig. 10a. The angle of internal friction was found in the range 65-71°. These high values are generally obtained because σ d is very low comparatively to σ u . Results of internal friction angle are not considered as intrinsic values of the material. They are linked to the test used. To represent internal friction angle of MCC from low to high pressures, the measured high values were associated to high densities and were extended to low densities by decreasing the internal friction angle down to 38°(internal friction angle of the powder MCC at low pressure using Jenike shear test). The resulting data were plotted in Fig. 10b. It should be noted here that the extension to low pressure is arbitrarily and could not be easily validated with experiment. Similar profiles of the cohesion and the internal friction angle were also proposed in [START_REF] Sinka | The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager Cap model[END_REF].

Eccentricity and hardening function

From the fully instrumented die-compression experiment, the maximum stress (p max ,q max ) of the die compressive cycle corresponds to the density reached in the compact. Then, for a given density, the cap surface parameters (the eccentricity R and the cap hardening function P b ) were calculated from the (p max ,q max ), the cohesion and the internal friction angle previously obtained [START_REF] Bonnefoy | Guidelines for modeling cold compaction behaviour of various powders[END_REF]. The resulting eccentricity R and hardening function P b were plotted in Figs. 11 and12. The above extension of internal friction angle gives an extension of the eccentricity at low densities. This extension has not been experimentally validated and the data of triaxial test of MCC could be useful in this case.

Finally, the resulting evolution of the yield surfaces with relative density and the corresponding die-compression data are plotted in Fig. 13.

Wall friction

Coulomb friction was assumed between the rolls and the powder. No instruments were available to measure the tangential force at the contact interface. The friction coefficient was therefore characterized from the die-compaction experiments.

During die compaction, the tangential force to the contact area (die-wall) F t = (σ uσ l )πR 2 is proportional to the normal force F r = 2πRhσ r . The normal pressure is considered uniform along the compact height h. The friction coefficient μ(σ r ) was then estimated as:

μ σ r ð Þ = σ u -σ l ð Þ πR 2 2πRhσ r ð1Þ
where the height h is obtained from the initial height of the powder bed (10 mm) and the measured punch displacement u u and R = D 2 with D = 11.28 mm the internal diameter of the die. Fig. 14 shows the evolution of the wall friction with the normal (radial) pressure in the condition of unlubricated die. As is seen in Fig. 14, the friction coefficient decreases down to 0.24 for low radial pressure (b5 MPa) and increases progressively up to 0.35.

Numerical procedure

The numerical simulation was performed using commercial software (Abaqus® explicit version 6.5). Behavior is assumed to be symmetrical between rollers and only the upper half of the domain was considered. The roll of 100 mm of diameter and 38 mm of width was assumed to be a rigid body. The powder was meshed with 9690 continuum elements C3D8R with reduced integration (Fig. 15). The gap was adjusted to 2.0 mm and the entry height of the material corresponds to 18°roll angle. The initial material density was fixed as 0.27 • 10 3 kg/m 3 . Because of the high volume reduction of mesh in the nip zone and the resulting mesh distortion, the Arbitrary Lagrangian Eulerian (ALE) procedure with adaptive meshing was employed with explicit integration using the Abaqus® Explicit code. The powder feed system was approximated by two conditions: uniform feed velocity (equal to the linear velocity of the roll 30 mm/s) and uniform feed pressure (1.0 MPa). Eulerian inflow and outflow boundary conditions were employed to ensure the entry and the exit of the material. For computational efficiency of the quasi-static problem, a mass-scaling factor was used and optimized based on the examination of the level of the kinetic energy that must be without oscillations and be small compared to the internal energy (both elastic and plastic strain energy of the deformable powder material). Coulomb friction was assumed for contact behavior between roll and powder. The friction coefficient was μ = 0.35.

A rigid surface was created in the feed zone to represent the feeder barrel. The presence of this surface avoids the severe distortion of meshes during the feeding. The friction coefficient between the deformable meshes in contact with this surface was fixed to 0. This condition, which can be physically questionable, ensures a uniform velocity profile in the feeder barrel and hence ensures for the powder, at the entrance between rolls, the same velocity than the linear velocity of the roll. The powder can then flow between the rolls without distortion problems. Also, the cheek plates preventing the powder loss from the sides of the rolls were materialized by null normal displacement.

As the material parameters change with the density, a user subroutine VUMAT was developed to reflect this evolution.

The homogeneity over the strip width was analyzed at the gap position through the distribution of the maximum principal stress (Fig. 16) and the density (Fig. 17) using the above two inlet feed conditions.

Discussion

As is shown in Fig. 5, the light transmission through compacts reveals a periodic heterogeneity shown by dark and light zones. The analysis of such heterogeneity with different screw velocities showed an identical period connected to the geometry of the screw. Similar results were observed on other powders in the work of Simon and Guigon [START_REF] Simon | Correlation between powder-packing properties and roll press compact heterogeneity[END_REF]. According to the above heterogeneity observed over the strip width, and in order to understand if the darker zone (respectively lighter zone) is associated to low densities (respectively higher densities zone), measurements of the porosity of samples cut from the compacted strip following the grid of Fig. 6, were determined by mercury intrusion porosimetry (Fig. 7a andb). These results show the heterogeneity of the porosity over the strip width and indicate that the higher density is in the central part of the strip. This is in agreement with the work of Lecompte et al. [START_REF] Th | Dry granulation of organic powders-dependence of pressure 2D-distribution on different process parameters[END_REF] and measurements of Miguelez-Moran et al. [START_REF] Miguelez-Moran | Characterisation of density distributions in roller compacted ribbons using micro-indentation and Xray micro-computed tomography[END_REF][START_REF] Miguelez-Moran | The effect of lubrication on density distributions of roller compacted ribbons[END_REF]. The weight of the sample used in these measurements is also plotted in Fig. 7a andb, which show little difference in the masses of the samples analyzed. In order to verify that high pressure mercury intrusion does not introduce errors in the measurements by modifying the microstructure of samples (local densification or destruction of internal pores) results obtained at low pressure (0.1 MPa) are plotted in Fig. 7b and also show a similar evolution to the results obtained at high pressure.

The heterogeneity revealed by the light transmission across the width of the strip and the heterogeneity obtained by mercury porosimetry (according to the grid of the Fig. 6) could not be completely correlated. The lighter zones (respectively darker zones) do not correspond to the denser zones (respectively less dense zones). The light transmission results are probably more related to the anisotropy of the microstructure which plays an important role of the light diffusion.

Finally, the two techniques used clearly show that there are heterogeneities across the width of roll compacted strips of MCC. However, questions remain on the effects of the pressure level, the roll width, the powder lubrication of reducing the heterogeneity that could be created by the screw. More investigations in this direction should clarify the impact of these parameters in the homogeneity of strips.

In the three-dimensional FEM (finite element method) simulations of roll compaction, two inlet feed conditions were considered to represent the feed system: a constant feed pressure and a constant feed velocity ensuring a variable entering flux of material. In the case of constant feed pressure, the resulting distribution (over the strip width) of the maximum principal stress and the density on the elements in contact with the roll are approximately uniform (Figs. 16a and17a). These results did not show the heterogeneity found in experimental investigations.

In the case of constant feed velocity the maximum principal stress over the strip width is higher in the central zone and decreases towards the sides (Fig. 16b). The corresponding density also presents the same profile along the width (Fig. 17b). Results of this simulation are more in agreement with the measurements of the mercury porosimetery (Fig. 7a) and the measured bulk density (Fig. 4). The decrease of the pressure and the density near the sides could be explained by a lesser mass of powder in these zones due to the forced flow of powder.

Conclusion

The objective of this work was to study the heterogeneity of the properties across the width of the compacted strips of microcrystalline cellulose powders produced by a roll press with a horizontal feed screw. Two experimental techniques; light transmission and mercury porosimetry and a three-dimensional finite element simulation were used in the study. The light transmission revealed periodic heterogeneity shown by darker and lighter zones of the compacted strip, whereas the mercury porosimetry showed higher density at the central part of the compact and lower density at the sides.

The three-dimensional FEM simulation of the roll compaction assuming a uniform feed pressure predicted a uniform maximal principal stress and density along the width. Simulations made by assuming uniform inlet feed velocity predicted a maximum principal stress and density across the width with the same profile: higher values at the central part of the compact and lower values on the sides. These predictions showed similar tendency to the measurement with mercury porosimeter and are in more agreement with the measured bulk density of strips.

Fig. 1 .

 1 Fig. 1. Cross section of the Komarek® B050PH roll press system. 1 -roll, 2 -roll shaft, 3 -bearing block, 4 -supporting hydraulic system, 5 -screw feeder, 6 -screw feeder drive, 7 -feeder barrel, 8 -cheek plate, and 9 -feed hopper.

Fig. 2 .

 2 Fig. 2. Picture of instrumented roll and slip ring assembly.

Fig. 3 .

 3 Fig. 3. Screw speed vs. roll speed. Zone 1: lack of feeding; Zone 3: overfeeding; Zone 2: good compaction. Line Vs/Vr = 9.2 corresponds to the speeds used to produce strips which bulk density is plotted in Fig. 4.

Fig. 4 .

 4 Fig. 4. Average bulk density of compacted strips of MCC according to roll speeds (Vr = 9.2Vs).

Fig. 5 .

 5 Fig. 5. Results of light transmitted through the compact of microcrystalline cellulose. Gap = 2 mm, roll speed = 6.4 rpm, and screw speed = 59 rpm.

Fig. 6 .

 6 Fig. 6. Grid of samples analyzed by mercury porosimeter.

Fig. 7 .

 7 Fig. 7. a. Mercury intrusion porosimetry (pressure 30 MPa). b. Mercury intrusion porosimetry (pressure 0.1 MPa).

Fig. 8 .

 8 Fig. 8. a. Axial stress vs. axial strain of the compression of MCC in a 11.28 mm diameter of unlubricated die. b. Equivalent stress vs. hydrostatic pressure (cycle loading and unloading of die-compression).

Fig. 9 .

 9 Fig. 9. a. Young's modulus vs. relative density E (ρ r = 1) = 10.30 GPa. b. Poisson's ratio vs. relative density ν (ρ r = 1) = 0.28.

Fig. 11 .

 11 Fig. 11. Hydrostatic yield stress vs. volumetric inelastic strain.

Fig. 12 .

 12 Fig. 12. Eccentricity vs. relative density.

Fig. 13 .

 13 Fig. 13. The evolution of the yield surfaces with the relative density ρ r of MCC. The dotted lines correspond to the die compressive cycles of MCC.

Fig. 14 .

 14 Fig. 14. Die-wall friction coefficient vs. radial pressure of MCC (fitted from Eq. (1)). Data from die compaction in unlubricated die of 11.28 mm of diameter.

Fig. 15 .

 15 Fig. 15. Meshed domain with continuum elements (C3D8R continuum element with reduced integration -Abaqus®).

Fig. 16 .

 16 Fig. 16. Distribution of maximum pressure across the width of the strip of MCC. (Predictions at neutral angle position). (a) Boundary conditions with uniform feed pressure. (b) Boundary conditions with uniform feed velocity.

Fig. 17 .

 17 Fig. 17. Density distribution across the width of the strip of MCC. (Predictions at neutral angle position). (a) Predictions using uniform feed pressure condition. (b) Predictions using uniform feed velocity condition.