Keywords: orientation client, réclamation, orientation réclamation, comportements défensifs Why Customer Orientation Does not Necessarily Stimulate Complaint Management customer orientation, complaint, complaint orientation, defensive behaviors

Pourquoi l'orientation client ne stimule pas nécessairement une gestion des réclamations efficace : le rôle négligé de l'orientation envers les réclamations Résumé en français de 100 mots au maximum : Nous avons recours à la théorie des perspectives pour expliquer comment l'orientation client peut éviter les comportements organisationnels défensifs vis-à-vis des réclamations. Lorsque les managers et les employés ont une conception négative des réclamations, ils développent des comportements défensifs, et cela se traduit par un management inefficace des réclamations. Les résultats d'une étude menée auprès de 137 responsables marketing montrent que des efforts en termes de gestion des réclamations sont efficaces si l'orientation client se traduit par une orientation envers les réclamations. Le rôle des managers est important pour favoriser une orientation réclamation dans la culture organisationnelle.

Introduction

Firms make extra efforts to manage complaint effectively, because managing complaint effectively is a key driver of customer satisfaction and loyalty [START_REF] Homburg | How organizational complaint handling drives customer loyalty: An analysis of the mechanistic and the organic approach[END_REF], and of a firm's financial returns [START_REF] Luo | Satisfaction, Complaint and the Stock Value Gap[END_REF]. Customer orientation should facilitate the implementation of an effective complaint management. Indeed, scholars and managers consider effective complaint management to be a critical evidence of a firm's customer orientation. [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF] state unambiguously that "complaint handling embodies the acid test of a firm's customer orientation" (p. 95). However, a customer orientation of corporate culture is not always sufficient to prevent against the defensive behaviors towards complaints that may arise in a firm [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. Indeed, even employees in customer-oriented firms may perceive complaints as threats to their selfesteem, reputation, autonomy, resources, and job security. The consequence is that those employees prevent against acquiring information about customers' complaints, disseminating this information in a firm, and using this information to increase customers' satisfaction.

Therefore, given the large investments needed to manage complaints effectively [START_REF] Fornell | Customer satisfaction and stock prices: high returns, low risk[END_REF]) and to successfully create a customer orientation [START_REF] Gebhardt | Creating a Market Orientation: A Longitudinal, Multifirm, Grounded Analysis of Cultural Transformation[END_REF], the important question arises as how a customer orientation of corporate culture can transmit into an effective complaint management.

Our objective in this communication is to address the following question: how customer orientation can prevent defensive organizational behaviors towards complaints? Research on the link between customer orientation and complaint management efficiency remains scarce, so that there is no explanation on why customer orientation has a small influence on the efficiency of complaint management. In this communication, we argue that prospect theory offers a relevant theoretical framework to address that question [START_REF] Kahneman | Prospect Theory: An Analysis of Decision Under Risk[END_REF]. A firm with strong customer orientation makes customers' satisfaction one of its preeminent objectives. It turns that managers and employees may consider customers' complaints negatively (as "losses") rather than positively (as "gains"). When managers and employees view complaints negatively, they are likely to exhibit defensive behaviors towards complaints, which results in an ineffective complaint management. We suggest that customer orientation stimulates complaint management when customer orientation results in a firm's will and ability to acquire information about customers' complaints, to disseminate this information in a firm, and to use this information, i.e., when a firm is oriented towards complaints.

Our communication makes three principal contributions. First, we show that investments into complaint management do not yield returns if customer orientation does not result in a firm's orientation toward complaints. Second, we contribute to the customer orientation literature.

Although complaint orientation is part of the measure of customer orientation [START_REF] Kohli | MARKOR: A Measure of Market Orientation[END_REF], we examine the possibility that a firm oriented to its customers may not be necessarily oriented to their complaints. Doing so, we offer an explanation for a small influence of customer orientation on complaint management efficiency. We provide a comprehensive framework to understand why and how a customer orientation of corporate culture transmits into an effective complaint management. Three, we show that senior management as a critical to play in implementing a complaint orientation of corporate culture.

Conceptual framework

In today's competitive markets, customer satisfaction is a universal goal: higher customer satisfaction leads to higher profits (e.g. [START_REF] Fornell | Customer satisfaction and stock prices: high returns, low risk[END_REF]. Complaint management offers several potential benefits to a firm whose objective is high customer satisfaction [START_REF] Luo | Satisfaction, Complaint and the Stock Value Gap[END_REF]. By effectively soliciting, handling, and analyzing customers' complaints, a firm can ensure high levels of customer satisfaction and loyalty [START_REF] Fornell | Increasing the organizational influence of corporate consumer affairs departments[END_REF][START_REF] Smith | A Model of Customer Satisfaction with Service Encounters Involving Failure and Recovery[END_REF][START_REF] Orsingher | A meta-analysis of satisfaction with complaint handling in services[END_REF]. Indeed, a firm that handles complaints efficiently can identify accurately the reasons for customers' dissatisfaction and can therefore prevent future dissatisfaction [START_REF] Fornell | The vicious circle of consumer complaints[END_REF].

However, there is ample evidence that most firms little invest in complaint management [START_REF] Tax | Customer evaluations of service complaint experiences implications for relationship marketing[END_REF][START_REF] Estelami | Competitive and procedural determinants of delight and disappointment in consumer complaint outcomes[END_REF][START_REF] Andreassen | From disgust to delight: do customers hold a grudge[END_REF][START_REF] Orsingher | A meta-analysis of satisfaction with complaint handling in services[END_REF]. Because of a lack of appropriate complaint channels (Andreassen 1988;[START_REF] Richins | A multivariate analysis of responses to dissatisfaction[END_REF][START_REF] Estelami | Competitive and procedural determinants of delight and disappointment in consumer complaint outcomes[END_REF] most customers refrain from complaining after a dissatisfying experience [START_REF] Oliver | Equity: How consumers interpret fairness[END_REF]). In the long-term, a "vicious circle" may arise. As a firm receives more complaints, managers in charge of complaint management become more isolated from a firm's decision making [START_REF] Fornell | The vicious circle of consumer complaints[END_REF]. It turns that the firm is less able to manage effectively new complaints, and complainers also tend to be more numerous.

Firms little invest in complaint management systems because most managers and employees consider complaints as "bad news" (Fornell and Westbrook 1984, p. 69). Because complaints are usually seen as bad news, managers and employees perceive them as a source of threat for their self-esteem. Therefore, to protect themselves, they exhibit defensive behaviors towards complaints, which refer to "organizational behaviors which avoid contact with dissatisfied customers, dissemination of complaint-related information within the organization, and responsiveness to complaints" (Homburg and Fürst 2007, p. 524).

Specific organizational behaviors and organizational cultures may help a firm to prevent defensive organizational behaviors towards complaints [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. A firm can prevent defensive behaviors with a supportive management of human resources that includes appropriate leaders' behaviors (e.g., setting a good example in terms of complaint management, or presenting the consequences of an effective complaint management), and training activities to efficiently deal with complaints. Because organizational behaviors are framed by a firm's organizational culture [START_REF] Homburg | A Multiple-Layer Model of Market-Oriented Organizational Culture: Measurement Issues and Performance Outcomes[END_REF], research has also considered which organizational cultures most help a firm in preventing defensive behaviors towards complaints. The customer orientation of corporate culture has received a particular interest.

Customers constitute the core of any definition of market orientation, whether cultural or behavioral. Thus, one key component of a market orientation culture is a customer orientation, defined as "the sufficient understanding of one's target buyers to be able to create superior value for them continuously" (Narver and Slater 1990, pp. 21-22). A customeroriented culture refers to a set of shared values and beliefs that puts the customer's interest first, and it often implies a "natural" predisposition to look for customer information [START_REF] Narver | The Effect of a Market Orientation on Business Profitability[END_REF][START_REF] Slater | Market Orientation and the Learning Organization[END_REF]. A firm with high customer orientation is likely to exhibit less defensive behaviors towards complaints [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. Indeed, a customer-oriented organization is the one that aims at delivering superior value to customers continuously, and is therefore the one who pays close attention to customers' complaints as a way to increase their satisfaction [START_REF] Jaworski | Market Orientation: Antecedents and Consequences[END_REF].

Hypothesis

A firm that is oriented towards the customers has not necessarily the will and ability to identify, analyze, and respond to customers' complaints [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. Prospect theory [START_REF] Kahneman | Prospect Theory: An Analysis of Decision Under Risk[END_REF] provides a relevant theoretical framework to understand why customer orientation may have a low influence on preventing defensive behaviors toward complaints. As we mentioned above, in today's competitive markets, the quest for satisfaction is omnipresent. According to prospect theory, this satisfaction quest creates (1) a positive prospect for satisfaction because satisfaction is seen as a gain (a positive "reference point"), and (2) oppositely, a negative prospect for dissatisfaction because dissatisfaction is seen as a loss. Therefore, because managers consider complaints as the visible expression of customers' dissatisfaction, complaints may be perceived as losses rather than gains. Consequently, managers are likely to develop psychological defense mechanisms towards complaints [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. These psychological defense mechanisms are "habitual and unconscious strategies used to deny, distort, or counteract sources of anxiety and to help maintain an idealized self-image" (Oldham and Kleiner 1990, p. 1), which extend to the organizational level [START_REF] Argyris | Strategy, change, and defensive routines[END_REF]Brown and Starckey 2000;[START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF].

Therefore, high levels of customer orientation can lead to a dual effect. On the one hand, a firm that is oriented towards customers may consider complaints as means to satisfy dissatisfied customers. In this first case, the firm sees complaints as "gains", and higher customer orientation should encourage complaint management. This is a gaining path. On the other hand, a firm that is oriented towards customers may consider complaints as the expression of customers' dissatisfaction. In this second case, complaints are perceived as "losses", and higher customer orientation should not encourage complaints handling. This is a loosing path. Therefore, customer orientation should favor complaints management when customer orientation results in a firm's will and ability to identify, analyze, and respond to customers' complaints to be able to create superior value for them continuously, i.e., when a firm is oriented towards complaints.

Our approach of a firm's complaint orientation clearly parallels Narver and Slater ' (1990) views on market orientation, and also corresponds to a cultural approach. We thus consider that any desired positive change on the behavioral level regarding complaints must be preceded by positive changes on the cultural level. Consistently with the behavioral approach of market orientation, we differentiate three distinct information processes: (1) information acquisition about customer complaints, (2) dissemination of information about customer complaints, and (3) use of information about customer complaints [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. We thus posit that: H1a: The effect of a firm's customer orientation on information acquisition about customer complaints is mediated by a firm's complaint orientation.

H1b:

The effect of a firm's customer orientation on dissemination of information about customer complaints is mediated by a firm's complaint orientation.

H1c:

The effect of a firm's customer orientation on use of information about customer complaints is mediated by a firm's complaint orientation.

Cultures are unlikely to appear spontaneously within an organization. This is because a new culture entails essential changes in the values and norms of the organization, which means changes in the organization's most fundamental ways [START_REF] Gebhardt | Creating a Market Orientation: A Longitudinal, Multifirm, Grounded Analysis of Cultural Transformation[END_REF]. Cultures rather result from a process of cultural transformation [START_REF] Kennedy | Implementing a Customer Orientation: Extension of Theory and Application[END_REF][START_REF] Gebhardt | Creating a Market Orientation: A Longitudinal, Multifirm, Grounded Analysis of Cultural Transformation[END_REF]. Senior management plays a critical role in this process, as research on creating a market orientation highlights. Narver, Slater, and Tietje (1998) posit that without appropriate leadership, implementing a customer orientation culture may be impossible. [START_REF] Frederick | The Rediscovery of The Marketing Concept[END_REF] states that "customer oriented values and beliefs are uniquely the responsibility of top management" (p. 37). Indeed, values are initially shaped through senior management actions that (1) exemplify desired values and (2) promote within the organization behaviors that are consistent with such desired values [START_REF] Gebhardt | Creating a Market Orientation: A Longitudinal, Multifirm, Grounded Analysis of Cultural Transformation[END_REF]. Lam, Kraus, and Ahearne (2010) confirm that senior managers' individual customer orientation enhances the individual customer orientation of middle managers. [START_REF] Kirca | Market Orientation: A Meta-Analytic Review and Assessment of Its Antecedents and Impact on Performance[END_REF] demonstrate in their metaanalysis that senior management emphasis significantly and positively influences the level of market orientation within the organization. Therefore, we posit that senior management commitment towards complaints drives a firm's complaint orientation. Hence, we posit: H2a: The effect of senior management commitment toward complaints on information acquisition about customer complaints is mediated by a firm's complaint orientation.

H2b:

The effect of senior management commitment toward complaints on dissemination of information about customer complaints is mediated by a firm's complaint orientation.

H2c:

The effect of senior management commitment toward complaints on use of information about customer complaints is mediated by a firm's complaint orientation.

Research methodology

Data collection

To ensure a high response rate, we collected the date in partnership with the AMARC. The AMARC is a French professional association whose mission is to help complaint managers by creating a professional network to share best practices about complaint management. The AMARC includes more than 200 firms; 90% are part of the CAC40 companies. Also, it was important for the key informants to have a broad overview of the firm strategies and of the firm complaint management system. Therefore, complaint managers of the firms were deemed suitable to act as key informants for this study. Respondents at this level are familiar with a broad organization-wide perspective, and deemed to be qualified to provide informed responses for the questions asked in the survey. Because the AMARC essentially includes complaint managers, the partnership with the AMARC allowed us to get the responses from the right manager within each firm. To motivate complaint managers to answer to the survey and to reduce inherent social desirability, we undertook to provide an individual feedback to each manager. This feedback was a benchmark of the practices in the manager's industry. To get a valid benchmark, respondents were thus truly motivated to not under-evaluate or overevaluate their responses. These executives were contacted via e-mail and sent a link to a webbased survey using Qualtrics. Respondents were assured of complete confidentiality and anonymity. After removing incomplete responses from the sample, the final sample included 137 firms, representing a response rate of 68,5%.

Validity and Reliability of Measures

We employ two types of constructs in our study: reflective constructs and formative constructs. Information acquisition about complaints, information dissemination and information utilization are formative constructs [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. We thus followed [START_REF] Diamantopoulos | Index Construction with Formative Indicators: An Alternative to Scale Development[END_REF] guidelines to index construction. We checked for potential multicollinearity, which did not seem to be a problem: the maximum VIF comes to 1.94, which is below the common cut-off threshold of 2.

We use reflective scales to measure all other variables. Where ever possible, existing scales were used in this study. Orientation towards complaints is an original scale. The initial set of items was developed from [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]' measure of human resources management, because three items from that scale clearly reflected a firm's complaint orientation. A qualitative phase with managerial experts was used to complete this initial list of items. To identify experts of the domain, we used a pyramiding approach, which proved to be efficient in identifying rare subjects [START_REF] Hippel | Pyramiding: Efficient Search for Rare Subjects[END_REF]. We first asked nine managers responsible for complaint management from firms with different sizes in various sectors to provide a list of renowned experts. Twelve experts finally participated.

They came (1) from 12 different sectors, including BtoB and BtoC settings, (2) from national and international companies, and (3) from small and large firms.

To assess the face validity of the scale, we followed the usual approach suggested by [START_REF] Zaichkowsky | Measuring the Involvement Construct[END_REF], slightly reframed by [START_REF] Obermiller | Development of a Scale to Measure Consumer Skepticism Toward Advertising[END_REF]. We first chose four scholars according to their previous publications in the field. Then, we asked them to evaluate the degree to which each item is a ''very good'', ''good'', ''fair'' or ''poor'' representation of our definition of a firm's complaint orientation. No expert rated an item as poorly representative of the concept. The experts agreed that all items provided at least "good" representations of the concept (PRL = 1; [START_REF] Rust | Reliability Measures for Qualitative Data : Theory and Implications[END_REF]. Therefore, according to Lichtenstein, Netemeyer and Burton (1990)' cut-off criteria, we concluded that our scale exhibited good face validity. Also, it is a parsimonious scale that avoids the risk of artificially raising the scale's validity indices [START_REF] Feldman | Self-Generated Validity and Other Effects of Measurement on Belief, Attitude, Intention, and Behavior[END_REF].

On the basis of our exploratory factor analyses, we removed several items from the scales because their communalities are less than .500. Results of a CFA show that the fit indices of a model that includes all the scales have satisfactory values: RMSEA = .075, SRMR = .064, CFI = .947, TLI = .926.

The Cronbach α for the scales range from .739 to .804, demonstrating satisfactory reliability for each scale. Convergent and discriminant validity were established using the procedure outlined by [START_REF] Bagozzi | Assessing Construct Validity in Organizational Research[END_REF]. We checked for the convergent and discriminant validity of customer orientation and orientation towards complaints. We found that they are different although related constructs with a correlation significantly different from zero (Unrestricted model Khi² = 43.18 with d.f. = 19; Khi2 with Restricted correlation of 0 = 83.69 with d.f. = 20; Khi² = 40.51 with d.f. = 1; p-value < 0.01) and with a correlation significantly different from 1 (Khi² with Restricted correlation of 1 = 82.70; d.f. = 20; Khi² = 39.52; d.f. = 1; p-value < 0.01). Therefore, all the scales used show good psychometric properties.

Results

In this research, we control for the effects of three variables on information acquisition about customer complaints, dissemination of information about customer complaints, and use of information about customer complaints: (1) interfunctional coordination because it is likely to facilitate information dissemination within an organization [START_REF] Jaworski | Market Orientation: Antecedents and Consequences[END_REF], (2) firm's size [START_REF] Menguc | Conflict, Leadership, and Market Orientation[END_REF], and (3) switching costs. It is important to control for the effect of switching costs. Indeed, when switching costs are high, customers can hardly change from one brand to another. It turns that organizations are less prompted to implement a market-oriented culture. In contrast, when switching costs are low, customers can choose among many alternatives to satisfy their needs. Organizations are also more prompted to adopt a market-oriented strategy. We measured firm size as a log-transformation of a firm's turnover. We measure interfunctional coordination with [START_REF] Auh | The Influence of Top Management Team Functional Diversity on Strategic Orientations: The Moderating Role of Environmental Turbulence and Interfunctional Coordination[END_REF]' scale and switching costs with one item from Chung-Yu (2010)'s scale.

We tested our hypotheses with path analysis. We employ two types of constructs in our study: reflective constructs and formative constructs. Information acquisition about complaints, information dissemination and information utilization are formative constructs [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. Therefore, we averaged the responses to the items to create indexes for each of these three constructs [START_REF] Diamantopoulos | Index Construction with Formative Indicators: An Alternative to Scale Development[END_REF]. Because the other constructs are reflexive ones, we calculated factorial scores for these constructs. We report our results in Table 1. .094 n.s.

.436*** .006 n.s.

-.084 n.s.

. .382***

Table 1. Results

We do not support our hypothesis H 2b . Indeed, we do not find support for a significant effect of a firm's orientation towards complaints on a firm's dissemination of information about complaints (p > .05). Consistent result is found with Zhao, Lynch and Chen (2010)'s approach. We discuss these results in next section.

Discussion and managerial implications

Scholars and practitioners do agree that higher customer orientation should result in higher customer satisfaction. Therefore, in today's competitive markets, firms have largely invested to implement customer-oriented cultures. However, recent research surprisingly reports mixed results regarding the actual impact of customer orientation on complaint management, which is considered to be a prevalent tool to increase customer satisfaction (e.g., [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]. Building on prospect theory [START_REF] Kahneman | Prospect Theory: An Analysis of Decision Under Risk[END_REF], our study

shows that investments into complaint management do not yield returns if customer orientation does not result in a firm's orientation toward complaints. More specifically, we propose three insights.

First, although complaint orientation has been embedded in the definition and in the measure of customer orientation [START_REF] Kohli | MARKOR: A Measure of Market Orientation[END_REF], we establish that these two concepts are actually distinct concepts. Indeed, we find that a firm that is oriented to the customer is not necessarily oriented to his/her complaints. We also establish the discriminant validity of both constructs. Second, we reveal the role of complaint orientation as a mediator of the effect of customer orientation on complaint management, which comprises the acquisition of information about complaints, the dissemination of information about complaints and the utilization of information about complaints. Third, we demonstrate that top management commitment toward complaints is critical to boost complaints handling in a firm.

By revealing the neglected role of complaint orientation, our study makes the following contributions to the marketing literature. First, we provide an explanation for the surprising results regarding the impact of customer orientation on complaint management; we therefore give new avenues for research concerning the return of investments into complaint management. For instance, [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF] surprisingly found a significant but very small impact of customer orientation on complaint management. In contrast, they found a significant and four times higher effect of HR supportiveness on complaint management. To explain these unexpected findings, we built on prospect theory to introduce the construct of complaint orientation, which was in prior research part of the measure of customer orientation (e.g., [START_REF] Kohli | MARKOR: A Measure of Market Orientation[END_REF]. Therefore, customer orientation can induce a positive view of complaints (a tool to gain higher customers' satisfaction) or a negative view (a consequence of customer's dissatisfaction). Our study reconciles these two views. By establishing that complaint orientation is not a dimension of customer orientation but a distinct concept, we offer a new framework that (1) explains why some studies report a weak effect of customer orientation on complaint handling behaviors and (2) offers avenues for research on both customer orientation and complaint handling literatures. First, one of the most promising perspective concerns the fact that, contrary to previous findings (e.g., [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF], our study shows that customer orientation is a driver of complaint handling behaviors. Our study gives a comprehensive framework to understand why and how, even organizations with negative prospects complaints, increasing customer orientation may actually have a strong impact on complaints handling behaviors. Second, by differentiating the effects of customer orientation on three complaints handling behaviors we extend [START_REF] Homburg | See no evil, hear no evil, speak no evil: a study of defensive organizational behavior towards customer complaints[END_REF]'s study that integrates these three behaviors into one single construct, which they call "defensive organizational behaviors". This distinction is important because our results show that complaint orientation fully mediates the link between customer orientation acquisition and utilization of information about complaints, but not the link between customer orientation and dissemination of information about complaints. Also, our findings suggest that loss realization and loss aversion may produce employees' psychological defenses against utilization of information about complaints. An important consequence is that implementing a strong customer orientation does not have any positive effect on complaints handling behaviors until complaints are not seen as gains. Therefore, since scholars unanimously recognize that effectively handling complaint drives satisfaction, loyalty and profit (e.g., [START_REF] Singh | Consumer Complaint Intentions and Behavior: Definitional and Taxonomical Issues[END_REF][START_REF] Smith | A Model of Customer Satisfaction with Service Encounters Involving Failure and Recovery[END_REF][START_REF] Luo | Consumer Negative Voice and Firm-Idiosyncratic Stock Returns[END_REF][START_REF] Orsingher | A meta-analysis of satisfaction with complaint handling in services[END_REF] Valentini and de Angelis 2010), the mediating role of complaint orientation will be an operative insight for future research to optimize complaint handling management. Our results regarding dissemination of information about complaints are unexpected. Contrary to our hypothesis, we find that higher customer orientation does not reduce the lack of complaints' dissemination within a firm. However, this result is consistent with prior literature. For instance, [START_REF] Homburg | A Multiple-Layer Model of Market-Oriented Organizational Culture: Measurement Issues and Performance Outcomes[END_REF] also find no evidence for an effect of market orientation on market-oriented behaviors, which encompass dissemination of market intelligence within an organization. Third, our findings suggest that senior management commitment towards complaints has direct and indirect effects on complaint management.

On the one hand, we contribute to the literature on the implementation of a customer-oriented culture in an organization. Despite the widely acknowledged links between market orientation and firm success [START_REF] Kirca | Market Orientation: A Meta-Analytic Review and Assessment of Its Antecedents and Impact on Performance[END_REF], few studies are devoted to the question of how a firm can be more customer oriented. Prior research indicates that senior management plays a critical role in promoting the behaviors that characterize firms with customer-oriented cultures [START_REF] Gebhardt | Creating a Market Orientation: A Longitudinal, Multifirm, Grounded Analysis of Cultural Transformation[END_REF]. We extend this research, and we show that senior management commitment is particularly effective in encouraging complaint management. When top management recognizes and promotes complaints handling, it makes complaints handling a salient and important objective for the employees. Senior management commitment towards complaints encourages all the behaviors that define an effective complaint management system: acquisition of information about complaints (γ = .189, p < 0.05), dissemination of information (γ = .436, p < 0.05), and utilization or information (γ = .203, p < 0.05). Senior management commitment has the strongest impact on dissemination of information about complaints. Top management is usually "unwilling to devote attention to communication about complaints, and even put the blame on the person trying to transmit the information" (Homburg and Fürst 2007, p. 527).

One consequence is that employees are likely to develop strong defense mechanisms against transmitting information about complaints. Therefore, when top management is committed to complaint management it encourages dissemination of information about complaints in the organization, because in this case employees feel free to transmit that information, or even they feel rewarded for it. On the other hand, senior management commitment towards complaints also has an indirect effect on complaint management: top management commitment encourages a firm's orientation towards complaints that in turn encourages complaints handling behaviors. Finally, we offer some contribution to market orientation theory. There's some empirical evidence that market orientation is not always beneficial to a firm's performance [START_REF] Kirca | Market Orientation: A Meta-Analytic Review and Assessment of Its Antecedents and Impact on Performance[END_REF]. The reasons for some unexpected results remain unclear. Our results suggest that prospect theory offers one possible explanation. The traditional hypothesis is that market orientation should result in higher customer satisfaction and, in turn, in higher firm's performance. But, because market orientation indicates that customer satisfaction is a norm to achieve superior performance, customer dissatisfaction is likely to be seen as a "loss" rather than a "gain". When dissatisfaction is seen as a loss, a vicious circle can arise.


Vibration damping in beams using the Acoustic Black Hole effect

Vivien Denis

To cite this version:

Vivien Denis. Vibration damping in beams using the Acoustic Black Hole effect. Acoustics


Marco Corneli

Pierre Latouche

Fabrice Rossi

Modelling time evolving interactions in networks through a non stationary extension of stochastic block models

Keywords: Random graphs, time clustering, stochastic block models, integrated classification likelihood

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Romain Guigourès
email: romain.guigoures@orange.com

Marc Boullé
email: marc.boulle@orange.com

Fabrice Rossi
email: fabrice.rossi@univ-paris1.fr

Guigourès

Rossi

A Study of the Spatio-Temporal Correlations in Mobile Calls Networks

For the last few years, the amount of data has significantly increased in the companies. It is the reason why data analysis methods have to evolve to meet new demands. In this article, we introduce a practical analysis of a large database from a telecommunication operator. The problem is to segment a territory and characterize the retrieved areas owing to their inhabitant behavior in terms of mobile telephony. We have call detail records collected during five months in France. We propose a two stages analysis. The first one aims at grouping source antennas which originating calls are similarly distributed on target antennas and conversely for target antenna w.r.t. source antenna. A geographic projection of the data is used to display the results on a map of France. The second stage discretizes the time into periods between which we note changes in distributions of calls emerging from the clusters of source antennas. This enables an analysis of temporal changes of inhabitants behavior in every area of the country.

Introduction

The telecommunication operators interest in investigating the behavior of the customers using the call detail records has continuously grown in recent years. Several studies has been performed, some of them focusing on clustering antennas using the call flows [START_REF] Blondel | Regions and borders of mobile telephony in belgium and in the brussels metropolitan zone[END_REF], [START_REF] Guigourès | Segmentation of towns using call detail records[END_REF]. They highlight a strong correlation between the retrieved clusters and the covered territories characteristics like the spoken languages, the metropolitan areas in country-wide studies ; or the socio-economic profile of the neighborhoods (e.g. student, upperor working-class) in local focuses. Such analysis are interesting for the telecommunication operators, particularly in developing countries where the needs in access to telecom services are becoming increasingly important while their usage are still unknown.

To go even further in the study of call detail records, clustering antennas from which the traffic similarly occurs over a studied time period could be investigated. A temporal analysis of the calls gives the means for understanding where excesses and lacks of traffic are located over the territory in function of the time period. Such a study provides information as well on the structure of the day, the week, the month or the year, as on the areas where the temporal phenomena are observed.

One major issue in the analysis of call detail records is the large amount of data. The data set we investigate in this article is a daily record of inter-antennas calls made in France from May 13, 2007to October 13, 2007. The number of antennas throughout the French territory is 17,895 between which 1.12 billions calls have transited. The calls originating from (resp. terminating) outside the french mobile network are not included in the data. In Section 2, we introduce methods dealing with this kind of analysis and justify the choice of them for our problem. Then in Section 3, results on the spatial correlations are investigated while in Section 4 the temporal correlations are explored. Finally, the last section provides an assessment of the analysis results.

Antenna Clustering based on Mobile Calls

The first concern is the data representation. Indeed, a call is described by the source antenna, the destination antenna and the day it has been made. In a previous work [START_REF] Blondel | Regions and borders of mobile telephony in belgium and in the brussels metropolitan zone[END_REF], an undirected graph is used to model a network of antennas linked by edges weighted by the calls frequency. In this paper, we choose to keep the matrix representation to exploit the natural direction of the calls.

Related Works

In [START_REF] Blondel | Regions and borders of mobile telephony in belgium and in the brussels metropolitan zone[END_REF], the authors build a partition of the graph by modularity maximization. This criterion [Newman, 2006] measures the quality of the segmentation of the graph into cliques (or community in the graph-theory terminology) that are groups of strongly connected vertices. The clusters of antennas obtained using this technique can be efficiently retrieved by employing algorithms that exploit the sparsity of the network [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] and thus modularity maximization is suitable for problems with a large amount of data, like clustering antennas. Numerous other graph clustering approaches have been proposed, see for instance [Schaeffer, 2007, Fortunato, 2010] for surveys. However, most of those approaches are based on some modularity or clustering assumptions: in terms of our context, those assumptions mean that the calls between antennas mostly occur within the groups of antennas. If it is the case, using e.g. modularity maximization or other fast graph clustering technique is very effective. If this is not the case, some patterns might be missed and the actual structure of the graph not retrieved. In our problem, we have no a priori knowledge that would justify a restriction to modular patterns. In fact we have even reasons to believe that non symmetric and non modular patterns might be present in the data: for instance, some antennas might be associated to specific locations (universities, popular touristic destinations) that lead to a significant amount of calls outside the area, while other antennas might exhibit more localized call destinations. We thus must find an alternative approach that enables the discovery of any kinds of patterns.

The concept of blockmodeling originates in the pioneering works on quantitative graph structure analysis conducted by sociologists in the 1950s in the context of social network analysis [Nadel, 1957]. To track the underlying structure of the network, a matrix representation of a graph is usually exploited, generally its adjacency matrix. Rows and columns represent the source and destination antennas, and the values of the matrix indicate the number of calls made between the antennas. Early sociological approaches suggested to rearrange the rows and the columns in order to partition the matrix in homogeneous blocks, a technique called blockmodeling. Once the blocks are extracted, a partition of the antennas of both source and destination subsets can be deduced. This type of simultaneous grouping is named co-clustering. Notice that the only way to produce non symmetric patterns (between source and target antennas) is to allow for two different clusterings (one for the source antennas and one for the destination antennas), thus leading to a co-clustering. Using this technique, we are able to track more sophisticated patterns than approaches based on a single clustering whose quality is judged by a density based measure such as the modularity. In fact such approaches can be considered as looking for a diagonal blockmodel in which off diagonal terms should be zeros.

Numerous methods have been proposed to extract satisfactory clusters of vertices. Some of them [START_REF] Doreian | Generalized blockmodeling of two-mode network data[END_REF] are based on the optimization of criteria that favor partitions with homogeneous blocks, especially with pure zero-blocks as recommended in [START_REF] White | Social structure from multiple networks: I. blockmodels of roles and positions[END_REF]. More recent deterministic approaches have focused on optimizing criteria that quantify how well the co-clustering summarizes the input data [START_REF] Reichardt | Role models for complex networks[END_REF] (see e.g. [START_REF] Wasserman | Social Network Analysis: Methods and Applications[END_REF] for details on such criteria). Other approaches include stochastic blockmodeling. In those generative models, a latent cluster indicator variable is associated to each vertex. Conditionally to the latent variables, the probability of observing an edge between two actors follows some standard distribution (a Bernoulli distribution in the simplest case) whose parameters only depend on the pair of clusters designated by the latent variables. In early approaches, the number of clusters is chosen by the user [START_REF] Nowicki | Estimation and prediction for stochastic blockstructures[END_REF]. More recent techniques automatically determine the number of clusters using a Dirichlet Process [START_REF] Kemp | Learning systems of concepts with an infinite relational model[END_REF]. Finally, some recent approaches consider non-boolean latent variables: cluster assignments are not strong and a vertex has an affiliation degree to each cluster [START_REF] Airoldi | Mixed Membership Stochastic Blockmodels[END_REF].

In addition to the diversity of structures that can be inferred from the network, co-clustering approaches are also able to deal with continuous variables [START_REF] Nadif | Model-based co-clustering for continuous data[END_REF], [Boullé, 2012]. Blocks are extracted from the data that yields a discretization of the continuous variables. For a further analysis, we are able to track temporal patterns: the source antennas are still the rows in the data matrix while the columns now model the time.

In the case of an analysis of a call detail record, the technique we employ must have some properties:

• Scalability: with nearly 18,000 antennas and 1.12 billion calls, we cannot afford to use methods with a too high algorithmic complexity, that is often an issue with co-clustering/blockmodeling techniques. • Genericity: the processed data are either nominal or continuous. This point is really important in our study because we focus on nominal attributes (the antennas label) and continuous (the time). • User-parametrization free: data are complex and their underlying structure is a priori unknown, giving parametrization of the co-clustering scheme (e.g number of clusters, etc.) might be an issue for the user with such a data set. • Reliability : the chosen approach must not yield spurious patterns, be resilient to noise and avoid overfitting. • Fineness and interpretability : the approach must exploit all the relevant data information in order to extract fine patterns. In addition, exploratory analysis tools must allow users to work with the results effectively.

Given the large amount of data, the majority of the co-clustering approaches is not applicable to the problem of antennas clustering. Sampling the data set might be possible. However with 17,895 antennas and 1.12 billion calls, the average frequency of calls between two antennas is approximately 3.5 and sampling the data would lead to a significant loss of information. Among the co-clustering approaches, we decide to use the MODL approach [Boullé, 2011] 1 .

Applying the MODL Approach

Before detailing the chosen approach, the problem must be formalized. The data set under study consists of calls which are the statistical units. Each call is described by three variables: the source antenna of the call, an element of V S (see Table 1); the destination antenna of the calls, an element of V D ; and the time at which the call started a real number (the observed call times form the set V T ⊂ R). Notice that while physical antennas are handling both incoming and outgoing calls, those two roles are separated in our model: V S and V D are completely distinct sets. This allows to build a directed model of the phone calls and therefore limits information loss.

The Table 1 lists the data features and the modelization parameters we want to infer. The analysis we perform can be divided into two steps. In the first one, we focus on the correlations between source and destination antennas while in the second, we concentrate on studying the time dimension of the calls. That is why, we introduce two distinct models: one is spatial M S and the other one is temporal M T . In both case, the MODL approach infers the parameters of the model M S (resp. M T ) from the data D.

In a first step, the model is based only on the antenna variables (source and destination). The co-clustering approach is applied to the call detail record to extract clusters of source antennas (in rows of the data matrix) and destination antennas (in columns of the data matrix). The objective is to group source antennas for which the calls are similarly distributed over the destination antennas and conversely for target antenna w.r.t. source antenna.

In a second step, the model is based again on two variables: the source antenna and the starting time of the call. As the time variable is continuous, the clustering has been constrained to respect the time ordering. This corresponds to a time quantization. The aim of the co-clustering in this case is to simultaneously group antennas and discretize the studied time period into segments during which the network is stationary. A higher order co-clustering (e.g. a tri-clustering approach like in [START_REF] Guigourès | A triclustering approach for time evolving graphs[END_REF]) could be applied in order to keep the three original variables. However, as will be become clear in Section 3, the source/destination coupling is very strong in this data set, up to a point where it hides the temporal patterns. By removing the destination variable, one can hope finding temporal structures.

MODL optimizes a criterion to find the co-clustering structure. The detailed formulation of the criterion as well as the optimization algorithms and the asymptotic properties are detailed in [Boullé, 2011] for a co-clustering with nominal variables and in [Boullé, 2012] for a co-clustering with heterogeneous variables, i.e nominal and continuous. The criterion is formulated following a MAP (Maximum a Posteriori) approach and is made up of a prior probability on the parameters of the co-clustering model and of the likelihood:

• The prior : denoted P(M S ) (resp. P(M T )), it penalizes the model by specifying the a priori distribution of its parameters. It is hierarchically and uniformly built in order to be the most weakly informative [Jaynes, 2003]. ..t : number of calls made during the t th time segments m i jt : number of calls made from the antenna v i to the antenna v j at time v t m M i j. : number of calls made from the i th cluster of source antennas to the j th cluster of destination antennas m M i.t : number of calls made from the i th cluster of source antennas during the t th time segment • The likelihood : Once the model parameters are specified, the likelihood P(D|M S ) (resp. P(D|M T )) is defined as the probability to observe the data given the parameters.

The product of the prior and the likelihood results in the posterior probability of the model. Its negative logarithm is the optimized criterion.

Definition 1. The spatial model M S , summarized representation of the data D, is optimal if it minimizes the following criterion:

c(M S ) = -log [P(M S )] -log [P(D|M S )] = log n S + log n C + log B(n S , k S ) + log B(n C , k D ) + log m + k -1 k -1 + ∑ c i ∈V M S log m M i.. + n M i. -1 n M i. -1 + ∑ c j ∈V M D log m M . j. + n M . j -1 n c
. j -1

(1)

+ log m! -∑ c i ∈V M S c j ∈V M D log m M i j. ! + ∑ c j ∈V M D log m M . j. ! -∑ v j ∈V D log m . j. ! + ∑ c i ∈V M S log m M i.. ! -∑ v i ∈V S log m i.. ! Definition 2.
The temporal model M T , summarized representation of the data D, is optimal if it minimizes the following criterion:

c(M T ) = -log [P(M T )] -log [P(D|M T )] = log n S + log m + log B(n S , k S ) + log m + k -1 k -1 + ∑ c i ∈V M S log m M i.. + n M i. -1 n M i.
-1

(2)

+ log m! -∑ c i ∈V M S V M D M T log m M i.t ! + ∑ V M D ∈V M T log m M ..t ! + ∑ c i ∈V M S log m M i.. ! -∑ v i ∈V S log m i.. ! B(|V S |, K S )
is the number of possible partitions of V S into K S potentially empty subsets.

The two first lines of the equation 1 and the equation 2 are the prior terms while the two last lines are the likelihood terms. In an information-theoretic point of view, a negative logarithm of a probability amounts to a Shannon-Fano coding length [Shannon, 1948]. Thus, the negative log of the prior probabilitylog(P(M )) is the description length of the model. As for the negative log likelihoodlog(P(D|M )), it is the description length of the data when modeled by the co-clustering. Minimizing the sum of these two terms therefore has a natural interpretation in terms of a crude MDL (minimum description length) principle [Grünwald, 2007]. The criterion c(M ) provides an exact analytical formula for the posterior probability of a model M . That is why the design of sophisticated optimization algorithms is both necessary and meaningful. Such algorithms are described by [Boullé, 2011].

The criterion is minimized using a greedy bottom-up merge heuristic. It starts from the finest model, i.e with one antenna per cluster and/or one timestamp per time segment. The merges of source and destination clusters or the merges of source clusters and time segments are evaluated and performed so that the criterion decreases. A post-optimization step improve this heuristic by making permutation between the clusters of antennas. This algorithm, that has a time complexity of O(m √ m log m), is detailed in [Boullé, 2011].

Analysis of the Spatial Correlations

First, we focus on the analysis of the calls between source and destination antennas. We obtain 2,141 clusters of source antennas and 2,107 clusters of destination antennas. The average number of antennas per cluster is between 8 and 9, which is very fine. The challenge lies in exploiting the results. The number of clusters is too important for a countrywide analysis of the antennas grouping but is suitable for local studies. We thus propose analysis at different geographic scales.

A Countrywide Analysis

First, we propose a countrywide analysis of the results and a projection on a map of France. The finest results do not provide a summarized enough view of the coclustering structure for such an analysis. That is why, we process an agglomerative hierarchical clustering of the clusters to reduce their number. Clusters are merged so that the criterion is the least decreased in order to obtain the most probable co-clustering model for a given number of clusters. This post-treatment allows a simplification of the model while handling its quality loss. In order to quantify this loss, we introduce a quality measure we call informativity rate.

Definition 3. The null model M / 0 S is the parametrization of the model, such that there is one single cluster of source and destination vertices or one single cluster of source and one time segment. The null model is the best representation of a data set with no underlying structure. Given the best model M * S obtained by optimizing the criterion defined in Definitions 1 and 2, the informativity rate of a model M S is:

τ(M S ) = c(M S ) -c(M / 0 S ) c(M * S ) -c(M / 0 S )
By definition, τ(M S ) ≤ 1 ; note that τ(M S ) < 0 is possible when M S is an irrelevant modelization of the data D (e.g. M S = M / 0 S when D are random data). The informativity rate allows the construction of a curve of the informativity of the model in function of the number of clusters. This aims at helping the user in finding a good trade off between a simple and an informative co-clustering.

The Figure 1 shows that the first merges have a weak impact on the model informativity. Hence, the number of clusters of both source and destination antennas can be significantly reduce from more than 2,000 to 85 while keeping 75% of the model informativity. This simplified model is used for the countrywide study: it is simple enough to be interpreted and informative enough to make a reliable analysis. Results are displayed in Figure 2.

The correlation between the clusters of antennas and their locations is strong despite the antennas positions are not taken in account in the optimized criterion. We can thus deduce that the people living in the same areas use to call the same destination and vice-versa. The map of the Figure 2 shows that the french territory can be segmented into several geographic areas that do not necessarily correspond to the regional administrative boundaries.

A local analysis

In a second step, we focus on a local analysis. To that end, we exploit the finest model (M * S ) and only consider a subset of antennas corresponding to the area of a french city. The antennas in Toulouse are segmented into seven clusters, displayed in Figure 3. The first cluster groups the antennas of the inner city (pale yellow circles), another cluster (neon green circles) groups antennas located in the west bank of the Garonne river, that corresponds to a largely residential area. The cluster of antennas pictured as pale pink circles takes place over the University of Toulouse campus and a disadvantaged neighborhood. As for the cluster grouping antennas displayed using pale green circles, it covers an area with the same characteristics than the previous one. The orange circles are located in the residential periphery of the city with different socio-economic profiles: upper-class toward South and working-class toward North. Finally the red squares are antennas located in the industrial areas adjacent to the international airport.

In order to understand why antennas have been grouped together, we focus now on the distribution of calls originating from the clusters. To that end, we study the contribution to the mutual information of each couple of source/destination stations.

Definition 4. This measure quantifies the dependence of two variables, here the partitions of the source and destination antennas. Let us denote it MI(V M S ,V M D ), defined as follows [START_REF] Cover | Elements of information theory[END_REF]:

MI(V M S ,V S D ) = ∑ c S i ∈V M S ∑ c D j ∈V M D p(c S i , c D j ) log p(c S i , c D j ) p(c S i )p(c D j ) (3)
Mutual information is necessarily positive and its normalized version is commonly used as a quality measure in the coclustering problems [START_REF] Strehl | Cluster ensembles -a knowledge reuse framework for combining multiple partition[END_REF]. Here, we only focus on the involvement to mutual information of a couple of Fig. 3: Projection of the clusters of source antennas on a map of Toulouse, there is one shape and color per cluster. source/destination clusters stations. This value can be positive or negative according to whether the observed joint probability of journeys p(c S i , c D j ) is above or below the expected probability p(c S i )p(c D j ) in case of independence. Displaying such a measure allows to quantify whether there is a lack or an excess of calls between two groups of antennas in comparison to the expected traffic. This is illustrated in the Figure 4. We focus on the traffic of calls originating from the pale pink cluster of the Figure 3. Antennas that are pictures as red circles are the ones to which an excess of traffic from the studied cluster is observed (p(V M S ,V M D ) > p(V M S )p(V M D )) while the antennas corresponding to the white circles are the ones to which the traffic is null or expected (p(V M S ,V M D ) ≈ 0 or p(V M S ,V M D ) ≈ p(V M S )p(V M D )). For this cluster of source antennas, there are no antennas to which we observe a significant lack of traffic. If any, their location would have been identified by a blue circle on the map. Note that the colors in the map represent the contribution to the mutual information, not the frequency of calls which logarithm is proportional to the diameter of the circles. Hence, we observe that the excess of traffic mainly occurs within the cluster we focus on, and slightly to the rest of the city.

Spatio-Temporal Analysis

In this second study, we propose to process a co-clustering on the source antennas and the time. In this study, we could have envisaged to apply a tri-clustering approach like in [START_REF] Guigourès | A triclustering approach for time evolving graphs[END_REF]. However, the previous analysis showed us a strong correlation between the partitions of source and destination antennas. Hence, we consider that both source and destination antennas bring the same information and we consequently use only one of them. The data are call detail records with 17,895 source antennas and 1.12 billion calls made over five month. The timestamps are the dates. The antennas grouping is different from the one we obtained in the Section 3. Here the antennas are grouped if the emerging calls are similarly distributed over the days. We obtain 6,129 clusters of source antennas and 117 time segments. Contrary to the source/destination antennas analysis, there is no correlations between the clusters of antennas and their locations: they are scattered over the entire french territory. As a consequence, a projection of the clusters on a map of France would not be interpretable, even for a reduced number of clusters. In order to investigate the phenomena that lead to such a result, we also study the contribution to the mutual information between the clusters of antennas and the time segments. To visualize this measure, we have simplified the co-clustering model in the same way as for the previous study and we have plotted a calendar of the excesses and lacks of traffic in Figure 5. From May 13 to July 5 and from September 1 to October 13, the discretization is periodic, highlighting the working days and the week-ends. During the working days at both these periods, the calls originate in excess from the antennas of the middle cluster and in deficit from the bottom cluster. The contrast between the clusters is reducing as summer approaches. As for the weekends, the inverse phenomenon is observed, but not to the same degree. These phenomena can be explained by the agglomeration of the economic activity on concentrated geographical areas, usually urban. Note that there is always a lack of calls originating from the top cluster at this period.

During the summer vacations, the periodicity working days/weekends is not observed anymore. The calls originating from the middle cluster are now in deficit while the ones made from the top cluster are significantly in excess compared to the usual traffic of the areas covered by the antennas and the traffic in the time segments. It is during this period that the contrast is the sharpest. That is the reason why, we focus on the segment from August 5 to August 15 and draw a map where the antennas are displayed and colored in function of the excess or deficit of calls made during this segment (see Figure 6).

During the summer vacations, there is a significant excess of calls made from the Atlantic and Mediterranean coasts. This means that during this period, the number of calls originating from these areas are more important than usually. These areas can truly be regarded as seasonal regions since they are characterized by an unbalanced distribution of calls over the year. Actually the population of the areas covered by this cluster have also an unbalanced population over the year: during the summer vacations, the population can be multiplied by more than ten, this has a direct impact on the calls distribution.

Conversely, the cities are colored in blue on the map of the Figure 6. This can be explained by the diminution of the economic activity due to the vacations. It is thus assumed that the populations move from the urban centers to the holidays Fig. 6: Contribution to the mutual information between the clusters of source antennas and the time segment from August 5 to August 15. In red, the antennas from which there is an excess of calls, in blue a deficit and in white the expected number of calls. resorts located on the seashore. We note however that the color only indicates the contribution to the mutual information, not the frequency: during this period, the antenna from which the most calls have been made is colored in blue and located in Paris.

Conclusion

In this article, we have proposed an analysis of a five month call detail record between 17,895 mobile phone antennas spread throughout the French territory. That represents a total of 1.12 billion calls. After having listed similar studies and introduced methods suitable for such analysis, we have discussed on the choices that conducted us to use the MODL approach. Two different types of analysis have been conducted while using one single approach, being generic and scalable enough to thoroughly investigate the data.

In a first study, the antennas have been grouped together if the calls originating from (resp. terminating to) them are distributed on the same groups of antennas. An analysis of a projection of the clusters on a map reveals a strong correlation between the geographic position of the antennas and the clusters they belong to, at the national or local levels. In a second study, we have lead a study in which the time is taken into account. Despite, the antennas belonging to a same cluster are not located in a welldefined area anymore, they cover nevertheless areas with common features: urban, rural or touristic. As for the time segmentation, this highlights different behaviors in terms of mobile phone usage during the summer vacations and the working periods, during which we observe a periodicity between the working days and the weekends. For example in August, there is an excess of calls in the touristic areas while there is a deficit of calls in the urban areas, where most economic activity is concentrated.

In future works, it might be interesting to lead a study in which several time features are embedded in order to characterize the behavior in terms of mobile phone usage, in function of the date, the day of the week and the time of the day.

  D : Data M S : spatial co-clustering model M T : temporal co-clustering model V S : source antennas V M S : partition of V S into clusters of source antennas V M S : partition of V S into clusters of source antennas V D : destination antennas V M D : partition of V D into clusters of destination antennas V T : time V M T : discretization of V T into time segments k S : number of clusters V M S k S : number of clusters of V M S k D : number of clusters of V M D k T : number of time segments of V M T k = k S k D : number of biclusters k = k S k T : number of biclusters n S : number of source antennas n M i. : number of source antennas in the i th cluster from the partition V M S n M i. : number of source antennas in the i th cluster from the partition V M S n C : number of destination antennas n M . j : number of destination antennas in the j th cluster from the partition V M D m: total number of calls m i.. : number of calls originating from the source antenna v i m M i.. : number of calls originating from the i th cluster from the partition V M S m M i.. : number of calls originating from the i th cluster from the partition V M S m . j. : number of calls terminating in the destination antenna v j m M . j. : number of calls terminating in the j e cluster from the partition V M D m M

Fig. 1 :

 1 Fig. 1: Informativity rate function of the number of clusters

Fig. 2 :

 2 Fig. 2: Projection of the clusters of source antennas on a map of France. There is one color and shape per cluster.

Fig. 4 :

 4 Fig. 4: Contribution to the Mutual Information between the cluster of the university campus and the clusters of the plotted antennas.

Fig. 5 :

 5 Fig. 5: Calendar of the excesses and lacks of calls from three clusters and 42 time segments.

Table 1 :

 1 Notations.

Software available on www.khiops.com


Arnaud De Myttenaere

Boris Golden

Bénédicte Le Grand

Fabrice Rossi

Study of a bias in the offline evaluation of a recommendation algorithm

Recommendation systems have been integrated into the majority of large online systems to filter and rank information according to user profiles. It thus influences the way users interact with the system and, as a consequence, bias the evaluation of the performance of a recommendation algorithm computed using historical data (via offline evaluation). This paper describes this bias and discuss the relevance of a weighted offline evaluation to reduce this bias for different classes of recommendation algorithms.

Introduction

A recommender system provides a user with a set of possibly ranked items that are supposed to match the interests of the user at a given moment [START_REF] Park | A literature review and classification of recommender systems research[END_REF][START_REF] Kantor | Recommender systems handbook[END_REF][START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions[END_REF]. Such systems are ubiquitous in the daily experience of users of online systems. For instance, they are a crucial part of e-commerce where they help consumers select movies, books, music, etc. that match their tastes. They also provide an important source of revenues, e.g. via targeted ad placements where the ads displayed on a website are chosen according to the user profile as inferred by her browsing history for instance. Commercial aspects set aside, recommender systems can be seen as a way to select and sort information in a personalized way, and as a consequence to adapt a system to a user.

Obviously, recommendation algorithms must be evaluated before and during their active use in order to ensure their performance. Live monitoring is generally achieved using online performance metrics (e.g. click-through rate of displayed ads) and several recommendation strategies can be compared using AB testing and online evaluation, whereas offline evaluation is computed using historical data. However putting an algorithm in production, collect and analyze data is generally a long process (many days or weeks). Offline evaluation allows to quickly test several strategies without having to wait for real metrics to be collected nor impacting the performance of the online system. One of the main strategy of offline evaluation consists in simulating a recommendation by removing a confirmation action (click, purchase, etc.) from a user profile and testing whether the item associated to this action would have been recommended based on the rest of the profile [START_REF] Shani | Evaluating recommendation systems[END_REF]. Numerous variations of this general scheme are used ranging from removing several confirmations to taking into account item ratings.

While this general scheme is completely valid from a statistical point of view, it ignores various factors that have influenced historical data as the recommendation algorithms previously used. Even if limits of evaluation strategies for recommendation algorithms have been identified ([3,6,8]), this protocol is still intensively used in practice.

We study in this paper the general principle of instance weighting proposed in [START_REF] De Myttenaere | Reducing offline evaluation bias in recommendation systems[END_REF] and show its practical relevance on the simple case of constant recommendation and on two collaborative filtering algorithms. In addition to its good performances, this method is more realistic than solutions proposed in [START_REF] Herlocker | Evaluating collaborative filtering recommender systems[END_REF][START_REF] Mcnee | Being accurate is not enough: how accuracy metrics have hurt recommender systems[END_REF] for which a data collection phase based on random recommendations has to be performed. While this phase allows one to build a bias free evaluation data set, it has also adverse effects in terms of e.g. public image or business performance when used on a live system, as random recommendations are obviously less relevant than personnalized recommendations got by an algorithm.

The rest of the paper is organized as follows. Section 2 describes in details the setting and the problem. Section 3 introduces the weighting scheme proposed to reduce the evaluation bias. Section 4 demonstrates the practical relevance of our method for the particular case of constant algorithms and present experimental results based on real world data extracted from Viadeo (professional social network3 ). Section 5 describes the results of our approach on two collaborative filtering and discuss the reduction of the bias for elaborated algorithms.

Problem formulation

Notations

We denote U the set of users, I the set of items and D t the historical data available at time t. As user are associated to items, D t can be represented as a bipartite graph. Let n U = #U and n I = #I be the cardinal of the set of users and items at time t, and A represents the adjacency matrix of the bipartite graph given by D t . Then A = 0 n I B B T 0 n U where 0 n represents the zero matrix of size n × n, and B is a n I × n U binary matrix. B is called biadjacency matrix and for each (i, u) in I × U , b i,u = 1 if the item i is associated to user u and 0 else. A representation of the data is presented on figure 1 A recommendation algorithm is a function g from U × D t to some set built from I. We will denote g t (u) = g(u, D t ) the recommendation computed by the algorithm g at instant t for user u. We assume given a quality function l from the product of the result space of g and I to R + that measures to what extent an item i is correctly recommended by g at time t via l(g t (u), i). We denote I u the items associated to a user u, and U i the set of users which are associated to the item i.

The classical offline evaluation procedure

Offline evaluation is based on the possibility of "removing" any item i from a user profile, which can be computed using stochastic or exhaustive sampling. Although exhaustive sampling gives more robust results, the stochastic approach is often prefered (especially for large systems) as it is faster and often precise enough to compare several algorithms. The user profile got after removing item i from user u is denoted u -i and g t (u -i ) is the recommendation obtained at instant t when i has been removed from the profile of user u.

Finally, offline evaluation follows a general scheme in which a user is chosen according to some probability on users P (u), which might reflect the business importance of the users. Given a user, an item i is chosen among the items associated to its profile, according to some conditional probability on items P (i|u). When an item i is not associated to a user u (that is i ∈ I u ), P (i|u) = 0. A very common choice for P (u) is the uniform probability on U and it is also very common to use a uniform probability for P (i|u) (other strategy could favor items recently associated to a profile). As the system evolves over the time, P (u) and P (i) depends on t.

The two distributions P (u) and P (i|u) lead to a joint distribution P (u, i) = P (i|u)P (u) on U × I. In other words, the classical offline evaluation consists in selecting a random node in user's part of the bipartite graph, and then a random node among the ones associated to the selected user. Many other graph sampling methods could be used (random edge selection, . . . )

Origin of the bias in offline evaluation

As presented in [START_REF] Li | Unbiased offline evaluation of contextualbandit-based news article recommendation algorithms[END_REF][START_REF] De Myttenaere | Reducing offline evaluation bias in recommendation systems[END_REF] the classical offline evaluation procedure ignores various factors that have influenced historical data as the recommendation algorithms previously used, promotional offers on some specific products, etc. Assume for instance that several recommendation algorithms are evaluated at time t 0 based on historical data of the user database until t 0 . Then the best algorithm is selected according to a quality metric associated to the offline procedure and put in production. It starts recommending items to the users. Provided the algorithm is good enough, it generates some confirmation actions. In other words, the recommendation campaigns introduce many new vertices in the bipartite graph representing the data (see figure 1). Those actions can be attributed to a good user modeling but also to luck and to a natural attraction of some users to new things. This is especially true when the cost of confirming/accepting a recommendation is low. In the end, the state of the system at time t 1 > t 0 has been influenced by the recommendation algorithm in production.

Then if one wants to monitor the performance of this algorithm at time t 1 , the offline procedure tends to overestimate the quality of the algorithm because confirmation actions are now frequently triggered by the recommendations, leading to a very high predictability of the corresponding items.

Finally, one can decompose the evolution of a recommendation system in two cycles represented in figure 2. On one hand there is a virtuous circle (also called lean circle) in three steps: first an algorithm is put in production and the data collection process starts, then the collected data are analyzed to measure the performance of the algorithm, and finally data are also used to select the best algorithm among several new ones by offline evaluation. On the other hand we also observe a vicious circle as the algorithm in production influences the users behaviors, which introduces a bias in historical data used for the offline evaluation procedure. This bias in offline evaluation with online systems can also be caused by other events such as a promotional offer on some specific products between a first offline evaluation and a second one. The main effect of this bias is to favor algorithms that tend to recommend items that have been favored between t 0 and t 1 and thus to favor a kind of "winner take all" situation in which the algorithm considered as the best at t 0 will probably remain the best one afterwards, even if an unbiased procedure could demote it. Indeed the score of an algorithm in production, given by the classical offline evaluation, tends to increase over time. More generally, the classical offline evaluation tends to overestimate (resp. underestimate) the unbiased score of an algorithm similar (resp. orthogonal) to the one in production.

More formally, the classic offline evaluation procedure consists in calculating the quality of the recommendation algorithm g at instant t as L t (g) = E(l(g t (u -i ), i)) where the expectation is taken with respect to the joint distribution:

L t (g) = (u,i)∈U ×I P t (i|u)P t (u)l(g t (u -i ), i). (1)
Then if two algorithms are evaluated at two different moments, their qualities are not directly comparable. Although as in an online system P (i|u) evolves over time 4 once a recommendation algorithm is chosen based on a given state of the system, it starts influencing the state of the system when put in production, inducing an increasing distance between its evaluation environment (i.e. the initial state of the system) and the evolving state of the system. This influence is responsible for a bias on offline evaluation as it relies on historical data.

A naive solution to correct this bias would be to compare algorithms only with respect to the original database at t 0 , but this approach is not optimal as it would discard natural evolutions of user profiles.

Impact of recommendation campaigns on real data

We illustrate the evolution of the P t (i) probabilities in an online system with a functionality provided by the Viadeo platform: each user can claim to have some skills that are displayed on his/her profile (examples of skills include project management, marketing, etc.). In order to obtain more complete profiles, skills are recommended to the users via a recommendation algorithm, a practice that has obviously consequences on the probabilities P t (i), as illustrated on Figure 3.

The skill functionality has been implemented at time t = 0. After 300 days, some of the P t (i) are roughly static. Probabilities of other items still evolve over time under various influences, but the major sources of evolution are recommendation campaigns. Indeed, at times t = 330 and t = 430, recommendation campaigns have been conducted: users have received personalized recommendation of skills to add to their profiles. The figure shows strong modifications of the P t (i) quickly after each campaign. In particular, the probabilities of the items which have been recommended increase significantly; this is the case for the green, yellow and turquoise curves at t = 330. On the other hand, the probabilities of the items which have not been recommended decrease at the same time. The probabilities tend to become stable again until the same phenomenon can be observed right after the second recommendation campaign at t = 430: the curves corresponding to the items that have been recommended again keep increasing. The purple curve represents the probability selection of an item which has been recommended only during the second recommendation campaign. Section 4.2 demonstrates the effects of this evolution on the evaluation of recommendation algorithms.

350 400 450 0.5 1.0 1.5 2.0 2.5 t % d'actions dues aux recommandations q q q q q q q q q q q q q q q q q q q q q 300 350 400 450 3 Reducing the evaluation bias 3.1 A weighted offline evaluation method to reduce the bias A simple transformation of equation [START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions[END_REF] shows that for a constant algorithm g (i.e. if recommendations are the same for every users): L t (g) = i∈I P t (i)l(g t , i).

As a consequence, a way to guarantee a stationary evaluation framework for a constant algorithm is to have constant values for the marginal distribution of the items, P t (i).

A natural solution would be to record those probabilities at t 0 and use them as the probability to select an item in offline evaluation at t 1 > t 0 . However, as the selection of users and items leads to a joint distribution, this would require to revert the way offline evaluation is done: first select an item, then select a user having this item with a certain probability π t (u|i) leading to a different probability of users selection. Finally this process lead to a similar problem on users, and as in most of systems #U > #I, it is more efficient to keep the classical evaluation protocol (see section 3.3 for more details).

Moreover, we will see that the recalibration of every items is not necessary to reduce the main part of the bias. Indeed in practice most of the time a few items concentrate most of the recommendations (very popular items, discount on selected products, ...). Thus one can reduce the major part of the bias by optimizing the weight of the p items such that the deviation given by |P t0 (i) -P t1 (i)| have the highest values. In practice the choice of p is done according to practical (time) or business constraints.

Thus the weighting strategy that we described in [START_REF] De Myttenaere | Reducing offline evaluation bias in recommendation systems[END_REF] consists in keeping the classical choice for P t (u) and weighting P t (i|u) by departing from the classical values for P t (i|u) (such as using a uniform probability) in order to mimic static values for P t0 (i) by :

P t (i|u, ω) = ω i P t (i|u)
j∈It ω j P t (j|u)

.

These weighted conditional probabilities lead to weighted item probabilities defined by: P t (i|ω) = u∈U P t (i|u, ω)P t (u).

Then we suggest to minimize the distance between P t1 (i|ω) and P t0 (i) by optimizing the Kullback-Leibler divergence, defined by :

D(ω) = i∈It 0 P t0 (i) log P t0 (i) P t1 (i|ω)
where I t0 represents the set of items present at t 0 . The asymmetric nature of this distance is useful in our context to consider time t 0 as a reference. Moreover this asymmetry reduces the influence of rare items at time t 0 (as they were not very important in the calculation of L t0 (g)).

Gradient calculation

We optimize D(ω) with a gradient based algorithm and hence ∇D is needed. Let i and k be two distinct items i = k, then

∂P (i|u, ω) ∂ω k = - ω i P (i|u)P (k|u)
j∈I ω j P (j|u) 2 = -P (i|u, ω) P (k|u, ω) ω k .

We have also

∂P (i|u, ω) ∂ω i = P (i|u, ω) ω i (1 -P (i|u, ω)) ,
and therefore for all k:

∂P (i|u, ω) ∂ω k = P (k|u, ω) ω k (δ ik -P (i|u, ω)) .
We have implicitly assumed that the evaluation is based on independent draws, and therefore:

P (i, k|ω) = u P (i|u, ω)P (k|u, ω)P (u). Then ∂D(ω) ∂ω k = i P t0 (i) ω k P t1 (i|ω) (P t1 (i, k|ω) -δ ik P t1 (k|ω)) .
And in the particular case of uniform selection, i.e. if P (u) ∼ U(U ) and P (i|u) ∼ U(I u ), then:

P (u) = 1 #U P (i|u) = 1 #I u • 1 i∈Iu P (i|ω) = 1 #U • u∈Ui ω i j∈Iu ω j P (i, k|ω) = 1 #U • u∈Ui∩U k ω i ω k ( j∈Iu ω j ) 2

Complexity

The value of p coordinates of the gradient can be computed with a O(p × n U ×I ) complexity, where n U ×I is the number of couples (u, i) wih u ∈ U and i ∈ I u (n U ×I = i∈I #U i ).

Indeed let us assume we have computed the beadjacency sparse matrix B of the bipartite graph twice: once indexed by raws, and once indexed by columns. Such matrix can be got in O(n U ×I ) and give access to every element in O(1). Then, in the particular case of uniform sampling it is possible to compute P (i|ω) for all i ∈ I in O(n U ×I ).

Then if u∈Ui ωi ( j∈Iu ωj ) 2 P (u) has been computed and saved for all i (complexity in O(n u×I )), we have P (i, k|ω) in O(#U i ) for all k.

So, after having computed and saved the values of P t0 (i) and P t1 (i) for all i, the quantity ∂D(ω) ∂ω k is a sum of #I elements computed in O(#U i ) and every coordinate of the gradient can be computed in

O( i∈I U i ) = O(n U ×I )
4 Illustration on constant algorithms

Data and metrics

We consider real world data extracted from Viadeo, where skills are attached to user's profile. The objective of the recommendation systems consists in suggesting new skills to users. The dataset contains 18294 users and 180 items (skills), leading to 117376 couples (u, i).

Both probabilities P t (u) and P t (i|u) are uniform, and the quality function l is given by l(g t (u -i ), i) = 1 i∈gt(u-i) where g t (u -i ) is a set of 5 items. The quality of a recommendation algorithm, L t (g), is estimated via stochastic sampling in order to simulate what could be done on a larger data set than the one used for this illustration. We selected repeatedly 20 000 couples (user, item) (first we select a user u uniformly, then an item according to P t (i|u, ω)).

The recommendation setting is the one described in Section 2.4: users can attach skills to their profile. Skills are recommended to the users in order to help them to build more accurate and complete profiles. In this context, items are skills. The data set used for the analysis contains 34 448 users and 35 741 items. The average number of items per user is 5.33. The distribution of items per user follows roughly a power law, as shown on Figure 4. q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q 1 2 5 10 20 50

Impact of previous recommendations campaigns

As described in section 2.2, the offline evaluation of a recommendation algorithm can by computed using stochastic or exhaustive approach. Here we will describe the impact of previous recommendation campaigns on the offline evaluation score and compute the score of offline evaluation by stochastic sampling on the sample data extracted from Viadeo, what permits to mimic the results which could be computed on bigger datasets. We first demonstrate the effect of the bias on two constant recommendation algorithms. The first one g 1 is modeled after the actual recommendation algorithm used by Viadeo in the following sense: it recommends the five most recommended items from t = 320 to t = 480. The second algorithm g 2 takes the opposite approach by recommending the five most frequent items at time t = 300 among the items that were never recommended from t = 320 to t = 480. In a sense, g 1 agrees with Viadeo's recommendation algorithm, while g 2 disagrees.

For each couple of selected user and item (u, i), the score given by the offline evaluation procedure of an algorithm g is given by l(g(u, i)). For the experiments we have selected 30 000 couples (u, i), where u is a user chosen uniformly on U , and i a skill chosen uniformly on I u (the set of skills associated to u). We will consider the quality function given by l(g(u, i)) = 1 i∈g(u,i) , where g(u, i) represents the top fives items suggested by the algorithm g after selecting the couple (u, i). Figure 5 shows the evolution of L t (g 1 ) and L t (g 2 ) over time. As both algorithms are constant, it would be reasonable to expect minimal variations of their offline evaluation scores. However in practice the estimated quality of g 1 increases by more than 25 %, while the relative decrease of g 2 reaches 33 %. As l is a binary function, it can be considered as a Bernoulli random variable of parameter p, where p corresponds to the expected probability that l(g(u, i)) = 1. Then, after n B simulations we have n B observations l(g(u (k) , i (k) )) k=1,...,n B (where u (k) and i (k) corresponds to the user and item selected during the k th step of the offline evaluation procedure) and the maximum likelihood estimator of p is given by

p = n B k=1 l(g(u (k) , i (k) )) n B
Thus p follows a binomial law which can be approximated by a gaussian random variable for n B big enough, and a 95% confident interval for p is classicaly given by

IC 95% ( p) = p -1.96 p(1 -p) n B ; p + 1.96 p(1 -p) n B

Reducing the bias

We apply the strategy described in Section 3 to compute optimal weights at different instants and for several values of the p parameter. Results are summarized in Figure 6.

The figures show clearly the stabilizing effects of the weighting strategy on the scores of both algorithms. In the case of algorithm g 1 , the stabilisation is quite satisfactory with only p = 20 active weights. This is expected because g 1 agrees with Viadeo's recommendation algorithm and therefore recommends q q q q q q q 300 350 400 450 items for which probabilities P t (i) change a lot over time. Those probabilities are exactly the ones that are corrected by the weighting method. The case of algorithm g 2 is less favorable, as no stabilisation occurs with p ≤ 20. This can be explained by the relative stability over time of the probabilities of the items recommended by g 2 (indeed, those items are not recommended during the period under study). Then the perceived reduction in quality over time is a consequence of increased probabilities associated to other items. Because those items are never recommended by g 2 , they correspond to direct recommendation failures. In order to stabilize g 2 evaluation, we need to take into account weaker modifications of probabilities, which can only be done by increasing p, as represented on figure 5.

Thus, the weighted offfline evaluation procedure reduces the bias for the very simple class of constant algorithms. In the next part we discuss the relevance of this procedure to reduce the offline evaluation bias on collaborative filtering algorithms.

Experimentations on a collaborative filtering

Collaborative filtering algorithms

Collaborative filtering is a very popular class a recommendation algorithms which consists in computing recommendation to a user u using the information available on other users, especially the ones similar to u. For example, a classical collaborative filtering consists in recommending the most frequent items among the ones associated to users having items in common with the user u.

More formally, let B t u be the vector of items of user u at time t (B t u ∈ {0, 1} #I ). Then B u (t) is a sparse vector as most of users are associated to only a few items, and corresponds to the u th column of the biadjacency matrix representing D t . The objective of collaborative filtering algorithms is to estimate B t u for t > t using the information known on other users. In this section we will discuss the efficiency of our method to reduce the offline evaluation bias on two different collaborative filtering algorithms:

a) B t u = v∈U \{u} B t u , B t v B t u • B t v • B t v b) B t u (i) = max j∈Iu(t) #(U i ∩ U j ) #U j
The equation a) is known as collaborative filtering with cosine similarity, whereas the equation b) computes the proportion of users associated to item i among the one associated to items possessed by u. Then we will note naive CF (Collaborative Filtering) the algorithm b).

Finally, the recommendation strategy consists in recommending the k items having the highest values in B t u .

Results

We apply the method described in Section 3 to compute optimal weights at different instants and for several values of the parameter p. The collaborative filtering algorithms are the one presented in section 5.1. Results are summarized in figure 7. Once again the analysis is conducted on a 201 days period, from day 300 to day 500, where day 0 corresponds to the launch date of the skill feature and it is important to notice that two recommendation campaigns were conducted by Viadeo during this period at t = 330 and t = 430 respectively. As we can see on figure 7, the scores strongly decrease after the first recommendation campaign (t = 330). Thus those campaigns have strongly biased the collected data, leading to a significant bias in the offline evaluation.

The figure 7 shows the influence of the value of p: the higher is p the more weights are optimized and the more the bias is corrected. However, the efficiency of the recalibration depends on the algorithms. The results show that the weighting protocol permits to reduce the impact of recommendation campaigns on offline evaluation results as intended. However it does not lead to the stabilization of the score of collaborative filtering algorithms (while it lead to constant scores for constant algorithms). This can be explained by the nature of collaborative filtering: we can't expect the score to be constant for such an algorithm as it depends on the correlation between users, which have been modified by the recommendation campaigns. In others words the bias can be decompose in two parts: one depending on the probability selection of each item, and the second one depending on the structure of the data (the vertices in the bipartite graph representing the data). Indeed the structure of the graph D t has been modified because since recommendation campaigns have increased the density of the graph by adding new vertices from targeted users to recommended items.

Conclusion

Various factors influence historical data and bias the score obtained by classical offline evaluation strategy. Indeed, as recommendations influence users, a recommendation algorithm in production tends to be favored by offline evaluation.

We have presented a new application of the item weighting strategy inspired by techniques designed for tackling the covariate shift problem. Whereas our previous results presented the efficiency of this method for constant algorithms, we have shown that this method also reduces the bias of more elaborate algorithms. However experiments on collaborative filtering shows that the bias can be decomposed in two part since previous recommendation campaigns change the probabilty selection of each item, but also modify the structure of the data.

Experiments shows that our is efficient to reduce the first bias. Future works will invesgate the correction of the structural bias.

Fig. 1 .

 1 Fig. 1. Representation of the data as a bipartite graph and notations

Fig. 2 .

 2 Fig. 2. The evolution of the recommendation system

Fig. 3 .

 3 Fig. 3. Impact of recommendation campaigns on the item probabilities: the left figure displays the percentage of observations induced by the recommendations, while the right figure shows examples of the evolution of P (i) through time.

Fig. 4 .

 4 Fig. 4. Distribution of items per user

Fig. 5 .

 5 Fig. 5. Evolution of Lt(g 1 ) (left) and Lt(g 2 ) (right) though time.

Fig. 6 .

 6 Fig.6. Evolution of Lt(g 1 ) (left) and Lt(g 2 ) (right) though time.

Fig. 7 .

 7 Fig. 7. Results on the collaborative filtering with cosine similarity and naive CF, respectively defined by equation a) and b) in section 5.1, for several values of p (the number of weights optimized).

See http://corporate.viadeo.com/en/ for more information about Viadeo.

even if P (u) could also evolve over time we do not consider the effects of such evolution in the present article.