Nadine Mandran
email: nadine.mandran@imag.fr

Laura Dupuis
email: laure.dupuis@imag.fr

Vanda Luengo
email: vanda.luengo@lip6.fr

DOP8_Qpp: Model to pre-process educational data

Keywords: data quality, data properties, pre-processing, reusing data, dashboard, data validation

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INTRODUCTION

This paper proposes one process to evaluate if data that come from educational context are reusable by research purposes in TEL domain.

Nowadays, masses of data are produced when students use learning platforms, when teachers produce dashboards of scores to follow up their students and when knowledge of these students are assessed through several tools such as a Multiple Choice Quiz (MCQ). These ecological data built by and for teachers in the field are valuable. They are crucial [START_REF] Berti-Equille | Quality awareness for managing and mining data[END_REF] to well understand the teachers' practices [START_REF] Bishop | Archiving your data: planning and managing the process[END_REF] to understand teaching and learning phenomena and (3) to provide valuable indicators to the TEL designers. For example, Márquez-Vera, C. & al [START_REF] Márquez-Vera | Predicting student failure at school using genetic programming and different data mining approaches with high dimensional and imbalanced data[END_REF] use ecological data to predict student failures. In their case they use information about high school students enrolled on Program II of the Academic Unit Preparation at the Autonomous University of Zacatecas (UAPAZ) for the 2009/10 academic year. All the information used in this study has been gathered from three different sources during the period from August to December 2010. Their dataset has 77 attributes, variables, or features for each student. The comprehensive dataset describes socioeconomics factors, personal factors and final scores from several knowledge domains (mathematic, physics, writing and reading Spanish, etc.).

Whether the data should be used for research purposes, it is necessary to ensure a good level of data quality. To achieve this, it is important to have criteria to know if data are reusable. Reusing data is difficult because data that come from educational field are not structured by a predetermined guideline, metadata to describe data production are not always available. Although Haugh & al [START_REF] Haug | The costs of the poor data quality[END_REF] indicate that poor data quality generates loss of data and also bad analysis, when data are produced without validation. So, before to analyse data it is useful to pre-process them. However, when data come from field, it is more difficult to pre-process them because researcher needs to interact with data producer, to understand how and why data have been produced. To preprocess this kind of data is important to identify some criteria to know if data are acceptable to lead data analysis. Several criteria are proposed by Berti Equille & Di Ruocco [START_REF] Berti-Equille | Quality awareness for managing and mining data[END_REF][START_REF] Di Ruocco | La qualité des données : concepts de base et techniques d'amélioration[END_REF] to ensure and assess data quality, such as the relevance, the ease of interpretation, the temporality coherence. These criteria are generic and there are not processes or tasks to reach them in the TEL context.

Our proposition is DOP8_Qpp: a model of process to shape decision making about data quality. It includes process with major tasks, data properties and set of quality criteria.

We evaluate our proposition by two ways. First, we instantiate our contribution using data come from MCQs realized in the first year in medical education in Grenoble, France (1800 students). Secondly, we have asked opinion of specialists with an online questionnaire to evaluate quality criteria and process.

First, paper presents related works about pre-processing of data and data quality criteria. Then, we described our proposal. Last part presents evaluation of our proposal.

RELATED WORK 2.1 Pre-processing steps in data mining

Lot of works focus on data processing and they propose several data life cycles [START_REF] Bishop | Archiving your data: planning and managing the process[END_REF][14][5] [START_REF] Romero | A Survey on Pre-Processing Educational Data[END_REF]. Several of them contains pre-processing steps precisely described. This first step is data transformation into an appropriate form for data analysis process. This step requires an important manualwork; it consumes 60-90% of the time, efforts and resources employed in the whole knowledge discovery process [START_REF] Miksovskỳ | Data pre-processing support for data mining[END_REF]. Romero & al [START_REF] Romero | A Survey on Pre-Processing Educational Data[END_REF] split the pre-process in 6 steps after data gathering: Data aggregation, data cleaning, user and session identification, attribute selection, data filtering and data transformation. Mandran & al [7] combined the data life cycle and operator life cycle to lead data analysis. In this scheme, pre-processing is composed with two steps: "Validate" to control data in relation to the reality and "Enrich" to enhance the meaning of data with the creation from new variables. If Romero et al. [START_REF] Romero | A Survey on Pre-Processing Educational Data[END_REF] define more precisely data pre-processing, arrangement of these steps could be a problem. For instance, data aggregation before data cleaning, without non-validated data, could introduce mistakes during data aggregation step. Moreover, there is no indicator to control validity of data and coherence of data files during data pre-processing. Regarding Mandran et al. [START_REF] Mandran | DOP8: merging both data and analysis operators life cycles for technology enhanced learning[END_REF], DOP8 was setup to follow data produced in a research context. That is, once research issue is defined, an experimental protocol is designed to lead data production. Thus, researcher controls data production. In DOP8, such as Romero et al. [START_REF] Romero | A Survey on Pre-Processing Educational Data[END_REF], there is no means to take into account data gathered outside research context and to qualify them in order to reuse these data.

DOP8 proposed by

Quality approach and data quality

Quality approach is process and tools to follow tasks which are needful to manage a project. During process, «actual results of an action are compared with a target or a set point » [START_REF] Sokovic | Quality improvement methodologies-PDCA cycle, RADAR matrix, DMAIC and DFSS[END_REF]. Brasseur [START_REF] Brasseur | Data Management: Qualité Des Données et Compétivité[END_REF] gives some features about data quality: "Data quality can address the needs of its users", "Data quality is dependent on their use", "The understanding of user needs is a prerequisite for defining and obtaining data quality requirement", "A big difficulty is that bad data quality is not easily detected. There are often some incidents or abnormalities during operational work, which reveal, here and there, inconsistencies on data." Therefore it is important to acquire methods and tools to control data throughout data life cycle.

In information systems, data quality proposes four approaches dedicated to improving data quality before analysis step: (1) preventive, (2) adaptive, (3) corrective and (4) diagnosed approaches [START_REF] Berti-Equille | Quality awareness for managing and mining data[END_REF]. Preventive approach allows upstream control before production. It is based on quality of the model and on quality development of the software. The adaptive approach allows data verification in real-time. The corrective and diagnosed approaches are conducted after data production. The corrective approach mainly includes: comparison with field reality, missing data imputation, and elimination of duplication. The diagnosed approach mainly includes: exploratory data mining, descriptive statistics, and metadata management.

Beside, [START_REF] Di Ruocco | La qualité des données : concepts de base et techniques d'amélioration[END_REF] Accessibility: ease of locating and accessing data • Ease of interpretation: ease of understanding data, their analysis and their use. Data must be understood without ambiguity.

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Uniqueness: a single object, a single record in the system, represents a real-world entity.

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Coherence: the absence of conflicting information.

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Conformity to a standard. Polańska and Zyznarski [START_REF] Polańska | Elaboration of a method for comparison of Business Intelligence Systems which support data mining process[END_REF] elaborate a framework to guide the quality of the data process. They propose measures to follow the data process. They describe them with "name, purpose, measurement method, type of the method, scale, type of scale, unit of measurements". These works about data quality are useful to lead a data reengineering. In the case of the data produced by teachers on the field, only the corrective and diagnosed approaches are possible, since data are produced before. Definitions proposed by Di Ruocco & al [START_REF] Mandran | DOP8: merging both data and analysis operators life cycles for technology enhanced learning[END_REF] are generic and all are not adapted in our case. Polańska and Zyznarski [START_REF] Wang | Design-based research and technology-enhanced learning environments[END_REF] provide a framework to describe and to adapt the data quality criteria. This framework is interesting but we have to adapt it. Indeed it only takes into account quantitative measures to produce quality criteria. But to understand why and how the data are produced and to lead a data re-engineering, it is crucial to interact with data producer; it is a qualitative measure.

To conclude on this point, data produced in real conditions by and for teachers are valuable and especially to lead a research in the context of the design based research methodology [START_REF] Wang | Design-based research and technology-enhanced learning environments[END_REF]. However, data life cycles do not clearly mention difference to take into account between data produced in a research context and out of it. Moreover, in data life cycles there are no data quality criteria to guide steps of these cycles and to validate data. Thus, our proposal has two targets: 1-Defining a process and tasks to help researcher to take up data and to co-operate with data producer and 2-Assigning a properties to data to qualify them and to ensure their reusability. Next section describes our proposal.

DOP8_Qpp

The name DOP8_Qpp means: data Quality indicators to lead Pre-Processing steps into DOP8 cycle. It help researcher whose needs ecological data produced on the field. In this article, we don't take into account data produced by researcher to address a research problem. Indeed, in this context, data production is controlled with a protocol. To lead our model with ecological data, it is necessary to distinguish two actors: 1-Data producer (DP) that produces data into an educational context and the Researcher-Analyst (RA), which needs to use data in another context (e.g. students marks produced by a teacher and data analyses lead by a researcher to find a model to describe students' learning evolution). Also two objects are useful: 1-Structure of Data (SD) which is the physical organization into an information system (e.g. data files, directory, folders, database, etc.) and 2-Data (D) which are a set of values that can be stored and used by a software.

DOP8_Qpp is described at figure 10. Before to explain it, we explain definition and concepts used, the process and dashboard concept.

Definition and concepts

Three elements are presented below: 1-Collect and preprocessing steps in DOP8 to reuse data, 2-UCUA Data properties, and 3-Quality criteria in relation with data properties.

Collect & Pre-processing in DOP8

DOP8 cycle begins by "Prepare", step where a protocol is designed to produce data. In this case, RA makes research questions and produces data to address these issues. After, it is step "Collect", where data are produced in an experiment field. Then, there are two pre-processing steps. First step "Validate" which goal is a validation of raw data to ensure coherence and relevance of SD and D. During this step, several major tasks are lead to control SD and D in link to the reality. Second step "Enrich" which goal is enrichment of data. During this step, several majors tasks are lead to add new data. The enrichment allows ease of understanding data, their analysis and their use.

Four data properties: UCUA

We propose to use four data properties.

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Utility: SD and D satisfy of user's needs now and in the future.

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Compliance: conformity between metadata provided by producer with SD and D.

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Usability: ability of SD to be used by a data analysis software and ability of D to enhance meaning of results.

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Acceptability: is synthesis of the previous three properties. If they are reached, SD and D are acceptable to lead data analysis.

Process (Figure 2) to control these three data properties is organized as follows: first "Utility" of SD and D are validated, then "Compliance " and "Usability" of SD, then "Compliance and Usability" of D. It is organized in this manner because it is necessary to collect metadata about SD and D and to meet RA's needs first, then to control data structure to provide exploitable structures by automated tools and finally to validate data and transform them to enrich data semantics. At the end, data can be qualified with "acceptability" or not.

Data properties and quality criteria

For each data properties, we use one or several quality criteria coming from data quality works [START_REF] Di Ruocco | La qualité des données : concepts de base et techniques d'amélioration[END_REF]. Usability of structure of data is controlled with "structural consistency".

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Usability of data is controlled with: 'Completeness of occurrences', 'Ease of interpretation'.

Operationalization

Quality criteria are a result of one or several major tasks, which produce values of measures. These values give making decision rules to qualify SD and D based in data properties.

Measures are qualitative and quantitative. Qualitative obtained by qualitative methods (e.g. interview with producer to gather metadata). Quantitative, obtained by an automatic process of data (e.g. ratio of missing values). Sequence of major tasks creates a process to qualify object (SD and D). This process helps to qualify data with the four UCUA properties.

Global process has two starting points: designing a research question and existence of a data produced in an educational context. To reach each property, major tasks allow production of quality criteria, then RA can takes a decision and qualify objects (SD and D). One or several dashboards must be designed to follow the process. Now, we detail process for each properties and each object.

To exemplify, our proposal we use a research question which is "There are a model which explain students' evolutions during two semesters in mathematics and biology?" and we have data produced by university during several years.

Utility of SD et D: main goal of tasks to reach first property "Utility" is to identify if all useful information has been provided by producer. That corresponds to data quality "relevance" of data. RA needs information about structure of data and data. RA must achieve two iterative majors tasks: 1interviewing DP to understand data and to obtain information about data 2-controlling if information is sufficient. If initial information is deficient, additional information must be requested to DP. A new interview can be made.

A dashboard is set up to record this information, such as a checklist where RA checks every needful element. In this part of process, quality criteria are set up with qualitative information. These information provided by data producer set up these data quality criteria and ensure validity of SD and D.

If all quality criteria are satisfied, data are qualified by "Utility"; else, the dimension of "Utility" is not reached. Data can not be used to address the research question.

About our example, RA needs data to identify: students, subjects taught but only mathematics and biology, date and scores of students. RA needs this minimal set of information to try to address research question. University gives data to RA with description of SD and description of D. Both descriptions exist and minimal set of information exists then the dimension "Utility" of SD and D is reached.

Compliance and Usability of SD:

To reach compliance and usability of SD, RA needs to realize 6 majors tasks: 1controlling conformity between SD and description of SD provided by producer, 2-controlling if all data structures exist, 3-controlling if relation between data structure are usable, 4-controlling if all variables exist in data structures, 5-transforming data structure into a new one if necessary (e.g. aggregation) and 6-controlling that data structure can be used with one of data analysis software. If information is deficient to assume the 6 tasks, additional information must be requested to data producer. Quality criteria used are 'Conformity', 'Completeness of entity', 'Completeness of attribute', 'Completeness of relation' and 'Structural consistency'. The five quality criteria are set up with qualitative information. At the end, SD is qualified with "compliance" and "Usability" dimensions. Process can be continued to qualified D with "compliance" and "Usability" dimensions. Otherwise, SD can not be used to address research question, process can not continue.

Compliance and Usability of D: Main goal of tasks to reach compliance and usability of D is validated value of data and enrich them. Quality criteria used are 'Accuracy', 'Uniqueness', 'Temporal precision', 'Consistency', 'Completeness of occurrences', 'Ease of interpretation'. RA needs to realize 8 majors tasks: 1-controls if values of each variable match with the description of coding scheme provided by producer, 2-controls that all records into data files are unique, 3-controls that there are sufficient data to ensure data temporality, 4-controls that data are not in contradiction with the others, 5-calculates number of occurrences, 6-calculates number of missing values, 7controls that data are in relation to constraints of data analyze methods, 8-creates new variables to enrich data, 9-controls that enrichment are sufficient. At the end, D is qualified with "Compliance" and "Usability" dimensions. Now, a first data analysis can be led. Data are qualified with properties "Acceptability".

In our example about students' evolution, one set of tasks can be: controlling that scores of students is between 0 to 20 controlling that several scores exist by students and controlling that ratio of missing values are less than 5%.

Nevertheless, one major difficulty still remained is the monitoring of these processes which combine major tasks, data properties and quality criteria. We propose to use dashboards to address these difficulties.

Dashboards

Advantage of dashboards is that traces of data pre-processing are kept. Thus, other RAs can reuse these data with a high level of trust. Dashboards are essential into a data quality process. To follow the pre-process, in addition to major tasks, is needful to add 3 elements in dashboard: 1-sub-tasks to refine major tasks, 2-type of measurement method to create quality criteria, and 3-decision criteria.

Major tasks are steps of process, when RA needs to reuse data; (s)he defines several sub-tasks that are needful to take over data structure and data. Sub-tasks are defined according to research question and type of datasets chosen. (See example in table 2). Type of measurement method indicates type of information produced by sub-tasks. We have adapted a typology proposed by Polańska and Zyznarski [START_REF] Wang | Design-based research and technology-enhanced learning environments[END_REF]: objective measures are produced in an automated way (such as ratio of missing values); subjective measures are produced by a human (such as a list of available data, or description of data production gathered in an interview with data producer). This distinction is important because an algorithm directly calculates objective measures whereas subjective measures can not be calculated by an automated way.

Table 2: Example of sub-tasks for the major task:

"control that all information to pick up data are available"`

Major tasks Sub-tasks

To control that all information to pick up data are available.

Interviewing produced

Listing data useful to address research question

Listing data available in data set

Obtaining metadata

Decision criteria indicate threshold that is accepted by RA to qualify data and to reuse them. It is fixed before to pick up data, (for instance RA fixed maximum value of missing value at 10%). Decision criteria are logical for subjective type of measurement method and numerical for objective type of measurement method. They validate quality criteria. If all quality criteria are validated, data properties are reached. Thus, data structure or data can be qualify or not with data properties. Dashboards proposed contains: Majors tasks, Subtasks, Type of measurement, Decision criteria, data quality criteria and appraisal of data properties. Sub-tasks, type of measurement, decision criteria are defined by RA in relation to research question and data that come from field. A first list of sub-tasks, type of measurement, decision criteria must be describe before picking-up data. This list can be completed and modified during process. Differences between these dashboards are major tasks, subtasks and quality criteria.

DOP8_Qpp enhancement of preprocessing in DOP8

With these different elements describe above, we enhance DOP8 to take into account data that come from educational fields without research protocol. It is enhanced with four data properties, a list of major tasks to qualify data, and a set of quality criteria. In figure 4, we present DOP8_Qpp; it contains specific steps to following data transformation, collecting existing data validating structure of data, transforming structure of data; validating values of data, validating data to sustain analysis and enriching data. Goal of each stage is to evaluate the data properties; a dashboard is produced at each stage. For example, when RA collects data, he uses dashboard described in figure 3.1 to reach criteria relevance. If these criteria are not reached, RA can follow (or not) recommendations made in purple boxes to enhance information to pre-process data. Thus, RA is guided during data pre-processing.

EVALUATION

Case study : common first year for health studies in France (PACES)

We tested our proposal with data that come from common first year for health studies in France. This case study is based on a set of data produced in a flipped classroom pedagogical approach.

With two research questions about students' evolution and data produced by PACES, our proposal is evaluated by testing dashboards with data properties, major tasks quality criteria. We instantiate a list of sub-tasks with, for each one, a type of measurement and a decision criteria. Table 4 presents instantiation of dashboard to monitor "Utility of SD & D" for PACES data. In this case, sub-tasks provided only subjective type of measurement. In the column remarks RA can notice interactions with data producer and encountered difficulties.

Comprehensive dashboards are available into a report "Process of data analysis in TEL research" 1 .

Table 4: Instantiation of dashboard to monitor utility of SD & D to analyze students' evolution with data PACES

This Table is the final version, but each sub-task could be iterative. Also dashboard to follow values of objectives were proposed. For example, descriptive statistics with thresholds for data usability.

This process allow us to follow and share this time consuming pre-process steps for a first research question. They was shared for a second linked research question with other context and others researchers.

To go beyond it is necessary refine criteria: "ease of interpretation". For instance, to study students' evolution and obviously: Iterative measures during time, Evaluation of students, Iterative measures for each topics, Time indicator, Topics and student indicator are useful. Future works on these criteria must be conducted to understand ecological data without ambiguity. How create useful data with a real semantic? Which sub-tasks can be made to enhance meaning of data?

Evaluation of criteria and of process by 36 data scientists

First, interviews administrated with data-scientists confirm process, four data properties and quality criteria. To evaluate our proposal with a more data-scientist and data-analyst, a survey was designed. Survey cannot present directly process and these elements; it is too hard to answer directly about properties and quality criteria. So, the survey measures level of agreement about useful elements to take over data, tasks and tools that we have proposed. Finally, we have evaluated DOP8_Qpp about data pre-processing steps (results are not presented here). Chosen methodology is an on-line questionnaire with Likert-scale. 51 answers are collected, (36 data scientist and 15 data analyst). Even if total numbers of answers are less than 100, results of questionnaire are presented in percentage, to ease reading.

Useful element to take over data in a comprehensive manner (Table 5). More than 75% of respondents agree the nine elements shown in table 5. The first five elements are crucial, more than 90% of respondent accept them. Then, the other four are important too, more than 75% of respondent accept them. Among these elements, the three most important are "list and description of variables" (77%), "coding scheme of data" (75%) and "description of data file contain" (73%). Useful tasks to take over SD & D (Table 7). About data structure, we propose 4 majors tasks. More than 90% of respondents agree the 4 tasks. About data, we propose 9 majors tasks. More than 90% of respondents agree the 9 tasks. Controlling uniqueness of data and describing variables are crucial (100% of respondents agreed). Mostly experts, more than 85% percent, agree on elements to take over data, actions and tools that we have proposed. Thus, these elements are essential to take reuse data by datascientist. So, these elements, tasks and tools can companion RA when he takes over data, then he can qualify data that come from educational field. So, he evaluates data quality and one of positive impact is quality of results. Beside, presented evaluation show the necessity to refine quality criteria and go beyond to support their application. Use sematic of data is one of the research tracks. For example, if we know that data represented "errors of students in several courses" we can apply reasoning methods to evaluate the utility of the data for a particular research question.

CONCLUSION

Also, when ecological data are qualified with four dimensions, we can ask question of positive impact of these data quality on quality of results and how to measure this quality? With which criteria?

Finally, in our proposal acceptability does not take into account social dimension, such as defined by [START_REF] Nielsen | Usability Engineering[END_REF]. In the case of student's evolution studies, one perspective is adding ethical criteria to control anonymization of data.

Figure 2 :

 2 Figure 2: Description of the model with four data properties and place in the process.

  Next tables present 5 examples of dashboards: 3.1 to reach Utility of SD & D, 3.2 to reach Compliance of SD, 3.3 Compliance of D and 3.4 Usability of data.

Table 3 . 1 :Table 3 . 2 :Table 3 . 4 :

 313234 Dashboard to monitor Utility of SD & D Dashboard to monitor compliance of SD Table 3.3: Dashboard to monitor Usability of D Dashboard to monitor Usability of D

Figure 4 :

 4 Figure 4: DOP8_qpp: enhancement of steps collect, validate and enrich with data properties and quality criteria.

Table 5 : Agreement about useful elements to take over data.

 5

		Strongly
	%	agree	Agree	Total
	description of data file
	contains	73	24	97
	Name and contact of
	producer	47	47	94
	Data production protocol	70	22	92
	List and description of
	variables	77	14	91
	Conceptual data model	59	31	91
	Coding scheme of data	75	14	89
	Data structure	64	24	88
	Research question	68	18	85
	Names of data files	57	20	77

Table 7 : Agreement about of set of tasks to take over data.

 7

	Major tasks to qualify	Strongly	Agree	Total
	data structure	Agree

  Using data produced by and for stakeholders such as teachers or students ensures relevance of the research results. Although, these data are valuable, it is difficult to reuse them without pre-processing steps and without properties and indicators to control data quality. Data quality research and data life cycle are combined here to target this problem and give means to refine the crucial pre-processing step. Thus important step more accessible to TEL domain. With respect of related work, our research brings process and concepts to take over ecological data. Add value of our proposal is four data properties Utility, Compliance, Usability and Acceptability of ecological data. Data quality indicators ensure that four dimensions are reached and then if data are reusable by other users, in other contexts. To elaborate data quality criteria a list of majors tasks are provided. RA uses this list and defines news subtask to take into account ecological data, which he wants used. Also, our research shows need to monitor data validation and transformation. To do this we proposed a set of dashboards. Finally, our process help RA to generate sets of data that have been qualified with four properties and quality criteria. Once, educational data field have been validated they can be reused with data mining tools.One instantiation shows the use of the process with data produced in a flipped classroom pedagogical approach. (1800 students, 12 scores for each students, and 12 MCQ tests). Right now, process and data properties are used with two other cases study (data come from MOOC: MOOCAZ [16], Serious games: TAMAGOCOURS [17]).Positive impact of our proposal is the qualification of data with four data properties: Utility, Compliance, Usability to reach Acceptability of ecological data to address and resolve research questions. Another add values are majors' tasks and dashboards to target problem and assist this activity.

French title « Processus d'analyse de données pour la recherche en EIAH » L.Dupuis.


Filippo Rebecchi

Marcelo Dias De Amorim

Vania Conan
email: vania.conan@thalesgroup.commarcelo.amorim@lip6.fr

Should I seed or should I not: On the remuneration of seeders in D2D offloading

Keywords: Data offloading, optimal control, device-todevice communications, epidemics

Traffic offloading using opportunistic device-todevice (D2D) communications is a new and exciting opportunity for cellular operators to cope with the unprecedented mobile data growth. A limitation of existing proposals is that they assume that all terminals are, by default, involved in the D2D forwarding process. In particular, they do not capture the need to reward seed users. For this reason, we include a rewarding cost in the design of the opportunistic offloading strategy. In our solution, we make the difference between nodes that receive content through the cellular channel only (leechers) and nodes that take part in the forwarding process (seeders). The key point for an operator is to design a global strategy to select which nodes act as seeders and which ones as leechers, in order to reduce the total dissemination cost. We formulate this question as a stochastic control problem that we solve using an application of Pontryagin's Maximum Principle. We provide a mathematical framework to devise the optimal strategy for opportunistic offloading under a generic cost model. First, we show that an optimal solution exists; then, from this policy, we extract some insights to develop heuristics. Finally, we discuss the advantages of the proposed model compared to the classic seeder-only model. We demonstrate that separating seeders/leechers leads to better incentive strategies in the most demanding cases of content with a large span of delivery delays.

I. INTRODUCTION

Device-to-device (D2D) communications are a well-timed strategy for operators to face the ever-increasing mobile data demand by offloading part of the traffic from their cellular infrastructure. Motivated by the delay-tolerance and redundancy of some types of content, operators may send data only to a subset of requesting users (seeders), which act as opportunistic forwarders to help propagate content using D2D communications. The combination of two complementary channels (cellular and D2D) provides extra capacity, helping reduce the impact of redundant traffic.

Several past works demonstrated that opportunistic D2D offloading does help relieve congested cellular networks [START_REF] Han | Mobile data offloading through opportunistic communications and social participation[END_REF], [START_REF] Whitbeck | Push-and-track: Saving infrastructure bandwidth through opportunistic forwarding[END_REF], [START_REF] Andreev | Cellular traffic offloading onto network-assisted device-to-device connections[END_REF]. Nevertheless, the effectiveness of this strategy revolves around the willingness of users to cooperate as forwarders. Incentives are then needed to stimulate participation, and to reward users that act as data relays. Existing proposals assume that all users are, by default, also seeders in the D2D domain (i.e., potential forwarders). Such an assumption may lead to suboptimal results when seeders are rewarded for transmitting content on behalf of the infrastructure. In those scenarios,

Interested Leecher Seeder (a) (b) (c)
Fig. 1. Offloading process: the infrastructure selects two nodes as content initial leechers (Fig. 1(a)), deciding that one of the leechers should be promoted as seeder (Fig. 1(b)). Later on, the infrastructure estimates that it is worth promoting another node because the D2D transmissions are not enough to guarantee sufficient dissemination (Fig. 1(c)).

uncontrolled D2D communications may generate additional costs without necessarily bringing dissemination gains. By borrowing the concept from the peer-to-peer jargon, we introduce a clear separation between leechers and seeders, as shown in Fig. 1. Leechers receive content via the cellular infrastructure, but only nodes promoted to the seeder state are allowed to forward the content opportunistically. The separation between leechers and seeders provides operators with an additional degree of freedom. The balance between instantaneous cost and future benefits of leech and seed decisions is strategic to data dissemination, given that the available resources (cellular bandwidth and rewards) are limited. Following the terminology introduced in [START_REF] Ning | Self-interest-driven incentives for ad dissemination in autonomous mobile social networks[END_REF], data dissemination follows a "push-pull" model, where receivers can be both consumers and disseminators of data.

We investigate the following problem: which fraction of leechers should be promoted as seeders and when should this happen? We provide an answer to this question using a mathematical support that leads to an optimal solution. Content diffusion in opportunistic networks is comparable to the spreading of a disease in a population. We model the dissemination process using a variant of the classic Susceptible-Infected-Recovered (SIR) epidemic model from . Since operators strive to optimize the distribution cost, we translate the possible decisions they can take into a cost function. We apply Pontryagin's Maximum Principle to minimize the cost function subject to the stateequations that govern content distribution. How to select the best seeders has been considered in the literature from the spatial re-use, throughput, and interference points of view, rather than from an incentive/economic perspective. As far as we know, no existing work on D2D offloading contemplates the difference between leechers and seeders, failing thus to quantify the trade-off that exists between performance and cost in more realistic systems. Performance evaluation of truly opportunistic forwarding has been treated extensively by means of ordinary differential equations (ODEs) [START_REF] Zhang | Performance modeling of epidemic routing[END_REF] or Markov chains [START_REF] Groenevelt | The message delay in mobile ad hoc networks[END_REF]. Instead, we propose an extended model that couples opportunistic dissemination with infrastructure control. Indeed, a central offloading coordinator controls the cellular injections and the promotion of users to the seeder state to reach optimal dissemination. The centralized scheme avoids the need for users to exchange periodically contextual information, such as meeting times, which instead can be directly extracted from the cellular infrastructure. Moreover, users are not required to continuously compute their seeding status, thus economizing battery and computation power. In summary, the main contributions of this paper are:

• Leecher-seeder model. We propose a D2D offloading model where the network operator controls content injections and the promotion of users to the seeder state. • Optimal control. We formulate D2D data offloading as an optimal control problem to minimize the dissemination cost in a hybrid scenario. A cost function trades off monetary and network resources consumed to reach a certain dissemination level and user satisfaction. • Cost functions. We prove how to solve the offloading problem for different cost functions. We show that under plausible cost functions, the control of the injection is continuous, while promotions have an on-off behavior. • Evaluation. We evaluate the sensitivity of the optimal controls to different contact rates and delay tolerance values. We evaluate the optimality of our strategy against other heuristics. Finally, we confirm the benefit of the proposed seeder-leecher model compared to the simplified seeder-only model currently employed in literature. The remainder of this paper is structured as follows: Section II introduces the leecher-seeder model. Cost-related issues are discussed in Section III. The optimal control problem is formulated and solved in Section IV. Section V presents numerical results. We postpone related work to Section VI. Finally, Section VII draws the conclusion and perspectives.

II. SYSTEM DESCRIPTION AND SCOPE

Overview. We formulate D2D offloading as an optimal control problem, modeling data dissemination using a variant of the classic SIR model for epidemics. In our model, some users request data and are referred to as interested. Initially, all the nodes are in the interested state waiting for content. At this stage, the operator can only use cellular transmissions to reach a subset of the interested users. Nodes that receive the content enter the leecher state, still not playing any active role in data distribution. At this point, the coordinator can promote a

(t) = 1 ∀t ∈ [0, T ].
Therefore, we can always represent the system using only two states. An interested node can receive the content whenever in contact with a seeder, but not with a leecher. For the sake of clarity, Table I provides a summary of the parameters used along the paper (some of them will be explained in the following sections).

Cost. Incentives to reward user participation in data dissemination can be offered by using virtual credit schemes or discounts.

For now, we do not consider any additional cost related to overhead, signalization, or maintenance of promotions, which are left for future work. Hence, promotion to the seeder state does not represent a cost in itself for operators, but allows users to be rewarded for their contribution in data dissemination. On the other hand, injections through the cellular infrastructure entail direct costs, in general related to resource availability in the access network. We will further elaborate the reflection on cost-related aspects in Section III.

Encounters and communication opportunities. In the real world, the system under observation can be described with discrete values (e.g., the number of users and the number of cellular transmissions performed). For the sake of modeling, we consider instead continuous values for the state and the control values. We assume that N is large and that encounters are homogeneous. Consistently with the literature, we use a mean field model that is accurate for a large population. D2D dissemination can be regarded as the spread of infective disease -not surprisingly, epidemic routing is a conventional forwarding strategy in opportunistic networks. As with a disease contagion in a population, in our model content spreads from seeders to interested nodes when such a pair enters in physical proximity. 2 We describe the state evolution with a system of ODEs along with a set of initial and terminal constraints. The contact rate λ(t) rules the encounter of any two nodes at time t. We assume that λ(t) includes also the uncertainties introduced by the wireless channel and the movement of nodes. At time t, we have n S (t) seeders capable of meeting n I (t) interested nodes. Consequently, interested nodes become seeders with rate λ(t)n I (t)n S (t) (the transitions are illustrated in Fig. 2).

Injections and promotions. In line with the existing literature on data offloading [START_REF] Rebecchi | Data offloading techniques in cellular networks: A survey[END_REF], we consider a central coordinator that manages the cellular injections, adding to its duties also the responsibility to decide the promotion of nodes. Leechers carry the content but need to be promoted in order to contribute to data dissemination. Operators can promote only the necessary fraction of seeders. This is done via a control channel binding users to the central coordinator. As a result, leechers shift to the seeder state with intensity regulated by the signal u P (t), a bounded Lebesgue integrable function with 0 ≤ u P (t) ≤ 1 ∀t ∈ [0, T ]. This increases the fraction of nodes in the forwarder state by a rate u P (t)n L (t). Therefore, the following system of ODEs controls the evolution of the interested, leecher, and seeder nodes in the system:

dn I dt = -λ(t)n I (t)n S (t) -u I (t)n I (t), (1a)
dn L dt = u I (t)n I (t) -u P (t)n L (t), (1b)
dn S dt = λ(t)n I (t)n S (t) + u P (t)n L (t), (1c)
with initial states n I (0) = i 0 , n L (0) = l 0 , and n S (0) = 1 -i 0 -l 0 . For the offloading problem we consider, we have i 0 = 1 and l 0 = 0, since we consider all users to be in the interested state at the beginning of content diffusion.

The equations above describe how the states n I , n L , n S change at time t as a reaction to the control signals u I , u L .

Note that ∂n I ∂t + ∂n L ∂t + ∂n S ∂t = 0; therefore, the model can always be expressed using only two out of the three equations.

III. DISTRIBUTING CONTENT IS COSTLY

The optimal offloading strategy consists in minimizing the amount of nodes still in the interested state at the end of content lifetime, while implementing a cost-savvy injection/promotion campaign. If cellular operators had no capacity limitations, then the optimal strategy would consist in injecting the maximum amount of data via the cellular channel. Instead, when capacity is limited, operators may seek to exploit D2D communication capabilities of their customers. Nevertheless, rewarding users can be costly, as both operational and budgetary constraints should be taken into account when serving content. Based on these aspects, Eq. 2 considers a cost function J that is general enough to grasp various types of cost incurred by operators:

J = Φ[n I (T )] payoff + T 0 f [u I (t) n I (t)] injection + g[λ n I (t) n S (t)] reward dt. (2)
In Eq. 2, Φ[n I (T )] represents the final payoff, i.e., the cost that the operator has to pay for having failed to satisfy the fraction n I (T ) of users by the deadline. This may lead to loss of earnings due to missed deliveries or to extra costs in terms of final injections [START_REF] Whitbeck | Push-and-track: Saving infrastructure bandwidth through opportunistic forwarding[END_REF], [START_REF] Rebecchi | DROiD: Adapting to individual mobility pays off in mobile data offloading[END_REF], [START_REF] Valerio | Adaptive data offloading in opportunistic networks through an actor-critic learning method[END_REF]. Word of mouth among angry consumers may also boost the commercial fallout of missed or late deliveries making this term highly non-linear. f [u I (t) n I (t)] captures the cost, in terms of network resources, of injections over the cellular channel. Despite advanced cellular technologies (e.g., LTE) increase the overall system capacity with multi-user diversity, each additional user to be served reduces the rate of improvement [START_REF] Kwan | Multiuser scheduling on the downlink of an lte cellular system[END_REF]. In other words, the radio resources allocated to transmit data do not follow a linear trend with the number of users. In addition, one must consider that overhead is larger when many users share the same bandwidth, rather than a single user benefiting from the whole bandwidth [START_REF] Ide | Influence of m2m communication on the physical resource utilization of lte[END_REF]. For these reasons, the injection term is likely another non-linear cost for operators. Finally, seeders are rewarded with g[λ n I (t) n S (t)] (instantiated as discounts or virtual credits), accorded each time they make an opportunistic transmission. Examples of possible incentive strategies in D2D networks are provided in [START_REF] Ning | Self-interest-driven incentives for ad dissemination in autonomous mobile social networks[END_REF], [START_REF] Zhang | Mobicache: Cellular traffic offloading leveraging cooperative caching in mobile social networks[END_REF].

The integral in Eq. 2 portrays the growing cost over time of these two latter terms. Note that promotion control u L does not appear inside the cost function. As outlined in Section II, promoting a node to the forwarder state does not directly generate a cost. However, seeders will be able to transmit data opportunistically, possibly increasing the rewarding cost for the operator. For physical reasons, Φ(.), f (.), and g(.) should be monotonically increasing functions, with Φ(0) = f (0) = g(0) = 0 (the cost for doing nothing is zero).

Relationship between control and cost. At any time, the offloading controller must choose the values of the control signals u I and u L . The decision is taken by assessing the fraction of nodes in each compartment (n I , n L , and n S ), the time remaining before the deadline, and the contact rate between nodes. Applied controls lead to two consequences: (i) direct effect, which generates the instantaneous costs f (.) and g(.) for the operator, and (ii) indirect effect, represented by the future change in states formalized by Eq. 1. The optimal offloading strategy requires the coordinator to plan its injection and promotion strategies by minimizing the cost for the operator while maximizing the rate of change of state variables.

Extreme cases. We aim to further clarify this concept by focusing on the injection control u I . We consider two extreme strategies that do not consider the evolution of dissemination. The first myopic strategy always injects the maximum amount of copies through the cellular channel. In this case, the offloading controller does not take into consideration the D2D capabilities of nodes. It turns out that this strategy is strongly suboptimal. The other extreme is a minimum injection strategy, where the coordinator injects only once. It is intuitive that an optimal decision avoids such extremes.

IV. OPTIMAL SOLUTION

We now have the sufficient background to derive the optimal offloading solution for our problem. By applying n S (t) = 1-n I (t)-n L (t), we can formulate the optimal control problem considering only the two state variables n I and n L . The system is controlled by the tuple < u I , u P >, which belongs to the set of all the admissible controls U = {u I , u P }, where u I , u P are Lebesgue integrable with u I , u P ∈ [0, 1]. The goal is to characterize the optimal controls < u * I , u * P > that minimize the cost function J, subject to the constraints defined in Eq. 1:

min u I (t),u P (t)∈U
J, subject to:

(3a)

dn I dt = -λ(t)n I (t)(1 -n I (t) -n L (t)) -u I (t)n I (t), ( 3b
)
dn L dt = u I (t)n I (t) -u P (t)n L (t), (3c)
n I (t) ≥ 0, n L (t) ≥ 0, n S (t) ≥ 0, n I (t) + n L (t) + n S (t) = 1, (3d)
where the initial states are i 0 = 1 and l 0 = 0.

A. Existence of an optimal control

The existence of an optimal solution can be determined by applying the Filippov-Cesari theorem (Theorem 4.1 in [START_REF] Fleming | Deterministic and stochastic optimal control[END_REF]). To prove this theorem, the following conditions should be satisfied: (i) the functions inside the integral in Eq. 2 are continuous, bounded, and convex in controls, with bounded derivatives, (ii) the control signals u I (t) and u P (t) take values in a closed set, and (iii) Eqs. 3b and 3c are linear in the controls. These conditions guarantee the existence of an optimal solution.

B. General solution -Pontryagin's Maximum Principle

Since an optimal control exists, we apply Pontryagin's Maximum Principle [START_REF] Boltyanskii | The theory of optimal processes. I. The maximum principle[END_REF] to derive necessary conditions on the optimal control (Theorem 3.4 in [START_REF] Grass | Optimal control of nonlinear processes: with applications in drugs, corruption, and terror[END_REF]). The conditions of Pontryagin's maximum principle reduce the computation of an optimal strategy to the solution of a boundary value problem for a system of differential equations. Let the tuple (n * I (.), n * L (.), u * I (.), u * P (.)) be an optimal solution to the problem formalized in Eq. 3. 3 There exist continuous and piecewise continuously differentiable adjoint functions p * I (t) and p * L (t) that maximize the present-value Hamiltonian function H.

For the sake of ease of mathematical manipulation, we transform the problem into a maximization problem by multiplying the Hamiltonian by -1. We also remove the dependence from time whenever possible, in order to make reading easier:

H(n I,L , u I,P , p I,L , t) = -f [u I n I ] -g[λ n I (1 -n I -n L )] (4) + p I [-λn I (1 -n I -n L ) -u I n I ] + p L [u I n I -u P n L ].
The Hamiltonian function, in analogy with the corresponding concept occurring in traditional mechanics, balances the rate of change of states and the cost incurred by operators. Indeed, the Hamiltonian is a generalized profit rate that includes both direct and indirect effects, and has to be maximized at each instant. The weights for the state variables are given by the adjoint functions p I and p L , which represent the marginal increase of H due to an increment in the state. Consequently, the adjoint equations p I and p L evaluated at the optimum are: with transversality (terminal) conditions that describe what must be satisfied at the end of the time horizon T :

dp * I dt = - ∂H (
p I (T ) = ∂Φ(n * I (T ), T ) ∂n I , p L (T ) = ∂Φ(n * I (T ), T ) ∂n L = 0. (6)
According to the maximum principle, there exist optimal controls, a tuple < u * I , u * P >∈ U of continuous and piecewise continuously differentiable functions, and their corresponding solutions n * I , n * L that maximize the Hamiltonian H satisfying Eqs. 5 and 6:

u * I,P (t) ∈ arg max u I,P ∈U H(n I,L , u I,P , p I,L , t). (7)
The canonical system, composed of four coupled ODEs (Eqs. 3b, 3c, 5a, 5b) and the transversality conditions (Eqs. 6), determines a boundary value problem (BVP).

C. The case of data offloading

Let us now focus on the case of the seeder-leecher model by solving the optimization problem for a class of cost functions Φ(.), f (.), g(.) in Eq. 2. The choice of these functions follows the discussion in Section III. However, thanks to the flexibility of the model, they can be replaced at will, to take into account the specificities and operating costs of certain networks. Besides the existence constraints discussed in Section IV-A, by physical reasons, Φ, f , and g should be monotonically increasing functions starting at zero.

We consider an exponential function for the final payoff Φ(x) = e x -1, a power-law function for the cellular injections f (x) = bx α , with α ≥ 2, and a linear function g(x) = cx to reward seeders that distribute content. The final payoff function Φ(x) starts at zero and then increases exponentially to model the cost for missing the delivery by the deadline T . f (x) represents the cost for injecting data on the cellular channel during the content lifetime. The power-law accounts for the cost of simultaneous data transmissions. Indeed, the more simultaneous cellular data transmissions, the more costly they become (in terms of consumed radio resources) at the base station. The power-law coefficient α depends on the considered network and on its overall congestion conditions. Since the domain of f ∈ [0, 1], lower values of α give the steeper curves. The cost function g(x) is linear, as we consider that the reward offered to forwarders for each opportunistic transmission they perform is fixed. By substituting the cost functions Φ(.), f (.), and g(.) in Eqs. 4, 5, and 6, the Hamiltonian, the adjoint functions, and the transversality conditions become:

H = -b(n I u I ) α -c(λ n I (1 -n I -n L ))+ + p I [-λn I (1 -n I -n L ) -u I n I ]+ (8a) + p L [u I n I -u P n L ], dp * I dt = b α n α-1 I u α I + c λ (1 -2n I -n L ) (8b) -p I [2λn I -λ + λn L -u I ] -p L u I , dp * L dt = -c λn I -p I λn I + p L u P , (8c)
p I (T ) = e n * I (T ) , (8d) p L (T ) = 0. (8e)
Injections. Given that Eq. 8a is strictly concave in the control variable u I , we extract u I (t) using the Hamiltonian maximization condition ( ∂H ∂u I = 0 evaluated at the optimum), along with the restriction on the maximum injection rate (n I (t)u I (t) ≤ I max (t) ∀t ∈ [0, T ]). By defining the function ψ(t) = α-1 p * I -p * L -αb , we can write the optimal solution as:

u * I (t) =      0, if ψ(t) < 0, ψ(t) n I (t) , if 0 ≤ ψ(t) ≤ I max (t), Imax(t) n I (t) , if ψ(t) ≥ I max (t). (9)
Equivalently, we have that u *

I (t) = min[max[ψ(t),0],Imax] n I (t)
.

Promotions. In the case of promotions, since the Hamiltonian is linear in the control variable u P , the maximization condition ∂H ∂u P = 0 is trivially satisfied and independent of u P . The control in this case is called singular (Definition 3.40 in [START_REF] Grass | Optimal control of nonlinear processes: with applications in drugs, corruption, and terror[END_REF]) with a bang-bang solution, i.e., a control that switches discontinuously between one extreme to the other. Since the maximization condition cannot help determine the optimal control, it is possible to rewrite Eq. 8a as:

H = -b(n I u I ) α -c(λ n I (1 -n I -n L ))+ + p I [-λn I (1 -n I -n L ) -u I n I ]+ (10) + p L [u I n I ] -u P [p L n L ] σ .
By defining a switching function σ = (p L n L ), to maximize H the control should take its maximum (minimum) value when σ < 0 (σ > 0). By construction, u P ∈ [0, 1], and:

u * P (t) = 0, if σ > 0, 1, if σ < 0. (11)
In order to be able to retrieve the evolution of the state and adjoint variables, we have to solve a system of coupled ODEs (respectively Eqs. 3b, 3c, 8b, 8c, 8d, and 8e), with a mix of initial and final conditions (boundary values). We solved it numerically by using the shooting method from the R package bvpSolve to compute the evolution of the state and adjoint variables as well as the optimal control [START_REF] Soetaert | Package bvpSolve, solving boundary value problems in R[END_REF].

V. NUMERICAL RESULTS

To identify the optimal offloading strategy, we conduct numerical analyses using the software R. Firstly, we execute a sensitivity analysis on the values of λ and T in order to understand their implications in the offloading strategy. Performance depends strongly on these two parameters, so do the injection and promotion strategies. Then, we explore under which circumstances the seeder-leecher model brings advantages over the classic seeder-only model. Finally, we address some implementation issues by comparing the optimal strategy with several heuristic strategies, investigating under which conditions and limits they can be adopted.

A. Injections and promotions

We investigate "when" the infrastructure should inject copies of the content and promote nodes as seeders. Figs. 3 and4 display the time evolution of both states and control variables for two different deadlines and different contact rates.

By comparing Figs. 3 and4, we discover that there are more promotions for short deadlines. We conclude that content lifetime strongly influences promotions. Short lifetimes (e.g., T = 5) are not sufficient to yield complete dissemination under the provided cost-function. Instead, for T = 10 we obtain complete data delivery (at least for the best contact rates). When content dissemination is incomplete by the deadline, the final payoff Φ(n I (T )) takes a large part of the cost function J. This behavior confirms a well-known phenomenon in opportunistic networking: increased delivery delays improve the fraction of nodes that receive content through opportunistic communications. The added dissemination time allows fewer injections and promotions, thus lowering costs for cellular operators. On the other hand, increased delivery times could hurt user satisfaction.

Besides the deadlines, low contact rates also lead to stronger injection and promotion controls. Focusing on injections (Figs. 3b and4b), we observe that the control is stronger in the beginning and in the end of the dissemination, following a nearly symmetrical pattern. This pattern depends on the interactions between n I and n S in Eq. 1a. Therefore, the injection rate is higher when a few nodes are in the seeder or the interested states -respectively at the beginning and at the end of the dissemination. Injections help overcome the slow start and the convergence time of opportunistic dissemination. Wang et al. previously pointed out the symmetric trend of the injection control, although for a simplified model [START_REF] Wang | Content dissemination by pushing and sharing in mobile cellular networks: An analytical study[END_REF].

Promotions (visible in Fig. 3c and4c) follow a completely different pattern. For T = 5, the control is always at its maximum. The shorter deadline is responsible for the poor dissemination. Indeed, even with an extreme promotion strategy (always on), there are still nodes in the interested state at the deadline. Longer deadlines, instead, allow devising more effective promotion strategies. For T = 10, u P (t) shows three different patterns depending on the value of the contact rate. For λ = 0.1, the control is always at its maximum for the entire dissemination. λ = 0.5 presents an on-off behavior, with promotions that stop when the amount of seeders reaches significant levels (in order to self-sustain without costing too much to operators). Finally, for λ = 1, promotions are activated only after half of the dissemination period. This collides with the desire to attain the widest possible dissemination of the content. Although at first sight this might seem counter-intuitive, we must not forget that operators pay a small fee for each opportunistic transmission performed by users. Under higher contact rates, opportunistic dissemination has to be limited in order to save monetary resources.

As anticipated in Section IV-C, u S takes the form of a bang-bang control with exactly one on-off switch. Injections performed when u S (t) = 0 serve only to satisfy the fraction of users that will likely not receive the content by the deadline, without further improving the dissemination (because these nodes are not promoted, remaining leechers). Moreover, we point out that the optimal strategy does consider moments where no additional seeders are needed (u S (t) = 0). This strengthens the idea that separating seeders and leechers is beneficial from a cost-benefit point of view.

B. Controlling seeders: Giving operators an edge

We examine in which cases it is worth separating leechers and seeders from the operator's point of view. Including the deadline T cost function J q q q q q q q q q q q 5 6 7 seeder-only:

J(T) Φ(T) F(T) G(T) : J(T) Φ(T) F(T) G(T)
Fig. 5. Cost functional J and its main components for the optimal strategy using two offloading models (seeder-seeder and two-state), varying the deadline T . Other parameters: λ = 0.5

Imax = 0.1, α = 2, b = 10, c = 1, i 0 = 1, s 0 = 0.
leecher state in the picture is motivated by the fact that not all users carrying the content may be required to forward data. These scenarios hinge on a combination of factors, such as the contact pattern or the delay tolerance of the content. Eventually, adding an intermediate state between interested and seeder becomes cardinal when operators want to reward user participation. As many works suggest, offering some kind of incentive (i.e., discounts or virtual credits) motivates user participation [START_REF] Zhuo | An incentive framework for cellular traffic offloading[END_REF]. However, current models in the literature consider only two states (interested and seeder).

The optimal solution for the seeder-only model will serve for comparison, and is detailed in Appendix A. In this case, the only control is the injection rate u I . We plot in Fig. 5 the evolution of the cost function J divided by its three main components Φ(T ),

F (T ) = T 0 f [u I (t) n I (t)]dt, and G(T ) = T 0 g[λ n I (t) n S (t)]dt.
Φ(T ) is the final payoff value, due to nodes that have not received the content by the deadline, F (T ) is the total cost of injections, and G(T ) is the total cost of rewarding.

Separating seeders and leechers is advantageous for operators if compared to the classic seeder-only model. In Fig. 5, we can observe that, as the deadline increases, the leecher-seeder model improves its aggregate performance compared to seeder-only. From our analysis, for T > 5, the number of uninfected nodes at the deadline decreases steadily, reducing the relative weight of the term Φ(T ) on the overall cost. Rewarding cost G(T ) takes the larger part of J, accounting for the cost to reward seeders. Seeding costs increase linearly for the seeder-only model as the deadline stretches, making up nearly the entirety of J. This confirms that an uncontrolled number of seeders can interfere with the will of operators to cut operational expenses. Instead, a separation between leechers and seeders offers improved flexibility in the control of the offloading evolution, bringing clear advantages in terms of distribution costs.

Note that for short deadlines (for T ≤ 5 in the example), all connection opportunities should be used as captured by the seeder-only model. Indeed, the cost-functional is dominated by the final payoff Φ(T ), whose value depends on missed data deliveries. The leecher-seeder model introduces an additional transition delay from the interested to the seeder state (the ODE formulation requires a non-null time to transit to a state); thus, its benefits come into play in the non-trivial situations of content with larger deadline requirements.

C. Implementation considerations

We first investigate how intuitive heuristics not requiring any optimization framework perform compared to the optimal strategy, and what lessons can be learned having the knowledge of the optimal injection and promotion controls. Finally, we consider the impact of realistic values of the contact rate.

Heuristics. The optimal solution requires to solve numerically the optimization problem, which is a complex task. This may not be possible, given the size of the network and time constraints, when the available processing power is limited. For this reason, we analyze the performance of simple heuristics. We compare the optimal strategy against three other heuristics. The first heuristic, named initial control, mimics an operator wanting to rely only on an initial subset of seeders. These seeders are the only way to distribute content by the deadline. This strategy relies on an initial injection at the rate I max , without performing any further injections. The second strategy, named constant control, steadily injects at a fixed rate of Imax 2 . These two strategies are static and do not require any knowledge of how the dissemination evolves. In both cases, the promotion control u R (t) is fixed at 1 for all the dissemination delay. Finally, we consider a more dynamic strategy, named pursue control, where both injection and promotion controls follow the evolution of the interested nodes n I (t). In this case, the control is strong at the beginning of the dissemination, gradually descending as the time passes by, following n I (t). The rationale behind this choice is that copies with a large tolerance to dissemination time are more effective in content dissemination. We compare these strategies in terms of the cost function introduced in Eq. 2, varying the deadline T and the contact rate λ.

In Fig. 6, we plot the cost functional J by varying the deadline. As expected, the functional J for the optimal control is always smaller than all the other heuristic strategies. However, for shorter deadlines (T < 6), the pursue strategy is very close to the optimal. This is the first lesson we can draw: with shorter deadlines, a control that follows the rate of interested nodes comes close to the optimal. As the deadline increases, the efficiency of the pursue strategy decreases. On the other hand, we note that the constant strategy approaches the optimum for larger deadlines. Indeed, the steady injection profile at rate Imax 2 is very similar to the one in the optimal strategy (depicted in Fig. 4b). Promotions are not adapted, concurring to increase the overall cost. Lastly, relying only on an initial set of seeders, without any additional injection during the dissemination duration (such as in the initial strategy), brings considerable efficiency drops. This happens because the set of initial leechers (which is inherently limited by I max ) cannot cover the entire network in T . Fig. 7 shows the trend of the cost functional J varying the contact rate λ. In general, the relative performance of the heuristics is the same as in the previous case.

Contact rate. Fig. 7 outlines the importance of the contact rate in the performance of the offloading strategy. λ is at the base of the opportunistic diffusion between mobile users. However, the contact rate depends on the mobility pattern of users, and can vary in time. λ may also include the uncertainties induced by the wireless channel and the movement of nodes. Operators should estimate the value of λ in order to adapt the optimal solution to current network conditions. In this context, offloading architectures that employ a feedback mechanism can prove very useful [START_REF] Valerio | Adaptive data offloading in opportunistic networks through an actor-critic learning method[END_REF], [START_REF] Whitbeck | Push-and-track: Saving infrastructure bandwidth through opportunistic forwarding[END_REF].

To give the reader an idea of the values at stake, the meeting time between nodes is nearly exponentially distributed when nodes move in a bounded region (of area A) according to a common mobility model [START_REF] Groenevelt | The message delay in mobile ad hoc networks[END_REF]. Under these assumptions, for the contact rate the following formula holds: λ N ≈ 8ωrv πA , where ω = 1.3683 for the random waypoint, ω = 1 for the random direction model, r is the transmission range, and v is the speed of the users. This means that, considering an area of 500 × 500 m 2 , 100 mobile users equipped with a transmission technology that allows direct transmissions up to 50 m and mobility of 25 km/h following the random waypoint model, we find a contact rate parameter λ = 0.488. If we consider, instead, real mobility traces from the Infocom dataset [START_REF] Scott | CRAWDAD data set cambridge/haggle (v. 2006-01-31)[END_REF], the average contact rate is λ = 0.14167.

VI. RELATED WORK

D2D offloading is an interesting method to reduce congestion in cellular networks [START_REF] Rebecchi | Data offloading techniques in cellular networks: A survey[END_REF]. Unicast and multicast D2D dissemination strategies are studied in [START_REF] Lin | Preference-aware content dissemination in opportunistic mobile social networks[END_REF], [START_REF] Boldrini | Contentplace: social-aware data dissemination in opportunistic networks[END_REF], [START_REF] Han | A new data transmission strategy in mobile d2d networks -deterministic, greedy, or planned opportunistic routing?[END_REF]. By adding the infrastructure in the picture, the focus shifts towards the selection of an optimal subset of users to kick-start dissemination [START_REF] Han | Mobile data offloading through opportunistic communications and social participation[END_REF], [START_REF] Barbera | Data offloading in social mobile networks through VIP delegation[END_REF]. However, these works consider only the optimal selection of the initial seed nodes without controlling the evolution of the dissemination. More related to our work, heuristics are presented in [START_REF] Whitbeck | Push-and-track: Saving infrastructure bandwidth through opportunistic forwarding[END_REF], [START_REF] Rebecchi | DROiD: Adapting to individual mobility pays off in mobile data offloading[END_REF]. A learning approach can help identify the best data carriers [START_REF] Valerio | Adaptive data offloading in opportunistic networks through an actor-critic learning method[END_REF] or broadcast modulation [START_REF] Rebecchi | A joint multicast/D2D learning-based approach to LTE traffic offloading[END_REF]. The optimization techniques presented in this work were initially developed as a solution of aeronautic problems, where engineers wanted to control a system by minimizing a certain performance index. Such techniques have not been sufficiently considered in the case of cellular data offloading. Instead, the works that are the closest to ours come from the literature on opportunistic diffusion [START_REF] Groenevelt | The message delay in mobile ad hoc networks[END_REF], [START_REF] Zhang | Performance modeling of epidemic routing[END_REF] and peer-to-peer networks [START_REF] Leibnitz | Modeling of epidemic diffusion in peer-to-peer file-sharing networks[END_REF], [START_REF] Fan | Stochastic differential equation approach to model bittorrent-like p2p systems[END_REF]. Optimal control has been employed to allocate resources (e.g., vaccines or security patches) to prevent the diffusion of epidemic diseases [START_REF] Behncke | Optimal control of deterministic epidemics[END_REF], [START_REF] Gaff | Optimal control applied to vaccination and treatment strategies for various epidemiological models[END_REF] or computer viruses [START_REF] Khouzani | Optimal dissemination of security patches in mobile wireless networks[END_REF]. The main difference to our work is that they aim at preventing the infection rather than encouraging it. Instead, applications of optimal control to boost epidemics exist in the area of marketing [START_REF] Kandhway | How to run a campaign: Optimal control of SIS and SIR information epidemics[END_REF], and opportunistic networks [START_REF] Khouzani | Optimal control of epidemic evolution[END_REF].

VII. CONCLUSION AND OUTLOOK

We proposed a novel analytical framework for opportunistic offloading capturing the differences between leechers and seeders. Following this approach, mobile operators are able to finely control the dissemination evolution through external controls such as cellular injections and promotions. We have shown the existence of solutions for the model. We applied Pontryagin's Maximum Principle to devise an optimal offloading strategy that minimizes the distribution costs. One of the main strengths of the model is that every parameter can be easily tuned. We analyzed the sensitivity to different values of contact rate and delay-tolerance, and evaluated the advantages of the proposed model over a simple two-state model. We provided evidence that when we have enough time flexibility, introducing a separation between leechers and seeders is strongly beneficial for the cellular operator.

Besides evaluating the optimal strategy in case of timevarying parameters, we believe that future developments can leverage the techniques presented in this paper to handle a more general case of stochastic diffusion processes following a Markov decision model. In this case, the evolution of the diffusion evolves following stochastic values, and the applied control depends on the observation of the system. The epidemic diffusion model can be also extended taking into account seeders that stop sharing content due to battery or storage constraints. Finally, a birth-death process can be included to represent the arrival and departure of users in the interest area.

APPENDIX TWO-STATE MODEL SOLUTION

The evolution of the state equations for the two-state model can be described using a single ODE to obtain the control u I that manages the cellular injections. Considering n I (t) = 1 -n S (t), the minimization problem becomes: . The injection control takes a similar form to the leechers-seeders model, with ψ(t) depending, this time, only on p * I (t).

Fig. 2 .

 2 Fig. 2. State transition rates for the leecher-seeder model.

  p I [λ (2n I -1 + n L ) -u I ] -p L u I , ∂n L -p I λn I + p L u P .

Fig. 3 .

 3 Fig. 3. Optimal offloading for different contact rates λ. T = 5s Other parameters: Imax = 0.1, α = 2, b = 10, c = 1, i 0 = 1, s 0 = 0.

Fig. 4 .

 4 Fig. 4. Optimal offloading for different contact rates λ. T = 10s. Other parameters: Imax = 0.1, α = 2, b = 10, c = 1, i 0 = 1, s 0 = 0.

Fig. 6 .

 6 Fig. 6. Cost functional J for different control strategies varying the deadline T . Other parameters: λ = 0.5 Imax = 0.05, α = 2, b = 10, c = 1, i 0 = 1, s 0 = 0.

Fig. 7 .

 7 Fig. 7. Cost functional J for different control strategies varying the contact rate λ. Other parameters: T = 10s, Imax = 0.05, α = 2, b = 10, c = 1, i 0 = 1, s 0 = 0.

  min u I (t)∈U J, subject to:(12a)∂n I ∂t = -λ(t)n I (t)(1 -n I (t)) -u I (t)n I (t), (12b)0 ≤ n I (t) ≤ 1, n I (t) + n S (t) = 1, n I (0) = i 0 , n S (0) = 1 -i 0 .(12c)We consider the same cost functions defined in Section IV-C. Employing Pontryagin's maximum principle, we find the Hamiltonian H, the adjoint function p I , and the transversality condition as follows:H = -b(n I u I ) α -c(λ n I (1 -n I )) (13a) + p I [-λn I (1 -n I ) -u I n I ], dp * I (t) dt =b α n α-1 I u α I + c λ (1 -2n I ) (13b) -p i [2λn I -λ -u I ] , p I (T ) =e n * I (T ) .(13c)By defining the function ψ(t) = α-1 p * I -αb , the Hamiltonian maximization condition (Eq. 7) along with the restrictions on the number of simultaneous transmissible copies through the infrastructure gives thatu * I (t) = min[max[ψ(t),0],Imax] n I (t)

TABLE I LIST

 I OF PARAMETERS.Network model. The system consists of N mobile nodes and one content to be distributed by the infrastructure to all nodes within the lifetime T . Intermediary nodes can be used as opportunistic relays. Following the notation introduced above, nodes can be in the interested, leecher, or seeder states. Their respective fractions are n I (t), n L (t), n S (t), and n I (t)+n L (t)+ n S

	Parameter Definition
	n I (t)	fraction of interested nodes at t
	n L (t)	fraction of leecher nodes at t
	n S (t)	fraction of seeder nodes at t
	λ(t)	contact rate at t
	u I (t)	direct injection rate at t
	u P (t)	promotion rate at t
	I max (t)	maximum injection rate at t
	T	content lifetime
	Φ(.)	final payoff function for interested nodes at T
	f (.)	instantaneous cost function for injection
	g(.)	instantaneous cost function for promotion
	fraction of them to the seeder state to diffuse the content. 1

Note that we do not align to the traditional SIR nomenclature. Instead, we use a nomenclature borrowed from peer-to-peer networks: "susceptible" users in the SIR model are analogous to interested in our model. Similarly, "infective" and "recovered" nodes are named respectively seeders and leechers.

As already mentioned, our system shares also several similarities with peerto-peer (P2P) networks, which features a large number of nodes downloading data from seeder nodes.

Throughout the paper, variables with the star superscript (e.g., n * I (t)) represent the optimum value.


Hilaire Pader'

Stéphanie Trouvé

Paolo Giovan

Lomazzo

Pader Like Lomazzo

Hilaire Pader

Hilaire Pader Tolosain

Peintre De L'altesse

Maurice De Savoye

Arnaud Toulouse

Colomiez

Gio Paolo Lomazzo

Lomazzo and

The Traicté de la proportion naturelle et artificielle des choses par Jean-Pol Lomazzo (1649), written by Hilaire Pader, is one of the earliest pieces of writing on art published in seventeenth century France. It is the first of the seven books that composed Lomazzo's Trattato della pitttura (1584). It concerns proportion. Pader hoped to translate the other six books of Lomazzo's treatise if the first was successful. Although the translation of the Trattato was never completed, Pader nevertheless followed up the ideas it contained in other works. In 1658, he published his Treatise on Painting in two parts: the Peinture parlante and the Songe énigmatique sur la peinture universelle. The aim of this article is to discuss the social and artistic stakes of Lomazzo's translation.

trained in the studio of the Municipal painter Jean Chalette (1581-1644), Pader went to Rome en 1635, where he worked in the workshop of the Sienese painter Niccolò Tornioli (1598-1652/1653), the official painter of Maurice of Savoy (1593-1657) from 1637 on. The Roman sojourn was decisive for Pader and transformed both his approach to his work and the way he thought about painting. It was during his five-year stay in the Eternal City that he discovered the writings of Lomazzo. Just as Lomaazzo's Trattato was dedicated to Charles Emmanuel, Duke of Savoy, Pader's translation was dedicated to his son, Maurice of Savoy. However, Pader was not the first to translate Lomazzo: at the very end of the sixteenth century, an Oxford scholar, Richard Haydocke had already translated the first five books of the Treatise into English. 3 Although Pader had hoped to translate the six following books of Lomazzo's Treatise, he stopped after the first, but developed the ideas it contained in his other works. He published a Treatise on Painting in two parts: the first in 1653, La Peinture parlante, 4 and the second in 1658, Le Songe énigmatique sur la peinture universelle. 5 The first is a didactic poem in which he develops the idea of proportion as the "first lesson" in painting. Pader was also largely inspired by the theory of passions (moti theory) used by Lomazzo. The second work used another of Lomazzo's books, the Idea del Tempio della pittura (1590). The diffusion of Lomazzo's ideas was, then, the result of an encounter between two men, the proportions of the human body, of women, of children, of horses and of architecture. Each of Lomazzo's precepts is based on the module of length of face. Depending on the canons used, there are figures whose body height is equivalent to that of six, seven, eight, nine or ten heads, according to principles established by Leonardo da Vinci and Albrecht Dürer. For Lomazzo, a human body equivalent to ten faces in height approached perfection. However, Pader attempted to make Lomazzo's theory more accessible by adding comments at the end of some chapters. He also simplified passages and made minor corrections in measurements. 6 Pader naively said that these mistakes had been made by the printer since Lomazzo was blind and could not read the proofs: Au demeurant, si dans la composition de ces mêmes figures, j'ai pris la liberté de corriger les faux préceptes qu'on a fait glisser dans l'impression de mon auteur, touchant les diverses proportions des corps ; ce n'est pas que j'ai oublié le respect que je dois à l'autorité d'un si grand homme ; ou que je veuille donner de nouvelles lumières à cet aveugle Soleil. Mais c'est parce que j'ai cru avec raison que les endroits où j'ai changé quelque chose, étaient contraires aux sentiments du Lomasse, qui n'avait pu corriger ces défauts à cause de celui de sa vue. 7 Finally, for didactic reasons, and to illustrate the different canons, he added engravings after Dürer, which allowed the reader to better understand Lomazzo's precepts. 8 The illustrations also show the practical application of theory and demonstrate that it was not merely a constraint.

Pader did not continue the literal translation of the Treatise, but in his other writings, he used the theory of Lomazzo to write a discourse on art based on his own experience and culture. In the Peinture parlante, whose title refers to the aphorism of Simonides as reported in Plutarch that painting is silent poetry and poetry talking painting, Pader, who was also a poet, tries to explain in verse, how the painter, like the poet, can successfully imitate human nature in action. Pader imagines a discussion with his son and student to whom he gives advice on how to reach perfection in painting. He structures his poem in two lessons. The first is devoted to the study of proportion, the second to the moti which are the "movement of the body and soul or figurative expression," considered by Lomazzo as "the most difficult to 6 Pader, Traicté de la proportion..., 1649, chapter VIII : De la Proportion extravagante de dix Testes, 26. 7 Pader, Traicté de la proportion..., 1649, Discours sur le sujet de cette traduction, unpaged. 8 Ibid.

attain in all art, and also the most important and necessary to know" (Trattato, p. 97-98).

Pader describes the physiological characteristics of many emotions (melancholy, obstinate anger, cruelty, avarice, invincibility, strength, constancy, justice, devotion…), illustrating each of them with an exemple taken from sacred history." [START_REF] Jean | Idea of the Temple of Painting[END_REF] In the Songe énigmatique sur la peinture universelle (An Enigmatic Dream about Universal Painting), Pader tells of a dream that turns into an initiatory journey in the palace of painting surrounded by a mystical garden. In the garden, there is a tree at every corner symbolising the artist most accomplished in each of the six parts of painting. This recalls Lomazzo's Idea del tempio della pittura, itself inspired by the Idea del theatro (Florence, 1550) by Giulio Camillo, a humanist working first in Venice and later in France at the court of King Francis the First in the 1530s. [START_REF]On the symbolic use of the temple of painting, see: Sylvie Deswarte-Rosa, "Idea et le Temple de la Peinture. I. Michelangelo Buonarrti et Francesco de Hollanda[END_REF] The association of an idea and a place in the creation of a pictorial narrative had, of course, already been employed by orators in antiquity as a mnemonic device. [START_REF]On the use of this method[END_REF] In his dream, Pader develops his argument as he leads us through each floor of the building, going from room to room, placing each idea in a part of the painting.

The narrator enters the palace accompanied by Saturn. He climbs a staircase that is tortuous and dark on the way up, but less obscure on the way down: the staircase is a metaphor for the ascent to knowledge. The narrator is then led into galleries. In the first, young boys are busy preparing colors. Another staircase leads to the next stage in a gallery bathed in light. Pader also renews the governors of the Temple of the Painting cherished by Lomazzo, and includes contemporary painters like Guido Reni or Nicolas Poussin, the "French Apelle," as well as Renaissance masters. If Pader's original purpose was to disseminate the ideas of Lomazzo through his writings, he must also have understood that theory was a way of promoting painting as a liberal art as well as a means of self-promotion.

Artistic Issues: Rules to Follow

At the beginning of the Treaty of Proportions Pader explains the reasons that led him to translate Lomazzo. [START_REF] Pader | Traicté de la proportion[END_REF] He says that the Trattato was the only book to use precepts on proportion, movement, color, light, and perspective in order to teach painting. It was the way forward that would bring the student "wisely to the Temple of Truth" and to "excellence in Painting." 13 It is the very complete nature of the Trattato, from the point of view of both practice and theory, that appealed to Pader because it allowed the young apprentice to perfect himself in all aspects of painting. 14 The question of the importance of mastering proportions had also been discussed in Paris in 1634 because it corresponded to a need to represent forms invented by the intellect. [START_REF] Thuillier | Doctrines et querelles artistiques en France au XVII e siècle : quelques textes oubliés ou inédits[END_REF] At the time as Pader was writing, the status of painters in France, and especially in Toulouse, was that of craftsmen and their craft was regulated by strict rules. Thus, by introducing new theoretical precepts, Pader offered painters an opportunity to escape a routine based on image reproduction and to distance themselves from their practice [START_REF]La construction de l'école de Toulouse XVII e -XVIII e siècles which will[END_REF] . The desire to bring a change to methods so markedly different from the practices of traditional workshops was perceptible as of 1641 when, returning from Rome, Pader established at Toulouse an Academy of drawing from life with the painter Jean André and the sculptor Guillaume Fontan. 17 There were no artistic treatises in French at the time even though Abraham Bosse had translated some artistic terms from the Italian, such as "vaghezza" translated to mean "union" in the Sentiments sur la distinction des diverses manières de peinture... . [START_REF] Le | D'acide et d'encre. Abraham Bosse (1604 ?-1676) et son siècle en perspectives[END_REF] It was only in 1651 that Pierre Daret translated of the Life of Raphael according to Vasari, [START_REF]Abrégé de la vie de Raphael Sansio d'Urbin, très-excellent peintre et architecte, où il est traitté de ses oeuvres, des stampes qui en ont été gravées, tant par Marc Antoine Bolognais, qu'autres excellents graveurs, traduit d'italien en françois par Pierre Daret[END_REF] a task that Pader had also undertaken (he translated seven lives of Vasari that remained unpublished and are lost today). The same year, Roland Fréart de Chambray published a translation of the Treatise on Painting by Leonardo da Vinci. These works were written in the particular context of the creation of the French Royal Academy of Painting and Sculpture in 1648, where they contributed to developing a doctrine. [START_REF] Marianne | Abraham Bosse et Léonard de Vinci. Les débats sur les fondements de la peinture dans les premiers temps de l'Académie[END_REF] But gradually the French Academy changed its teaching so that it was more focused on direct observation of masterpieces, unlike Italian art theory, which was based on concepts. 21 In addition to practical rules, Pader defended the idea of the liberal conception of painting, saying that ideas should come "before the painter took pencil in hand." 22 This ability to compose a concetto was reserved for the "great minds among scholars." ("beaux esprits des hommes seulement litterez"). The next phase, being the execution of the idea by the hand was deemed exclusively reserved for painters. In Pader's own practice, this led him to a write projects for two paintings for the decoration of the chapel of the Black Penitents of Toulouse.

The first picture represented the Flood (now lost), while the other represents the Triumph of Joseph (Toulouse, St. Stephen's Cathedral). Before painting this work of art, Pader formulated a concetto. He meant to explain the outline of the composition, the arrangement of figures, the facial expressions to be represented, and the dominant colors. 23 Following Lomazzo's theory, Pader wanted to recount the story faithfully respecting the disposition of the figures. In the Triumph of Joseph, a frieze was required for the representation of triumph and it had to be as readable as possible. The painter also had to take into account the anatomy, age, dress appropriate to the wealth and rank of Joseph, and to include the various participants in the scene (musicians, common people, and citizens) with their expressions of triumph, joy, good cheer, and majesty.

The theoretical and artistic issues of translating and disseminating Lomazzo was also a way for Pader to affirm his status as scholar-painter in a quest for recognition from his peers at the Royal Academy.

Social Issues

This new interest in theory contributed to a rise in the social status of the artist. Pader was not merely attempting to train painters. He also wanted to provide the keys to understanding his own paintings, and he continued to paint as he developed his theory of painting. Theory, then, was a way to justify the quality of his paintings and place himself as a connoisseur. Thus, in a letter written to the Brotherhood of Black Penitents of Toulouse who had criticized one of his paintings representing the Flood [le 'Déluge'], he tried to explain that for a work of art the subject was not the only important thing. For example, to justify his use 21 Jacqueline Lichtenstein, "De l'idée de la peinture à l'analyse du tableau : une mutation essentielle de la théorie de l'art," Revue d'esthétique, 31/32, 1997, 17-35. 22 Pader, La Peinture parlante, Explication des mots et termes..., 1653, unpaged 23 Ibid.

of dark colors in the painting he turned to the theory of propriety, claiming that colors should be appropriate to the subject. In the case of the Flood, which took place during the night, a black tone was the most suitable. Using Lomazzo's theory of proportion, he also wanted to convince spectators to observe the figures in the middle ground and background, rather than simply seeing the foreground. 24 Unlike other painters, Pader distinguished himself as a specialist in painting because he was well-acquainted with Lomazzo's theory.

He was thus one of the first artists to be admitted to the Royal Academy of Painting and Sculpture in 1659 with the presentation of a painting, The General Peace under the Reign of Augustus (which has disappeared), as well as his written works. The title of Academician gave him the status of a true artist ("vrai artiste") but he did not make a career in Paris. It was in Toulouse that he needed to display his knowledge of Lomazzo's theory and to claim his status as scholar-painter. Pader aspired to the title of Municipal painter, a prestigious position in a provincial town that offered privileges, such as a salary, housing, and a workshop. In 1661, he was hired by the city to do the portraits of its representatives thanks to his sojourn in

French

  artist Pader and his first patron Maurice of Savoy. It also raises a number of questions as to artistic and social issues. By what means did Pader make Lomazzo's theory accessible and how did he disseminate Lomazzo's ideas? Did knowledge of the rules developed by Lomazzo guarantee the quality of a painting? Finally, how was Lomazzo's theory used by the artist to invent himself as a scholar-painter? Translation and Assimilation of the Ideas of Lomazzo Pader mastered the Italian language well enough to translate the technical terms and specific vocabulary of art. The Book of Proportions is a largely faithful translation of the thirty two chapters of Lomazzo's book which deal, in an encyclopedic manner, with the licentia de'Superiori. According to R. Klein (1974, II, 468). There are four printed versions of the first edition. Pader translated the 1585 version.

  Italy and his reception at the Academy.25 To conclude, what is important is to show how Lomazzo's texts were transformed, what they became, and how they were reappropriated. Thus to translate Lomazzo was not only to find equvalents for Italian words in French, but to convey a new, liberal, conception of painting to French readers. By translating the first book of the Trattato and using passages from other books like the Trattato or the Idea del Tempio della pittura to build his own work, Pader contributed to the changing status of the painter that began in the middle of the seventeenth century in France. Pader's books initially aimed to disseminate Lomazzo's theory in France, but subsequently served his own artistic ambitions. This must be seen in the specific context of the beginnings of the French Royal Academy and the Toulousain circle.Because his status as an artist could not be established by his painting alone, Pader also used books and words.24 Hilaire Pader, Plan ou Dessein idéal pour le tableau du Deluge qui doit être representé dans la Chapelle de Messieurs les Penitens Noirs de Tolose, undated; reprinted Geneva, Minkoff 1973, 9. 25 Tlse, Arch. Mun., BB 37, fol. 173-174, "Délibération du 17 janvier 1661"; exhibition catalogue. Toulouse, 1953, 23. See Stéphanie Trouvé, "La leçon d'Hilaire Pader sur le portrait selon Lomazzo : l'introduction des albums des Portraits et armes des Présidents et Conseillers au Parlement de Toulouse (1664)," Studiolo, 7, 2009, 139-160.

3

  Tracte Containing the Artes of Curious Paintinge Carvinge & Buildinge, Written First in Italian by Jo-Paul

	Lomatius painter of Milan and Englished by R[ichard] H[aydocke] Student in Physik, Oxford, Joseph Barnes,
	1598.
	4 Hilaire Pader, La Peinture parlante, Toulouse, A. Colomiez, 1653; reedited 1657; reprinted Geneva, Minkoff,
	1973, as La Peinture parlante, suivi du Songe énigmatique sur la peinture universelle, suivi du Contrat du
	Déluge.
	5 Hilaire Pader, Le Songe énigmatique sur la peinture universelle, Toulouse, A. Colomiez, 1658; reprinted
	Geneva, Minkoff, 1973.