Marco Büchler

Laurence Mellerin

Intertextuality in Ancient Languages JDMDH Special Issue on Computer-Aided Processing of Intertextuality in Ancient Languages

HAL is

Part 1: Towards a Digital Ecosystem: NLP.

Corpus infrastructure. Methods for Retrieving Texts and Computing Text Similarities

• Methods for the detection of intertexts and text reuse, manual (e.g. crowd-sourcing) or automatic (e.g. algorithms); • Infrastructure for the preservation of digital texts and quotations between different text passages; • Linguistic preprocessing and data normalisation, such as lemmatisation of historical languages, root stemming, normalisation of variants, etc.

1) Identification of Parallel Passages Across a Large Hebrew/Aramaic Corpus

Authors: Shmidman, Avi and Koppel, Moshe and Porat, Ely

We propose a method for efficiently finding all parallel passages in a large corpus, even if the passages are not quite identical due to rephrasing and orthographic variation. The key ideas are the representation of each word in the corpus by its two most infrequent letters, finding matched pairs of strings of four or five words that differ by at most one word and then identifying clusters of such matched pairs. Using this method, over 4600 parallel pairs of passages were identified in the Babylonian Talmud, a Hebrew-Aramaic corpus of over 1.8 million words, in just over 30 seconds. Empirical comparisons on sample data indicate that the coverage obtained by our method is essentially the same as that obtained using slow exhaustive methods.

2) Interactive Tools and Tasks for the Hebrew Bible

Authors: Winther-Nielsen, Nicolai

Computer-aided processing of intertextuality offers new promising tools for the visualization of the sources and for the performance of tasks that would barely be possible to do without a digital solution. This contribution explores how the corpus of the Hebrew Bible created and maintained by the Eep Talstra Center for Bible and Computer can support new ways in which we can learn from our ancient texts as modern knowledge workers. It first describes how the corpus was used for the development of Bible Online Learner as a persuasive technology to enhance language learning with, in, and around a database that drives interactive tasks for learners. The achievements obtained through this project so far are very promising. Interactive corpus-technology also has an important bearing on the task of textual criticism as an increasingly specialized area of research that depends on the availability of digital resources.

Commercial solutions developed by software companies like Logos offer advanced digital scholarly resources from the German Bible Society as a useful alternative to often inaccessible and expensive print versions. Corpus-driven learning and new digital resources will also allow scholars to do new academic tasks in textual criticism and interpretation, and we already now see promising tools for text categorization, analysis of translation shifts, and interpretation emerge as the potential models for the future. The main goal in the future will be to provide easier and more affordable global access for these new tools.

3) Preprocessing Greek Papyri for Linguistic Annotation

Authors: Vierros, Marja and Henriksson, Erik

Greek documentary papyri form an important direct source for Ancient Greek. It has been exploited surprisingly little in Greek linguistics due to a lack of good tools for searching linguistic structures. This article presents a new tool and digital platform, " Sematia " , which enables transforming the digital texts available in TEI EpiDoc XML format to a format which can be morphologically and syntactically annotated (treebanked), and where the user can add new metadata concerning the text type, writer and handwriting of each act of writing. An important aspect in this process is to take into account the original surviving writing vs. the standardization of language and supplements made by the editors. This is performed by creating two different layers of the same text. The platform is in its early development phase. Future developments, such as tagging linguistic variation phenomena as well as queries performed within Sematia, are discussed at the end of the article.

4) Text Alignment in Ancient Greek and Georgian: A Case-Study on the First Homily of Gregory of Nazianzus

Authors: Pataridze, Tamara and Kindt, Bastien

This paper discusses the word level alignment of lemmatised bitext consisting of the Oratio I of Gregory of Nazianzus in its Greek model and Georgian translation. This study shows how the direct and empirical observations offered by an aligned text enable an accurate analysis of techniques of translation and many philological parameters of the text.

5) Recurrent Pattern Modelling in a Corpus of Armenian Manuscript Colophons

Authors: Van Elverdinghe, Emmanuel

Colophons of Armenian manuscripts are replete with yet untapped riches. Formulae are not the least among them: these recurrent stereotypical patterns conceal many clues as to the schools and networks of production and diffusion of books in Armenian communities. This paper proposes a methodology for exploiting these sources, as elaborated in the framework of a PhD research project about Armenian colophon formulae. Firstly, the reader is briefly introduced to the corpus of Armenian colophons and then, to the purposes of our project. In the third place, we describe our methodology, relying on lemmatization and modelling of patterns into automata. Fourthly and finally, the whole process is illustrated by a basic case study, the occasion of which is taken to outline the kind of results that can be achieved by combining this methodology with a philologico-historical approach to colophons.

6)

7) Integrated Sequence Tagging for Medieval Latin Using Deep Representation Learning

Authors: Kestemont, Mike and De Gussem, Jeroen

In this paper we consider two sequence tagging tasks for medieval Latin: part-of-speech tagging and lemmatization. These are both basic, yet foundational preprocessing steps in applications such as text re-use detection. Nevertheless, they are generally complicated by the considerable orthographic variation which is typical of medieval Latin. In Digital Classics, these tasks are traditionally solved in a (i) cascaded and (ii) lexicon-dependent fashion. For example, a lexicon is used to generate all the potential lemma-tag pairs for a token, and next, a context-aware PoStagger is used to select the most appropriate tag-lemma pair. Apart from the problems with outof-lexicon items, error percolation is a major downside of such approaches. In this paper we explore the possibility to elegantly solve these tasks using a single, integrated approach. For this, we make use of a layered neural network architecture from the field of deep representation learning.

8) Measuring and Mapping Intergeneric Allusion in Latin Poetry using Tesserae

Authors: Burns, Patrick J.

Most intertextuality in classical poetry is unmarked, that is, it lacks objective signposts to make readers aware of the presence of references to existing texts. Intergeneric relationships can pose a particular problem as scholarship has long privileged intertextual relationships between works of the same genre. This paper treats the influence of Latin love elegy on Lucan's epic poem, Bellum Civile, by looking at two features of unmarked intertextuality: frequency and distribution. I use the Tesserae project to generate a dataset of potential intertexts between Lucan's epic and the elegies of Tibullus, Propertius, and Ovid, which are then aggregrated and mapped in Lucan's text. This study draws two conclusions: 1. measurement of intertextual frequency shows that the elegists contribute fewer intertexts than, for example, another epic poem (Virgil's Aeneid), though far more than the scholarly record on elegiac influence in Lucan would suggest; and 2. mapping the distribution of intertexts confirms previous scholarship on the influence of elegy on the Bellum Civile by showing concentrations of matches, for example, in Pompey and Cornelia's meeting before or during the affair between . By looking at both frequency and proportion, we can demonstrate systematically the generic enrichment of Lucan's Bellum Civile with respect to Latin love elegy.

9) A Hackathon for Classical Tibetan

Authors: Almogi , Orna and Dankin , Lena and Dershowitz , Nachum and Wolf , Lior

We describe the course of a hackathon dedicated to the development of linguistic tools for Tibetan Buddhist studies. Over a period of five days, a group of seventeen scholars, scientists, and students developed and compared algorithms for intertextual alignment and text classification, along with some basic language tools, including a stemmer and word segmenter.

Part 2: Managing different types of text reuses

This part focuses on the conceptual definitions, the modelling of the unstable idea of "quotation" and the XML-TEI encoding to implement for its characterization.

1) TEI-encoding of text reuses in the BIBLINDEX Project

Authors: Hue-Gay, Elysabeth and Mellerin, Laurence and Morlock, Emmanuelle

This paper discusses markup strategies for the identification and description of text reuses in a corpus of patristic texts related to the BIBLINDEX Project, an online index of biblical references in Early Christian Literature. In addition to the development of a database that can be queried by canonical biblical or patristic references, a sample corpus of patristic texts has been encoded following the guidelines of the TEI (Text Encoding Initiative), in order to provide direct access to quoted and quoting text passages to the users of the https://www.biblindex.info platform.

2) Encoding (inter)textual insertions in Latin "grammatical commentary"

Authors: Bureau, Bruno and Nicolas, Christian and Pinche, Ariane

The ancient commentaries provide a large sample of quotations from classical or biblical texts for which Latin gramamrians developed a complex system of insertion of quoted texts. The paper examines how to encode these places using XML Tei, and focuses on difficult cases, such as inaccurate quotations, or quotations of partly or wholly lost texts.

3) A Classification of Manuscripts Based on A New Quantitative Method. The Old Latin Witnesses of John's Gospel as Text Case

Authors: Pastorelli, David

A new method for grouping manuscripts in clusters is presented with the calculation of distances between readings, then between witnesses. A classification algorithm (" Hierarchical Ascendant Clustering "), achieved through computer-aided processing, enables the construction of trees illustrating the textual taxonomy obtained. This method is applied to the Old Latin witnesses of the Gospel of John, and, in order to provide a study of a reasonable size, to a chapter as a whole (chapter 14). The result basically confirms the text-types identified by Bonatius Fischer, founder of the Vetus Latina Institute, while it invalidates the classification adopted by the current edition of the Vetus Latina of the Gospel of John.

Part 3: Visualisation of intertextuality and text reuse 1) Version Variation Visualization (VVV): Case Studies on the Hebrew Haggadah in English

Authors: Cheesman, Tom and Roos, Avraham,

The 'Version Variation Visualization' project has developed online tools to support comparative, algorithm-assisted investigations of a corpus of multiple versions of a text, e.g. variants, translations, adaptations (Cheesman, 2015(Cheesman, , 2016;;Cheesman et al., 2012Cheesman et al., , 2012Cheesman et al., -13, 2016;;Thiel, 2014;links: www.tinyurl.com/vvvex). A segmenting and aligning tool allows users to 1) define arbitrary segment types, 2) define arbitrary text chunks as segments, and 3) align segments between a 'base text' (a version of the 'original' or translated text), and versions of it. The alignment tool can automatically align recurrent defined segment types in sequence.Several visual interfaces in the prototype installation enable exploratory access to parallel versions, to comparative visual representations of versions' alignment with the base text, and to the base text visually annotated by an algorithmic analysis of variation among versions of segments. Data can be filtered, viewed and exported in diverse ways. Many more modes of access and analysis can be envisaged. The tool is language neutral. Experiments so far mostly use modern texts: German Shakespeare translations. Roos is working on a collection of approx. 100 distinct English-language translations of a Hebrew text with ancient Hebrew and Aramaic passages: the Haggadah (Roos, 2015).

2) Visualizing linguistic variation in a network of Latin documents and scribes

Authors: Korkiakangas , Timo and Lassila , Matti

This article explores whether and how network visualization can benefit philological and historical-linguistic study. This is illustrated with a corpus-based investigation of scribes' language use in a lemmatized and morphologically annotated corpus of documentary Latin (Late Latin Charter Treebank, LLCT2). We extract four continuous linguistic variables from LLCT2 and utilize a gradient colour palette in Gephi to visualize the variable values as node attributes in a trimodal network which consists of the documents, writers, and writing locations underlying the same corpus. We call this network the "LLCT2 network". The geographical coordinates of the location nodes form an approximate map, which allows for drawing geographical conclusions. The linguistic variables are examined both separately and as a sum variable, and the visualizations presented as static images and as interactive Sigma.js visualizations. The variables represent different domains of language competence of scribes who learnt written Latin practically as a second-language. The results show that the network visualization of linguistic features helps in observing patterns which support linguisticphilological argumentation and which risk passing unnoticed with traditional methods. However, the approach is subject to the same limitations as all visualization techniques: the human eye can only perceive a certain, relatively small amount of information at a time. The software programs generally used with the TLG (Thesaurus Linguae Graecae) and the CLCLT (CETEDOC Library of Christian Latin Texts) CD-ROMs are not well suited for finding quotations and allusions. QuotationFinder uses more sophisticated criteria as it ranks search results based on how closely they match the source text, listing search results with literal quotations first and loose verbal parallels last.

2) Digital Greek Patristic Catena (DGPC). A brief presentation

Authors: Paparnakis, Athanasios and Domouchtsis, Constantinos

The project is to develop a database, which is planned to include all available information on the use of the Bible in the patristic works of Migne's Patrologia Graeca. Utilization of the data will be available through a web page equipped with necessary tools for developing data mining techniques and other methods of analysis. The main aim of the project is to revive the catenae, the ancient exegetical tool for biblical interpretation.

3) Dealing with all types of quotations (and their parallels) in a closed corpus: The methodology of the Project The literary tradition in the third and fourth centuries CE: Grammarians, rhetoricians and sophists as sources of Graeco-Roman literature

Authors: Rodríguez-Noriega, Lucía
The Project The literary tradition in the third and fourth centuries CE: Grammarians, rhetoricians and sophists as sources of Graeco-Roman literature (FFI2014-52808-C2-1-P) aims to trace and classify all types of quotations, both explicit (with or without mention of the author and/or title) and hidden, in a corpus comprising the Greek grammarians, rhetoricians and " sophists " of the third and fourth centuries CE. At the same time, we try to detect whether or not these are first-hand quotations, and if our quoting authors (28 in all) are, in turn, secondary sources for the same citations in later authors. We also study the philological (textual) aspects of the quotations in their context, and the problems of limits they sometimes pose. Finally, we are interested in the function of the quotation in the citing work. This is the first time that such a comprehensive study of this corpus is attempted. This paper explains our methodology, and how we store all these data in our electronic card-file.

4) Editing New Testament Arabic Manuscripts in a TEIbase: fostering close reading in Digital Humanities

Authors: Clivaz, Claire and Schulthess, Sara and Sankar, Martial

If one is convinced that " quantitative research provides data not interpretation " [Moretti, 2005, 9], close reading should thus be considered as not only the necessary bridge between big data and interpretation but also the core duty of the Humanities. To test its potential in a neglected field -the Arabic manuscripts of the Letters of Paul of Tarsus -an enhanced, digital edition has been in development as a progression of a Swiss National Fund project. This short paper presents the development of this edition and perspectives regarding a second project. Based on the Edition Visualization Technology tool, the digital edition provides a transcription of the Arabic text, a standardized and vocalized version, as well as French translation with all texts encoded in TEI XML. Thanks to another Swiss National Foundation subsidy, a new research project on the unique New Testament, trilingual (Greek-Latin-Arabic) manuscript, the Marciana Library Gr. Z. 11 (379), 12th century, is currently underway. This project includes new features such as " Textlink " , " Hotspot " and notes: HumaReC.

5) Bioinformatics and Classical Literary Study

Authors: Chaudhuri, Pramit and Dexter, Joseph P.

This paper describes the Quantitative Criticism Lab, a collaborative initiative between classicists, quantitative biologists, and computer scientists to apply ideas and methods drawn from the sciences to the study of literature. A core goal of the project is the use of computational biology, natural language processing, and machine learning techniques to investigate authorial style, intertextuality, and related phenomena of literary significance. As a case study in our approach, here we review the use of sequence alignment, a common technique in genomics and computational linguistics, to detect intertextuality in Latin literature. Sequence alignment is distinguished by its ability to find inexact verbal similarities, which makes it ideal for identifying phonetic echoes in large corpora of Latin texts. Although especially suited to Latin, sequence alignment in principle can be extended to many other languages.

6) Computer -Assisted Processing of Intertextuality in Ancient Languages

Authors: Hedges, Mark and Jordanous, Anna and Lawrence, K. Faith and Roueché, Charlotte and Tupman, Charlotte

The production of digital critical editions of texts using TEI is now a widely-adopted procedure within digital humanities. The work described in this paper extends this approach to the publication of gnomologia (anthologies of wise sayings) , which formed a widespread literary genre in many cultures of the medieval Mediterranean. These texts are challenging because they were rarely copied straightforwardly ; rather , sayings were selected , reorganised , modified or re-attributed between manuscripts , resulting in a highly interconnected corpus for which a standard approach to digital publication is insufficient. Focusing on Greek and Arabic collections , we address this challenge using semantic web techniques to create an ecosystem of texts , relationships and annotations , and consider a new model -organic , collaborative ,

interconnected , and open-ended -of what constitutes an edition. This semantic web-based approach allows scholars to add their own materials and annotations to the network of information and to explore the conceptual networks that arise from these interconnected sayings .

As this special issue allows continuous updates, it is still possible to add a contribution if you are working on these topics.

7) Using the Text Alignment Network for Scholarship on Intertextuality

Authors: Kalvesmaki, Joel

The Text Alignment Network (TAN) is a suite of XML encoding formats intended to serve anyone who wishes to encode, exchange, and study translations, paraphrases, adaptations, quotations, and other varieties of text reuse. This article briefly introduces TAN, and in the spirit of the special issue of this journal focuses on the syntax of its intertextual pointers, which are styled to be both human-readable and -interoperable. Because TAN is at present an experimental format, this report notes progress, promise, and future prospects.

8) Processing Tools for Greek and Other Languages of the Christian Middle East

Authors: Kindt, Bastien

This paper presents some computer tools and linguistic resources of the GREgORI project. These developments allow automated processing of texts written in the main languages of the Christian Middel East, such as Greek, Arabic, Syriac, Armenian and Georgian. The main goal is to provide scholars with tools (lemmatized indexes and concordances) making corpus-based linguistic information available. It focuses on the questions of text processing, lemmatization, information retrieval, and bitext alignment.

From manuscript catalogues to a handbook of Syriac literature: Modeling an infrastructure for Syriaca.org

  Despite increasing interest in Syriac studies and growing digital availability of Syriac texts, there is currently no up-to-date infrastructure for discovering, identifying, classifying, and referencing works of Syriac literature. The standard reference work (Baumstark's Geschichte) is over ninety years old, and the perhaps 20,000 Syriac manuscripts extant worldwide can be accessed only through disparate catalogues and databases. The present article proposes a tentative data model for Syriaca.org's New Handbook of Syriac Literature, an open-access digital publication that will serve as both an authority file for Syriac works and a guide to accessing their manuscript representations, editions, and translations. The authors hope that by publishing a draft data model they can receive feedback and incorporate suggestions into the next stage of the project.

Authors

: Gibson, Nathan P. and Michelson, David A. and Schwartz, Daniel L.

and the University of Göttingen and SUB Göttingen (e-TRAP), with the support of the National Research Agency (ANR Biblindex) and the Partner University Fund (PUF). establishing an exhaustive statement of the biblical references found in the texts of the Late Antiquity and the Middle Ages.

Contact

Marco Büchler: mbuechler(at)etrap(dot)eu Laurence Mellerin: laurence.mellerin(at)mom(dot)fr See http://jdmdh.episciences.org/volume/view/id/158


Gwen Maudet
email: gwen.maudet@imt-atlantique.fr

Mireille Batton-Hubert

Patrick Maille

Laurent Toutain

Emission scheduling strategies for massive-IoT: implementation and performance optimization

In today's monitoring solutions, each application involves custom deployment and requires significant configuration efforts to accommodate sensor changes. In contrast, in this paper, a massive deployment of battery-powered sensors is considered and we propose a more versatile monitoring solution that is not tied to physical device deployment.

We characterize a monitoring strategy by formally defining a function that modifies the transmission period of a sensor that has just transmitted. Such a function can be customized to manage the tradeoff between overall monitoring accuracy and sensor energy consumption, for which we suggest a formalization through a generic metric of monitoring accuracy (to be weighed versus the monitoring network lifespan).

We introduce a specific two-parameter instantiation for the period update function, that ensures strictly periodic emissions from sensors even when new sensors join the system over time. We show through simulations how the two parameters-target emission period and number of jointly used sensors-can be chosen according to the objectives for the monitoring, by highlighting the Pareto front for accuracy and energy-efficiency.

Advances in electronics and signal processing have enabled the miniaturization of hardware. Thanks to the emergence of new energy-saving communication techniques, a new class of applications has arisen: miniaturized devices deployed on a large scale capable of sensing their environment [START_REF] Carminati | Trends and paradigms in the development of miniaturized sensors for environmental monitoring[END_REF], [START_REF] Imam Nashiruddin | Coverage and capacity analysis of LoRa WAN deployment for massive IoT in urban and suburban scenario[END_REF], [START_REF] Varsier | Capacity limits of LoRaWAN technology for smart metering applications[END_REF].

In a classical approach, a few highly reliable sensors are placed at the points of relevance, so as to provide meaningful information. This solution is static and rigid in time. In contrast, the so-called massive Internet of Things (mIoT) considers a large quantity of cheap energy-autonomous sensors, with no prior information about their quality or position. This paradigm shift is a game-changer for the development of monitoring solutions, as it requires great flexibility in sensor management [START_REF] Gubbi | Internet of things (IoT): A vision, architectural elements, and future directions[END_REF], [START_REF] Hurlburt | The internet of things: A reality check[END_REF], [START_REF] Puccinelli | Wireless sensor networks: applications and challenges of ubiquitous sensing[END_REF]. This allows the development of versatile solutions, independent of the physical development.

Overview of research on wireless sensor energy efficiency

Sensors are usually battery-powered, whose energy is consumed during the transmission of information. The more the data transmitted, the more accurately the environment is monitored but at the same time, the faster the energy is consumed. Due to these energy limitations, it is necessary to propose efficient management of sensor emissions, finding the compromise between monitoring quality and resource consumption. First, from the early 2000s, energy-saving functionalities for devices have been proposed [START_REF] Alippi | Energy management in wireless sensor networks with energy-hungry sensors[END_REF], [START_REF] Raghunathan | Emerging techniques for long lived wireless sensor networks[END_REF]. Triggered and adaptive sensing methods allow the sensor's sampling rate to be adapted to variations in the environment.

In 2018, the authors of [START_REF] Preeth | An adaptive fuzzy rule based energy efficient clustering and immune-inspired routing protocol for WSN-assisted IoT system[END_REF] propose to aggregate the information provided by multiple surrounding sensors to a single collector. The paper uses the fuzzy methods AHP and TOPSIS to determine cluster-heads that forward the aggregated information to the gateway. This energy-efficient solution significantly reduces the gateway traffic load. Other information aggregation methods, based on index tree structures, are proposed in [START_REF] Tang | An energy efficient hierarchical clustering index tree for facilitating time-correlated region queries in wireless sensor network[END_REF], [START_REF] Tang | EGF-tree: An energy efficient index tree for facilitating multi-region query aggregation in the internet of things[END_REF]. Those last approaches are particularly suitable for heterogeneous geographical distributions of sensors.

In 2014, [START_REF] Akgül | Self-organized things (SoT): An energy efficient next generation network management[END_REF] introduces the concept of "Self-Organized Things", where sensors are self-managed, following energy optimization mechanisms. Those mechanisms allow sensors to be put into sleep mode if their spatial coverage is already guaranteed by other active sensors. In 2017, [START_REF] Kaur | An energy-efficient architecture for the internet of things (IoT)[END_REF] proposes a complete energyefficient hierarchical architecture for IoT. The paper develop a continuous modification of the sensors' sleep times, based on the battery level, the standard variation of the returned values and the distance to other sensors. New results of [START_REF] Kaur | An energy-efficient architecture for the internet of things (IoT)[END_REF] are proposed in 2019 [START_REF] Gupta | Energy-efficient model for deployment of sensor nodes in IoT based system[END_REF], validating the relevance of the sleep mode usage.

The solutions presented above lack applicability in a context such as mIoT monitoring. Indeed, it is important to consider the following characteristics:

• mIoT solutions are mostly based on LPWANs, as the least energy intensive network model. This is a star architecture, where direct sensor-to-sensor communications are not allowed.

• Sensors are built to send periodic messages, and their transmission period can only be changed during a short window-time after a message is sent to the gateway.

• One of the main goals we can expect from mIoT is automatic deployment. Therefore, it is needed to handle dynamic integration of sensors without having any prior knowledge about location.

To the best of our knowledge, there is no solution for the problem of optimally monitoring an environment under these hypotheses.

Positioning and development of the proposed solution

In this paper, we consider a large amount of "things" deployed in an area, that we want to use for monitoring purposes. The goal is to minimize the gap between the environment to monitor and its representation by the system. This is in general at the cost of more measures being sent by sensors, hence a tradeoff between the accuracy of the monitoring and the energy consumption.

The sensors we consider are transmitting messages periodically and are in sleep mode between each message, to save energy. Following the behavior of LoRaWAN class A objects, a listening window is open after each message sent, during which the monitoring system can modify the emission period. Sensors have limited energy in their battery, that is consumed over time. Note that the system does not know how many sensors will be in use, nor their respective positions, and will have to integrate them in the monitoring.

In this paper, we first introduce a standard formulation of a monitoring strategy, through the formal definition of a period update function, modifying the emission period of a sensor that has just transmitted. In addition, to measure the relevance of these functions, we propose a generic metric of monitoring quality, somehow quantifying the informative value of the messages received by all sensors, with the value of data depleting over time. By choosing the total monitoring duration as the overall energy efficiency indicator, this leads us to a clear definition of a multi-objective problem.

We apply our approach to a specific case where an additional constraint is imposed: taking into consideration that the simple but effective method for tracking a physical quantity over time is to obtain information at regular time intervals [START_REF] Gruijter | Sampling for Natural Resource Monitoring[END_REF], we want the sensors to be programmed so that a message is sent periodically by one of them. We then develop a period update function that ensures the periodic transmission of messages from one (and only one) of the sensors, and dynamically adapts to new sensors entering the monitoring system. The function is configured by two parameters, namely the number of jointly emitting (through a cycle) sensors and the target emission period. We observe that the greater the number of sensors transmitting jointly, the greater the diversity of the data stream, but the more energy-consuming the solution. Therefore, we show that a compromise must be made between the diversity of the information received (i.e., the quality of the monitoring) and the total monitoring duration (i.e., the energy efficiency).

The contributions of this paper can be summarized as follows.

• We formalize the definition of a period update function.

• We propose a generic monitoring quality indicator, which together with the total network lifetime characterize a multi-objective problem.

• We construct a 2-parameter period update function, ensuring the strict periodic emission by one of the sensors. We compute analytical bounds on the monitoring duration according to the parameters used.

• We perform a numerical comparison in order to illustrate the approach and discuss the fitting parameters of the developed period update function.

The rest of the paper is organized as follows. Section 2 defines the studied mathematical problem and proposes a formal definition of the period update function. We propose a definition of the monitoring quality, which together with the total monitoring duration forms the two performance indicators of a period update function. In Section 3, we introduce a two-parameter period update function ensuring periodic emissions, and derive bounds for the total monitoring time according to the chosen parameters. Numerical investigations applying the proposed monitoring strategies are carried out in Section 4. Finally, we conclude and highlight the limits of the studied model in Section 5, suggesting directions for future works.

2 Problem statement and model

Assumptions and notations on sensors

We are interested in the monitoring of an environment with IoT sensors. Sensors are dynamically integrated in the management system at the time of their first emission, also called instant of activation. We consider a total of n sensors on battery indexed by their order of activation: 0 being the first sensor to be activated and n -1 the last.

The sensor i activates at time t i , with an initial energy e i . It sends messages periodically. Following each transmission, it is possible to modify the transmission period of the sending sensor, through a downlink message sent during the sensor reception window.

This paper revolves around how to define the new emission period to assign to each sensor after its emission, so as to ensure periodic emissions overall, while managing a tradeoff between the monitoring quality (through a measure of diversity) and energy efficiency (through the total lifetime of the monitoring network). We therefore use the notion of period update function f , defined in more detail hereafter, for that purpose.

A sensor is said to be active at a time t if, at that time, it is activated and has enough energy to emit again. Conversely, a sensor that is not active anymore at time t is said to be dead. We talk about the end of monitoring when there are no more active sensors. Thus, we consider for the whole paper that the n sensors activates without the monitoring stopping in the meantime.

In the strategy developed hereafter, we also consider a specific consumption model: only the consumption related to emissions and period changes is taken into account (with respective energy cost of c e and c r ), since the other consumptions can be considered as negligible [START_REF] Bouguera | Energy consumption modeling for communicating sensors using LoRa technology[END_REF], [START_REF] Bouguera | Energy consumption model for sensor nodes based on LoRa and LoRaWAN[END_REF]. Finally, we assume that each sensor spends its energy until its battery is exhausted.

f : H t - → R + * , (1)
where f (H t ) represents the new transmission period for a sensor that has just sent a message at time t.

For the function developed hereafter, each message sent by a sensor contains: its ID, the remaining energy in the battery, its transmission period; all that information is added to H t for each new message sent.

The function f is used for each new received message. In particular, f defines the initial period of sensors. In practice, if the function f returns a different period from the sensor's current one, a downlink transmission from the gateway (with an energy cost c r to the sensor) takes place to modify that period, so that after sending a message at time t, a sensor's period always equals f (H t ).

Defining a monitoring quality metric

In most monitoring applications, the objective is to obtain a spatio-temporal coverage of the study area thanks to adapted sampling [START_REF] Castello | Optimal sensor placement strategy for environmental monitoring using wireless sensor networks[END_REF], [START_REF] Pourshahabi | Spatio-temporal multi-criteria optimization of reservoir water quality monitoring network using value of information and transinformation entropy[END_REF]. However, here we assume that we do not know the position of the sensors. This is why we introduce the notion of average diversity: the amount of information from different sources weighted by their relative relevance. We characterize here the relevance of a data by its aging.

The freshness of a message evaluated at t [START_REF] Bouzeghoub | A framework for analysis of data freshness[END_REF], [START_REF] Even | Utility-driven assessment of data quality[END_REF], [START_REF] Sun | Sampling for data freshness optimization: Nonlinear age functions[END_REF] will represent the relevance of the transmitted information as a function its age. We model it through a positive decreasing function taking as argument the difference between the observation time t and the message sending time t ′ , i.e., ∆ t = t -t ′ > 0.

Sensors send messages to the management system, updating their emission period after an emission if told so by the gateway. We apply the notion of freshness to a sensor by considering its most recent emission, in order to propose the following definition of diversity. Definition 2. The diversity at time t is defined as the sum of the freshnesses of all sensors that have been activated at that time.

For a given period update function f , we define the average diversity as the average of the diversities over the entire monitoring duration. The average diversity related to a period update function f is denoted by D(f ).

Below are two examples of freshness functions:

• u T (∆ t ) = 1 ∆t<T , for some value T > 0, meaning that the value of some received data remains constant during T then suddenly drops to 0.

• v T (∆t) = exp (-∆t T )
, with a smoother depletion of the information value over time.

The parameter T characterizes the relevance time of data: if T is large, then we consider that "old" data remains relevant.

To illustrate the meaning of the diversity measure, consider the freshness function u T (t): if a period update function f induces a diversity X, then that means that over a sliding window of size T , messages are received on average by X different sensors.

A multi-objective problem

In this paper, we are also interested in the overall energy efficiency, rather than in local efficiency: our metric for energy efficiency will be the total monitoring duration, that is the time between the activation of the first sensor and the end of the monitoring.

In this way, we characterize a bi-objective problem: we can quantify and compare the qualities of period update functions through our two performance metrics, for energy efficiency and monitoring quality.

Ensuring periodic emissions from at most M sensors

In this section, we develop a strategy to guarantee, by defining the period update function, that there is one (and only one) periodic emission, with a period τ , and that at most M sensors emit in turn (M and τ are chosen by the monitoring manager).

Definition of effectiveness

An effective method for tracking an average physical quantity over time is to collect regular samples [START_REF] Gruijter | Sampling for Natural Resource Monitoring[END_REF]. Hence, starting from the instant of the first message received at time t 0 , we want in this paper to receive exactly one message at regular time intervals from one of the active sensors. By introducing the time step parameter τ , that property will be named effectiveness, as formalized below.

Definition 3. A period update function is said to be effective over the instants of period τ if the sensor emissions verify that:

• Starting at t 0 , one and exactly one emission is made at each time step τ as long as there are active sensors.

• Apart from the activation, no sensor emits a message between each time interval τ .

For such periodic update functions, we can quantify the energy efficiency relative to a fixed τ , called the sample span. Definition 4. Given an efficient period update function f , its sample span L(f ) is defined as the number of consecutive emissions at time steps τ until the end of the monitoring.

The monitoring duration is then simply defined as τ L(f ). We develop below an analytical upper bound for the span (and thus, duration) of efficient period update functions.

Proposition 1. If no sensor activates exactly at an instant of the form t 0 + kτ for an integer k, then any effective period update function will have a sample span upper-bounded by:

L max := n-1 i=0 e i -nc e -(2n -1)c r c e ( 2
)
The proof is developed in Appendix B.

In the rest of this section, we develop a specific function, that we will denote by f M,τ , and that we will show is effective over the instants of period τ , while jointly using up to M sensors to provide some diversity.

Toward an effective period update function with controlled diversity

Considering the scenario of tracking an average physical quantity, we want to develop a periodic update function allowing to receive messages at regular intervals; the target time τ between two emissions will be the first parameter of our function.

In a context such as the one predicted for mIoT, it is possible to have some misplaced or faulty sensors. Moreover, in some cases, it may be necessary to receive spatially diversified information to get a more accurate global view. It may then be necessary to receive information from various sources (quantified by the average diversity). The second parameter of the function, that we will denote by M , will be the number of sensors transmitting in a round-robin fashion (when possible), to take into account this possible requirement.

For given parameters M and τ , we therefore want to define a period update function f M,τ such that at most M sensors transmit in turn, will overall periodic emissions with period τ . If any, the other sensors will be set in sleep mode, and successively take over the dead sensors. Hence, • when the number of active sensors is below M , all active sensors emit in turn. In that case, the active sensors have an emission period set to τ times the number of active sensors. If a new sensor activates, all the sensors then change their emission period to maintain that property.

• As soon as they are more than M active sensors, our proposed scheme works differently: M sensors emit periodically, with a period of M τ , and the period of all the other sensors is set so that they successively take over dead sensors. When one such sensor takes over the death of another one, its period is set to M τ , to ensure the same role.

An illustrative example of sensor emissions using the period update function is shown in Fig. 1. As was the initial goal, there is exactly one emission on each blue vertical line, meaning that we receive exactly one message every τ . Note that when there are at least 3 active sensors, the emissions are only shared between 3 sensors emitting periodically.

Detailed definition of the period update function

First of all, we define the function in the case where the emitting sensor is already active. When the total number of active sensors does not exceed M i.e. an insufficient number of sensors has activated or a too important number of sensors is out of battery. In this case, the function defines for each sensor a period f (H t ) = |Π(t)|τ . When there are enough active sensors (|Π(.)| > M ), then f (H t ) = M τ . Thus, for already active sensors, the period update function is defined by:

f M,τ (H t ) = min(M, |Π(t)|)
When a sensor activates, there are 2 different cases. If the number of active sensors is less than M , then new sensors get included in the round-robin. The activating sensor should emit τ after the sensor that has emitted just before. We firstly consider that the previous emission corresponds to a sensor already active. Thus, the period update function is defined by:

f M,τ (H t ) = τ |Π(t)| -(t -last emission)
The previous sensor emits on the instants of period τ . Thus, denoting "%" the operator remainder of the division algorithm, (t-last emission) = (t-t 0 )%τ , so that:

f M,τ (H t ) = τ |Π(t)| -(t -t 0 )%τ (3)
This last formula generalizes to the case where the previous emission does not come from an already active sensor.

We consider now the second case: there are already at least M active sensors. Then, a sensor that activates is put to sleep until a sensor dies. The sensor takes over from a sensor whose relay is not taken i.e. emits M τ after its last emission:

f M,τ (H t ) = death time of a sensor -t + M τ
We now introduce an object allowing to keep in memory the deaths of each sensor. We define death-date the list sorted by ascending date of the dead sensors whose relay is not assured. The list is updated with each new message from a sensor.

Algorithm 1 defines the death update algorithm, updating the list deathdate. This algorithm is executed each time a message is received, following the use of the period update function.

This lead us toa formal definition of f M,τ :

Definition 5. The period update function f M,τ used for a sensor just after it sent a message is defined by:

• if first message received from that sensor , f M,τ (H t ) = τ |Π(t)| -(t -t 0 )%τ if |Π(t)| ≤ M death-date[0] -t + M τ if |Π(t)| > M • Else, f M,τ (H t ) = min(M, |Π(t)|)τ (4)
The death-date list, whose size is at most M , is updated to always contain the sorted list of sensor death instants whose relays are not covered. In particular, when a sensor activates while |Π(t)| ≥ M , the death date of the sensor whose relay has just been taken is replaced by the predicted death of the new one.

Note that for the special cases M = 1 and M = n, combinatorial and memory space simplifications can be used to implement the function f M,τ . Algorithm 1 Death update algorithm i corresponds to the sensor ID, e i the energy of the sensor just after it has emitted the message and p i the period which is set by f M,τ . The function called "add" (and "update") add (and update) elements to the list while sorting it in ascending date order. The function "remove" is working if the element is still in death-date. Require: death-date,time t , sensor index i, period p i , remaining energy

e i 1: if |Π(t)| ≤ M then 2:
if e i < c e then 3:

Remove sensor i from death-date 4:

else if p i ̸ = M τ then 5:
Update death-date of sensor i with value t + p i + M τ ei-ce-cr ce 6:

else if p i = M τ then 7:
Update death-date of sensor i with value t + p i + M τ ei-ce ce 8:

end if 9: else 10:

if fst emission from i then end if 14: end if

Properties and performance of f M,τ

The following propositions establish that f M,τ behaves as we wanted it to and guarantees a minimum monitoring duration.

Proposition 2. The f M,τ period update function is effective on the instants of period τ . Proposition 3. Considering sensors with the same initial energy e, the sample span of f M,τ is at least:

L min (f M,τ ) := ne -nc e -(2n -1 + M (M -1))c r c e (5)
and at most:

L max (f M,τ ) := ne -nc e -(2n -1 M =1 )c r c e (6)
Proof (sketch). As long as there are no more than M active sensors, in the worst scenario each new sensor that activates disrupts the existing schedule, forcing all other sensors to consume energy to change their emission period.

To get the upper bound, we on the contrary consider the most favorable scenario, that is when the first M sensors activate in the same time interval of length τ , and n is a multiple of M . If all sensors are activated (almost) simultaneously, the scheduling is only disturbed once, which has little impact on the overall system energy. In that case, the solution is close to the global optimum L max of Proposition 1. Except in that condition, at a fixed τ , an increase in the M parameter generally has a negative influence on the monitoring duration.

Formal demonstrations of Propositions 2 and 3 are respectively developed in Appendices C and D In addition to reducing monitoring time, choosing a large value of M can induce high downlink consumption, which in the worst case increases quadratically with respect to M . Downlink consumption congest the network, and can have a significant impact on Quality of Service [START_REF] Vincenzo | Improving downlink scalability in lorawan[END_REF].

Performance results

This section discusses the experimental analysis of the period update function f M,τ , carried out through simulations. We propose to study the performance by using the function f M,τ for different values of the number of sensors jointly emitting M and of the target emission period τ . From the initial conditions defined in Table 1, we apply the period update function f M,τ for each emission of sensor until the end of the monitoring, in order to determine monitoring duration and average diversity performance indicators.

Parameter

Influence of the number M of sensors jointly transmitting

For a fixed target emission period τ , the parameter M influences both performance metrics, as Fig. 2 illustrates. The sample span (thus, the monitoring duration) decreases when M increases (Fig. 2

"Diversity versus duration" trade-offs

We represent in Fig. 3 the monitoring time and average diversity metrics obtained with f M,τ , for different values of (M, τ ). Of course, one would like to be as north-east as possible in the figure (high diversity and high monitoring duration). Interestingly, the Pareto front is not always attained with the same value of M : if the network designer preferences (or the application needs) favor the monitoring duration, smaller values of M should be preferred, while larger values should be chosen if diversity matters most.

If the need for diversity is not very strong, then choosing a small value of M and a relatively large τ time step (compared to the relevance time of data T ) allows to extend considerably the total monitoring duration ( and induce a low consumption of the downlink). On the other hand, if the need for diversity is more important, it is necessary to choose a larger value of M , and a small time step τ , leading to a more frequent energy consumption, at the price of a shorter monitoring duration. As an example, if the diversity requirement is D > 10, then choosing M = 44 and τ = 1.97, ensure a monitoring duration of 2.9 * 10 5 .

Hence the methodology leading to Fig. 3 can be adapted to the specific parameters of a new scenario, and applied to determine the best-performing parameters M and τ for the needs of the application.

conclusions and future works

This paper presents a method for scheduling sensor emissions in an initially uncoordinated environment, thanks to a 2 parameters emission scheduling strategy. Considering a massive number of deployed battery sensors, we show that a trade-off has to be made between a precise tracking and an extended monitoring time. We present a method to set the appropriate parameters of the proposed solution to find a compromise according to the preferences (energy efficiency vs. monitoring quality).

In addition, we provide new general bases for the development of centralized approaches based on requirement one can expect for mIoT. Starting from the first approach proposed in this paper, the forthcoming challenge will be to look for more dynamicity in monitoring strategies: (i) the hypothesis of strict regular message transmission should be relaxed in order to gain flexibility and robustness, (ii) In addition to the dynamic inclusion of sensors already managed in the proposed strategy, the solution must also adapt to unplanned departures, (iii) We supposed here that all the sensor data where equivalent. In a real system, messages can be very disparate; it is important to characterize this information heterogeneity to include it in monitoring policies.

Acknowledgement

This work has been sponsored by the ValaDoE Chair.

A Github repository

It is possible to access the github repository that helped us to create the different figures of the concerned document:

https://github.com/gwenmaudet/emission_strategies_to_optimize_monitoring. git B Upper bound of all the period update functions effective over the instants of period τ

The following notation is introduced: Definition 6. Let f be an effective period update function. Then we denote by r i (f ) the number of period changes of the sensor i.

Proposition 4. The sample span of a period update function f , effective over the instants of period τ is:

L(f ) = n-1 i=0 e i -c e -c r r i (f ) c e (7)
Proof. We denote by d i (f ) the number of emissions (excluding activation) on time steps τ without collisions with other emissions. The consumption of a sensor is divided between: it first emission c e , each period change c r r i (f ), and since f is effective, the collision-free emissions on the time steps τ c e d i (f )

e i ≥ c e (1 + d i (f )) + c r r i (f )
By definition of the behavior of the sensor, it sends messages until the battery is empty:

d i (f ) = e i -c e + c r r i (f ) c e
The total number of emissions on the time steps τ corresponds to the number of emissions (excluding activation) of each sensor without collision i.e. by definition of d i (f ):

L(f ) = n-1 i=0 d i (f ) Lemma 1.
Let f and g be 2 effective period update functions. If ∀i, r i (f ) ≥ r i (g), then the sample span of g is greater than that of f : L(f ) ≤ L(g).

Proof. We use directly Proposition 4.

Demonstration of Proposition 1(page 7):

Proof. We will show that for a period update function, the minimum number of period changes for a sensor is:

r i (f max ) = 1 + 1 i>0
In order to exclude particular cases, let us suppose that the sensors do not activate at the instants of period τ i.e.

t i ̸ ≡ t 0 [τ ] (8)
Then, let us consider g an effective period update function. We will show that ∀i, r i (g) ≥ r i (f max ).

-i = 0. Necessarily, g change at least one time the period of the sensor 0. Then r 0 (g) ≥ r 0 (f max ).

-i > 0. Let us suppose r i (g) < 2, then r i (g) = 1. We note p its period of emission. Since g is effective, its first emission after activation is on the time steps τ i.e.

t i + p ≡ t 0 [τ ]. But from (8), t i ̸ ≡ t 0 [τ ], so p ̸ ≡ 0[τ ].
Looking at the following emission, t i + 2p, we come across an absurdity. Thus r i (g) ≥ r i (f max ).

Hence, f max gives a minimum number of changes for each sensor. From Lemma 1, f max gives a maximum bound to the sample span. The sample span is equal to:

L(f max ) = n-1 i=0 e i -c e -c r (1 + 1 i>0 ) c e

C demonstration of the effectiveness of the proposed period update function

We first introduce some tools to track the behavior of sensors over time, and then propose a proof of the effectiveness of the function f M,τ .

C.1 Sensor representation set and characterization over the instants of period τ Definition 7. A sensor i active at time t is represented by E i (t) such that:

E i (t) :=    e i (t)
≥ c e : remaining energy at time t p i (t) > 0 : emission period δ i (t) > 0 : time before sending a message after t

E i (t) := ∅ if the sensor i is not active at time t (didn't activate yet or is dead). We denote by Π(t) the set of active sensors at time t, |Π(t)| is the quantity of active sensors at time t i.e.

i ∈ Π(t) ⇔ E i (t) ̸ = ∅
We define E(t), so that:

E(t) := (E i (t)) i∈Π(t)
The formalization of the state E(.) will help to prove that the function f M,τ is effective, i.e. the emissions of the sensors (except activations) are on the instants of period τ .

First of all, we can define E(.) until the first emission of the first sensor:

t < t 0 , E(t) = () E(t 0 ) = (E 0 (t 0 )),E 0 (t 0 ) =    e 0 (t 0 ) = e 0 -c e -c r p 0 (t 0 ) = f (H t0 ) δ 0 (t 0 ) = f (H t0 ) (10)
The initial emission period of the sensor is defined by f . The sensor consume energy for its first emission and the setting of its period: c e + c r . Then, from time t 0 , the next emission occurs after a duration of f (H t0 ).

Moreover, E(.) evolves over time, for each emission. We characterize the variations from state E(t) to state E(t + ∆t).

A sufficient condition for the sensors to transmit at most once between t and t + ∆t is that their period is larger than the time step ∆t, or more generally that the minimum of the period update function is greater than ∆t.

f ≥ ∆t (11)
We now characterize the evolution from t to t + ∆t under condition [START_REF] Gubbi | Internet of things (IoT): A vision, architectural elements, and future directions[END_REF]. If i is a sensor that is active at time t, i.e. i ∈ Π(t). Moreover, if the sensor does not transmit between t and t + ∆t, δ i (t) > ∆t. Then, the periods and energies states don't change. The duration before the next emission at time t + ∆t is decreased by ∆t:

E i (t + ∆t) =    e i (t + ∆t) = e i (t) p i (t + ∆t) = p i (t) δ i (t + ∆t) = δ i (t) -∆t (12)
If i emits between t and t + ∆t: 0 < δ i (t) ≤ ∆t. Then, it consumes an energy c e . Moreover, the function f is used to determine the new emission period of the sensor. It will consume an additional energy c r if the defined period is different from the current one. The sensor is represented in E(t + ∆t) only if it is active at t + ∆t i.e. if it has enough energy to transmit again. In order to define a simple form of E i (t + ∆t), we define the energy remaining in the sensor i at time t + ∆t. e i (t + ∆t) = e i (t) -c e -c r 1 f (H t+δ t )̸ =pi(t) (1 is the indicator function).

Then:

E i (t + ∆t) =     ∅ , if e i (t + ∆t) < c e    e i (t + ∆t) p i (t + ∆t) = f (H t+δt ) δ i (t + ∆t) = δ i (t) -∆t + f (H t+δt )
, else [START_REF] Hurlburt | The internet of things: A reality check[END_REF] Finally, if a new sensor i activates between t and t + ∆t, i.e. i ̸ ∈ Π(t), i ∈ Π(t + ∆t) and t i ∈]t, t + ∆t]. In this case, f defines the emission period of i and:

E i (t + ∆t) =    e i (t + ∆t) = e i -c e -c r p i (t + ∆t) = f (H ti ) δ i (t + ∆t) = t i -(t + ∆t) + f (H ti ) (14)
In all these cases, if [START_REF] Gubbi | Internet of things (IoT): A vision, architectural elements, and future directions[END_REF] is verified, then δ . (.) > 0. As explained above, the notations are used here to help the proof of efficiency of f M,τ . Thus, we use the following notations allowing simplification in the writings.

Definition 8. We define the characterization of the sensor set over the instants of period τ :

E k := E(t 0 + kτ )
In the same way, we define (Π k ) k∈N := {Π(t 0 + kτ ), k ∈ N}.

For i ∈ Π k :

E i,k := E i (t 0 + kτ ) =    e i,k = e i (t 0 + kτ ) p i,k = p i (t 0 + kτ ) δ i,k = δ i (t 0 + kτ )
.

From Definition 8, we mathematically define the effectiveness: Definition 9. A period update function is said to be effective over the instants of period τ if, using characterization of the sensor set over the instants of period τ :

∀k ∈ N, |Π k | > 0 ⇒ ∃!i ∈ Π k : δ i,k = τ ∀j ∈ Π k , j ̸ = i ⇒ δ j,k > τ (15)
We then define its sample span L by:

L := max{k, |Π k | > 0} + 1 (16)
C.2 Proof of the effectiveness of the function Demonstration of Proposition 2 (page 10)

Proof. First, we consider that the description of the algorithm is sufficient to assert that Assertion 1. death-date is a list updated at each new emission such that it corresponds to the list sorted by ascending order of death of sensors whose relay is not already provided by other sensors.

We will show by induction the statement called P (k): Looking at the instant t 0 + kτ :

• If the number of active sensors does not exceed M , then the active sensors emit exactly on the next consecutive jτ instants, 1 ≤ j ≤ |Π k |.

• Otherwise, the jτ instants are covered for the first M instants by M sensors. The next emission of the other sensors occurs at least M τ after the death of the next sensor. There is exactly one that emits M τ after the death of the next sensor.

Mathematically, introducing

I k = {i ∈ Π k , δ i,k > M τ }, P (k) means:
-

If |Π k | ≤ M, ∀j, 1 ≤ j ≤ |Π k |, ∃!i ∈ Π k : δ i,k = jτ -Else,    ∀j, 1 ≤ j ≤ M, ∃!i ∈ Π k : δ i,k = jτ ∃!i ∈ I k : δ i,k = next-death(k) -(t 0 + kτ ) + M τ ∀i ∈ I k , δ i,k ≥ next-death(k) -(t 0 + kτ ) + M τ
Where next-death(k) represents the death time of the next sensor after t 0 + kτ . Initialization: We want to prove P (0). |Π 0 | = 1 ≤ M . f M,τ sets the period to τ , so that δ 0,0 = τ , hence P (0) is true.

Heredity: Assume P (k) is true for some k ≥ 0. We define n :

= |Π k |. Disjunction of cases. -If |Π k | ≤ M and |Π k+1 | ≤ M .
• If there is no variation in the set of active sensors, |Π k | = |Π k+1 | = n ≤ M . By using P (k), ∃!i ∈ Π k : δ i,k = τ so that from the definition of f M,τ , we have δ i,k+1 = nτ . Moreover, ∀j, 1 < j ≤ n, ∃!i : δ i,k = jτ so δ i,k+1 = (j -1)τ . Finally ∀j, 1 ≤ j ≤ n, ∃!i : δ i,k+1 = jτ, which means P (k + 1) is true.

• We now consider that m new sensors indexed (l + r) r∈ [1,m] activate between states E k and E k+1 and no sensor dies. l represents the number of sensors activated from the beginning of the monitoring. Since |Π k+1 | ≤ M , n + m ≤ M . We consider without loss of generality that t 0 + kτ < t l+1 < t l+2 ... < t l+m ≤ t 0 + (k + 1)τ . Π k+1 = Π k {(l + r) r∈ [1,m] } so that |Π(t l+r )| = n + r. Since t l+r -t 0 -(t l+r -t 0 )%τ = kτ , from Eq. ( 14) we can simplify δ l+r,k+1 :

∀r, 1 ≤ r ≤ m, δ l+r,k+1 = t l+r -(t 0 + (k + 1)τ ) +(τ |Π(t l+r )| -(t l+r -t 0 )%τ ) = τ (|Π(t l+r )| -1) = (n + r -1)τ Now from induction hypothesis, ∃!i ∈ Π k : δ i,k = τ , so that δ i,k+1 = (n + m)τ . Moreover, 1 < j ≤ n, ∃!i ∈ Π k : δ i,k = jτ ⇒ δ i,k+1 = (j -1)τ . Finally: ∀j, 1 ≤ j ≤ n + m, ∃!i ∈ Π k+1 : δ i,k = j
Which means P (k + 1) is true.

• If a sensor indexed l dies between t 0 + kτ and t 0 + (k + 1)τ , Π k+1 = Π k |{l} and δ l,k = τ . Since j > 1, ∃!i, δ i,k+1 = (j -1)τ , P (k + 1) is true. It can be further shown that if between the states E k and E k+1 , one sensor dies and several activates, P (k + 1) remains true.

-If |Π k+1 | > M . • If there is no variation in the active sensors Π k = Π k+1 . From P (k), ∀j, 1 ≤ j ≤ M, ∃!i ∈ Π k : Π k = jτ .
Then, since the period of these sensors is M τ (definition of f M,τ ), from the same reasoning as above ∀j, 1 ≤ j ≤ M, ∃!i ∈ Π k+1 : Π k+1 = jτ . No sensor dies, so next-death(k) = next-death(k+1). Thus, about the other sensors (which do not emit on the first M instants): from Eq. ( 12), ∃!i ∈ I k+1 : δ i,k+1 = next-death(k+1) -(t 0 + (k + 1)τ ) + M τ and ∀i ∈ I k+1 , δ i,k+1 ≥ next-death(k+1) -(t 0 + (k + 1)τ ) + M τ . Hence, P (k + 1) is true.

• If m sensors, indexed (l + r) r∈ [1,m] activates and no one dies. Then, if |Π k | < M , the sensors activates in the same way as a case already studied above, until |Π k | = M . Without loss of generality, we consider now that (l + r) r∈ [1,m] activates while there are already M active sensors. Then ∀j, 1 ≤ j ≤ M, ∃!i ∈ Π k+1 : δ i,k+1 = τ j. Moreover, f M,τ fix the period of l + 1 to death-date[0]t l+1 + M τ and delete death-date[0] from death-date. The same thing is done for all l + r. In that case, δ l+1,k+1 = death-date[0] -(t 0 + (k + 1)τ ) + M τ . and it is the same for the other sensors indexed l + r. Hence, P (k + 1) is true.

• If a sensor l ∈ Π k dies. Then, δ l,k = τ and Π k+1 = Π k /{l}. Then next-death(k) = t 0 +(k +1)τ . We consider,from hypothesis P (k), i so that δ i,k = next-death(k)- (t 0 + kτ ) + M τ = (M + 1)τ and δ i,k+1 = M τ . Hence : ∀j, 1 ≤ j ≤ M, ∃!i ∈ Π k+1 : Π k+1 = jτ
Moreover, next-death(k+1) is updated to the death date of the next sensor, and from assertion 1, f M,τ fix the period of a sensor so that it emit M τ after the death of the next sensor i.e. ∃!i ∈ I k+1 : δ i,k+1 = next-death(k+1) -(t 0 + (k + 1)τ ) + M τ . Moreover, all the other sensors will emit after that time: ∀i ∈ I k+1 , δ i,k+1 ≥ next-death(k+1) -(t 0 + (k + 1)τ ) + M τ . P (k + 1) is true. P (k+1) remains true if multiple sensors activates and one sensor die between t 0 + kτ and t 0 + (k + 1)τ .

D Proofs of the bounds of the sample span of the defined function

Proposition 5. The lower bound of the period update function f M,τ , for sensors with same energy e is:

n 2 > M, L(f M,τ ) ≥ M -1 i=0 e-ce-(i+1+1i>0)cr ce + (n -2M ) * e-ce-2cr ce + M -1 i=0 e-ce-(i+2)cr ce (17a) M ≥ n 2 , L(f M,τ ) ≥ n-M -1 i=0 e-ce-(M -i+1i>0)cr ce + M -1 i=n-M e-ce-(2M -n+1i>0)cr ce + n-1 i=M e-ce-(i-n+M +2)cr ce (17b)
Proof. Let a sensor indexed i. It modifies 1 + 1 i>0 times its emission period to adjust itself with respect to the other sensors during its first two emissions. |Π(.)| varies at each activation and death of a sensor. We will study the case where sensor i is subject to the most period changes. If i activates while |Π(.)| ≤ M , it can modify its emission period for any new sensor activating, until there are M active sensors i.e. at most M -i -1 additional times. In that case, the sensor i emits at least once between each activation. This is the case if the time between two activations is greater than the period of i. Focusing on the sensor indexed 0: ∀i, 1 ≤ i ≤ M -1, t i -t i-1 > τ i Furthermore, if we consider that a sensor i activates when there are no more than M remaining active sensors, it can change its emission period for each sensor death from the time when there are M remaining sensors. Considering that the sensors are activated at sufficiently spaced instants, and that they have the same initial energy, then the activation index corresponds to the death index of the sensors. Thus, the sensor i will change its emission period at the death of sensors if i ≥ n -M . It will thus change at most i -(n -M ) additional times.

However, depending on the chosen value of M , the results are different. It is necessary to split in 2 cases: If M is small relative to n i.e. M < n 2 . The sensors will have a different number of period changes following these 3 intervals Fig. 4: If M is close to n i.e. M ≥ n 2 . In this case, the intervals [0, M -1] and [n -M, n -1] overlap. We study the 3 following intervals (Fig. 5):

• 0 ≤ i ≤ M -1, r i = M -i + 1 i>0 . • M -1 ≤ i ≤ n -M, r i = 2. • n -M ≤ i ≤ n -1, r i = i -(n -M ) + 2.
• si 0 ≤ i ≤ n -M, r i = M -i + 1 i>0 .

• si n -M ≤ i ≤ M -1, r i = 2M -n + 1 i>0 .

• si M -1 ≤ i ≤ n -1, r i = i -(n -M ) + 2 Proof. We consider that the first M sensors are activated in the first time interval τ i.e. ∀i,

1 ≤ i ≤ M, t i ∈ [t 0 , t 0 + τ ]
Let us consider the first M sensors. In this case, considering the first sensor of index 0 : f M,τ modifies its emission period a first time to τ , then modifies it to M τ if M ̸ = 1. For the sensors of index 0 < i < M , their first emission period is τ |Π(t i )| -(t -t 0 )%τ . Their second period is exactly τ M . Each sensor performs exactly r i = 2 period changes. Moreover, for the following sensors, they also change their emission period a first time to emit following the death of a sensor, then the period is fixed to M τ . Finally, since all the sensors turned on at the same time and consumed a similar amount of energy, if n ≡ 0[M ], then the cycles of M sensors will be renewed each time at the same time, and thus the M last sensors will die at the same time in turn (no additional period change consumption). In this case, all sensors must change their emission period twice except for the 0 sensor if M = 1. In the latter case, the upper bound on monitoring time is reached with respect to the period update functions effective over the instants of period τ .

Hence:

L(f M,τ ) ≤ n-1 i=0
e -c e -c r (1 + 1 M >1 or i>0 ) c e This lead us to the approximation Eq. ( 6), and then we proved Proposition 3 (page 10)

  : flexible and generic solutions for the future of monitoring

  sensor Representation of the sensor emission over time footstep change of period instruction

Figure 1 :

 1 Figure1: Illustration of sensor emissions using f M,τ for M = 3 and τ = 1. 7 sensors, of equal battery-capacities e = 15 and with emission and period change consumption c e = c r = 1, randomly activate between time 0 and 40. The emissions of sensor i appear as dots on the y = i line. Black points indicate period changes after an emission, through the function f M,τ . As was the initial goal, there is exactly one emission on each blue vertical line, meaning that we receive exactly one message every τ . Note that when there are at least 3 active sensors, the emissions are only shared between 3 sensors emitting periodically.

11 : 12 :

 1112 add death-date of sensor i with value death-date[0] + M τ (1 + ei-cr ce ) remove death-date[0] from death-date 13:

Figure 2 :

 2 Figure 2: Representation of some performance indicators using the period update function f M,τ , varying the number of jointly emitting sensors , for a few target emission periods τ . (a) corresponds to the sample span with analytical bounds, (b) the diversity.

Figure 3 :

 3 Figure 3: Performance of the update period function f M,τ , for several values of M (given in the legend) and τ (from 0.5 to 10 by 0.1 increments). Each point corresponds to the two-dimensional performance metrics (Diversity on the xaxis and monitoring duration on the y-axis), for fixed parameters M, τ of the update function.

Figure 4 :

 4 Figure 4: Representation of the number of period changes depending on the index of the sensor when M < n 2

Figure 5 :Proposition 6 . 1 i=0e

 561 Figure 5: Representation of the number of period changement depending on the index of the sensor when M ≥ n 2 We can then get the minimization: L(f M,τ ) ≥

Table 1 :

 1 Simulation parameters

		Meaning	Value
	n	Number of sensors	300
	e i = e	Battery capacity	500
	c e = c r	Emission and reception energy cost	1
	t i -t i-1	Time between 2 consecutive activations 15π
	T	Relevance time of a data	20
	Freshness function Depletion of the data over time	v 20