text:
Zoé Mesnil
email: zoe.mesnil@u-pec.fr

A reference for studying the teaching of logic

Keywords: Logic, teaching, didactic transposition, quantifiers

This paper presents a work developed in my thesis on the teaching of logic in high school in France. The current official instructions specify that teachers don't have to make a mathematical logic course, but have to help their students develop a relevant use of some notions of logic as tools. Therefore, I point out in this paper that in order to take this constraint into account in the study of didactic transposition process, it is relevant to describe it from a reference knowledge for logic, but that such knowledge has never been established by the mathematics community. In order to offer such a reference, I base myself on a double epistemological and didactical study in which I favor the links between logic and language. I will also explain the choices made for this reference and its use as a methodological tool through the example of quantifiers.

Introduction

In France, mathematics syllabuses for high school (students between 15 and 18) mention explicit goals concerning certain notions of logic. For example, they recommend that "students [be] trained on examples to wisely use the universal and existential quantifiers (the symbols ∀ , ∃ are not required) and to identify the implicit quantifications in some propositions, particularly in conditional propositions". These current recommendations come after twenty years during which the logic was excluded from syllabuses. This exclusion itself is a reaction to the too abstract and formal aspect of modern mathematics (taught in France between 1969 and1981) for which mathematical logic and set theory were basic elements. They strongly highlight the logical notions as tools and even show distrust against their feature as objects. This distrust can be interpreted as a resistant mark of turning down modern mathematics. The syllabuses specify that "the concepts and methods of the mathematical logic should not be subject of specific courses but must naturally take place in all chapters of the syllabus". I described more precisely the characteristics of the conditions and constraints due to these official instructions in a contribution to CERME 8 [START_REF] Mesnil | New objectives for the notions of logic teaching in high school in France: a complex request for teachers[END_REF]: teaching goals are ill-defined and subjected to strong constraints. Moreover, mathematics teachers do not share a common reference for interpreting these syllabuses. Indeed, mathematical logic is not part of the domains necessarily studied by a mathematics student, bringing a diversity of teachers' knowledge, that is usually not filled during their training in which teaching of logic is often only superficially and quickly addressed.

In this contribution, I would like to pursue this issue of a reference for the teaching of logic. In my thesis, I studied the teaching of logic in high school as the result of a didactic transposition process, and I'll explain first why the nature of mathematicians' logical knowledge requires thinking this transposition not from a mathematical knowledge like mathematical logic, but from a reference knowledge, nonexistent for the moment in the mathematical community [START_REF] Mesnil | La logique: D'un outil pour le langage et le raisonnement mathématiques vers un objet d'enseignement[END_REF]. Secondly, I will justify the choice I made then to construct a methodological tool to conduct the analysis of syllabuses and textbooks. This tool is a reference in which the notions of logic are presented from three points of view that root them deeply in mathematics, in mathematical activity, and in the classroom. I will conclude by illustrating the choices I made for this reference, and its use through the example of quantifiers.

The need for a reference

During the didactic transposition process [START_REF] Chevallard | La transposition didactique[END_REF], a mathematical object is identified in a body of knowledge (in French, savoir savant), and a first succession of adaptations will make it able to become an object of teaching in a particular institution. It is then identified by mathematics teachers in the knowledge to be taught (in French, savoir à enseigner) and is subjected to a second succession of adaptations to become a taught object.

But except in some university courses, teaching logic does not mean teaching mathematical logic, but the logic at work in mathematical activity, which supports mathematicians' expression and reasoning. We can consider that mathematicians have a logical knowledge, which may be the subject of a didactic transposition process, but this knowledge is more visible in practices than in treaties. Mathematical logic can be perceived as a description of the principles of this logic. It was then explicitly taught during the time of modern mathematics, in France and elsewhere, but these experiments showed that this teaching, conducted formally and isolated, did not help pupils to express themselves and to reason [START_REF] Adda | L'évolution de l'enseignement de la logique en France[END_REF]. Several researchers in mathematics education now agree that such teaching should be explicitly linked together with mathematical activity in which the logic is omnipresent (see for example [START_REF] Epp | The role of logic in teaching proof[END_REF][START_REF] Durand-Guerrier | Recherches sur l'articulation entre la logique et le raisonnement mathématique dans une perspective didactique : Note de synthèse[END_REF]. However, there are few research studies on the effect of teaching logic on reasoning and expression capacities, which is still an open issue.

Thus, to take into account this particular nature of the logical knowledge and its connection to mathematics, and to appropriately study the didactic transposition process of this knowledge, I propose to describe this transposition not from a mathematical knowledge, but from a reference knowledge. I take this notion from Rogalski and Samurçay (1994) who thus characterize a knowledge produced by practices, but decontextualized from situations where knowledge is apparent into action. These authors state that it is necessary that this reference knowledge can "be expressed with its concepts, its methods, its systems of representation and its language" (ibid, p. 46). However, concerning notions of logic, such a reference knowledge does not appear in that there is no corpus collecting the logical knowledge necessary to mathematical activity and about which everyone agree the choices of concepts which are featured and of their representation.

I then conducted a study to answer the following research question: what kind of reference knowledge would be epistemologically and didactically relevant to the logic teaching? This study allowed me to construct a reference, which I then used to analyze the knowledge to be taught in high school in France. I call this analysis tool the reference, and not reference knowledge, because the production of a knowledge falls under a long and collective process.

The importance of language for reasoning, and its links with logic

The study of various logical systems across the ages allowed me to identify invariants and differences in the role assigned to logic and in the ways that are given to it to fulfil it. The study of didactic studies shows how issues relating to the teaching of logic meet the concerns and choices of these logicians.

All these logical systems are built from a work on language. The concept of proposition is primordial in them all. Aristotle describes it in terms of subject-copula-predicate, and it was not until Frege at the end of the XIX e century that this analysis was to be replaced by an analysis in terms of function and argument allowing two things essential for mathematical language: on one hand to consider predicates with several arguments, on the other hand to pull out the act of quantification from the proposition by making it expressed by quantifiers which act on variables. Predicate logic that is then born is able to model mathematical propositions. From a didactic perspective, several research studies have shown many examples of situations in which the predicate logic is a relevant reference for didactic analysis which allows to highlight the importance of issues on quantification [START_REF] Durand-Guerrier | Recherches sur l'articulation entre la logique et le raisonnement mathématique dans une perspective didactique : Note de synthèse[END_REF].

The current language of mathematicians is inspired by Frege's formalism, but it isn't a strict use of a formal language. Focusing particularly on problems of language in mathematics teaching, [START_REF] Laborde | Langue naturelle et e ́criture symbolique: deux codes en interaction dans l'enseignement mathe ́matique[END_REF] showed that there is a particular use of language in mathematics, due to the interaction of the two codes of symbolic writing and natural language. This interaction allows mathematicians to use reformulations useful for conceptualization. Teachers are familiar with the particular features of mathematicians' language, but they can cause difficulties for students who "discover together the concepts and the way we talk about them" (Hache, 2015, p.28).

The common goal of the studied logical systems is to ensure the validity of reasoning, with a preliminary work on language. But for the authors of The Logic of Port Royal1 , logic above all needs to be trained and the formalization of reasoning is seen as an obstacle to the use of intuition, whereas for Leibniz and Frege on the contrary, logic must provide a system of signs in which reasoning can be expressed, and this formal expression guaranteeing its infallibility. [START_REF] Gandit | Preuve ou de ḿonstration, un the me pour la formation des enseignants de mathe ḿatiques : Premie re partie[END_REF] denounces the excessive place taken by the formal aspect in the beginning of proof learning. But being careful about formalization at the time of the discovery of deductive reasoning does not mean that it cannot subsequently help those who begins to have a good practice of it. Thus, in higher education, [START_REF] Selden | Unpacking the logic of mathematical statements[END_REF] suggest presenting theorems and definitions in an informal formulation, which allows intuitive understanding, and in a formal language, which allows linking structure of the statements and structure of its proof.

Organization of the reference

In the reference I proposed, I decided to give a broad place to language. Entering into logic by the language is consistent with the epistemological study, granting it an important place is consistent with didactic studies. Of course, in the same way as it is done in the studied logic systems, especially in a didactic perspective, the study of notions of logic as components of mathematical language has to be articulated with their use in reasoning.

Finally, these epistemological and didactic studies led me to propose a reference in which the presentation of logical concepts combines three approaches:

 The mathematical logic. It is a recent branch of mathematics that can be considered as a result of what has been sought by different logical systems creators since Greek antiquity. Mathematical logic seems so particularly suitable as a formal reference to describe the logic at work in mathematics.

 The study of the language practices of mathematicians. In this way, the presentation of the logical concepts is rooted in mathematical activity taking into account how they are expressed in mathematical discourse, using predicate logic to uncover some complex and sometimes ambiguous formulations that are yet a part of the language practices of mathematicians, widely imported in the mathematics classroom.

 The research in mathematics education. In this way, the presentation of the logical concepts is rooted in mathematics classroom, taking into account the difficulties that the complexity of these notions can bring for students.

In this reference, the components of mathematical language are shown, beginning with the primordial notions of proposition and variable. Then for the connectives AND and OR, implication, negation, quantifiers, I have consistently adopted the three approaches mentioned2 . Although the focus is on the language, reasoning is of course not absent from the reference. A difficulty for pupils and students is to distinguish, in a text of a proof, mathematical propositions concerning mathematical objects, and parts of the text which allow to follow the progression of reasoning, such as variable introductions, or justification of an inference. The confusion between implication and deduction falls under this type of difficulty.

The example of quantifiers in the reference

As announced, the reference contains first a presentation of logical concepts from mathematical logic. Predicate logic uses two quantifiers: applied to a variable x, and from a proposition P, the universal quantifier allows to obtain the proposition ∀x P, and the existential quantifier allows to obtain the proposition ∃x P (description of the syntactic aspect of the quantifiers: they operate on a variable and a proposition to build a new proposition).

Let E be a set in which the variable x can take values. The proposition ∀x P[x] is true3 when for any element a of E, the proposition P[a] is true. The proposition ∃x P[x] is true when there is at least one element a of E such that P[a] is true (description of the semantic aspect of quantifiers: truth conditions of a quantified proposition). Quantifiers have an important effect on the variables: a variable that is in the scope of a quantifier is a dummy variable in the quantified proposition, and this proposition does not give information on the object designated by the variable, but on the set in which it can take its values.

Some important results on quantified propositions may be established by a semantic way, using the sense, as well known equivalence between NOT(∀x P[x]) and ∃x NOT(P[x]), or the fact that if ∃y ∀x P[x,y] is true, then ∀x ∃y P[x,y] is true. These results are then used in syntactic manipulations, independent of the sense, in the same way we manipulate algebraic equalities.

In mathematical language, the quantifiers are a way to express the quantification, but there are many others. We can see this through some examples of mathematical propositions4 :

1) Le carré d'un nombre réel est positif (The square of a real number is positive) 2) Le carré d'un nombre réel est toujours positif (The square of a real number is always positive) 3) Tous les réels ont un carré positif (The square of any real number is positive 4) Tout réel x est tel que x 2 est positif (Any real number x is so that x 2 is positive) 5) Pour tout réel x, x 2 est un réel positif (For all real number x, x 2 is a positive real number) 6) ∀x∈ℝ, x 2 ≥0 They are several formulations of the same property, but universal quantification is expressed very differently. In proposition (1), quantification is implicit, implied by the word un (translated with a). We frequently use the indefinite article un to mark a universal quantification, in everyday language as in mathematics. But un is also sometimes used to mark an existential quantification, which is obviously confusing! Sometimes, the two usages coexist in the same proposition, such as in "un réel positif possède une racine carrée" ("any positive real number has a square root", in English, the first un is rather translated with any and the second une is rather translated with a, and there is no confusion). In proposition (2), the adverb toujours (always) is used to explicitly mark this universal quantification, as the word tous (any) in proposition (3). Propositions ( 4) to ( 6) are distinct from the first by the use of a variable. Furthermore, one can identify in each of these propositions an expression which express the quantification (here universal quantification) and which has the property that it can be separated from the proposition "x 2 is positive" (or equivalent formulation). Such expression works as quantifiers of mathematical logic, and I therefore also calls them quantifier.

Finally, we saw that in the language practices of mathematicians, quantification can be implicit or explicit, and in the second case, possibly marked by a quantifier which is an expression observing syntactic rules of use. Propositions (4) and ( 5) may seem closer to propositions (1) to (3) as they are formulated "with words" contrary to the proposition (6) which only uses mathematical symbols, and that may seems much more formal. I would like to stress that such a vision hide formalization still existing in these propositions, in the sense of a shaping according to certain rules, even if that formalization is not accompanied by a symbolization.

I will conclude by mentioning some of the difficulties of high school students or senior students in related to the use of quantifiers. First, the implicit quantifications are not always perceived by students. The case of the universal quantification associated with the implications and the formulation if ... then ... is highlighted for a long time (Durand-Guerrier, 1999). Quantification is often encapsulated in stiffened structures (for example, "un is as big as we want by taking n big enough") that the expert mathematician knows how to reformulate by explaining the quantifications, but these reformulations in more formal language tend to disappear from the language used in high school, and are a source of difficulty when students meet them in higher education. Another difficulty concerns the failure to take the order of quantifiers into account when there is an alternation. We know that students have rather an interpretation for all... there exists... even though they are facing a proposition there exists... for all... [START_REF] Dubinsky | On student understanding of AE and EA quantification[END_REF]. Furthermore, [START_REF] Chellougui | L'utilisation des quantificateurs universel et existentiel en premie ̀re anne ́e universitaire entre l'explicite et l'implicite[END_REF] showed student difficulties with the use of an existential proposition. In a proof text, there is generally a confusion between the affirmation of the existence of an element checking a property, and the act which consist to consider it and give it a name. Likewise, mathematicians do not get trapped by the "dependence rule" in the statements for all... there exists... and identify easily this error in a student production. However, they do not necessarily explain this error to the students by linking it to formal rules of manipulation of variables and quantifiers [START_REF] Durand-Guerrier | An epistemological and didactic study of a specific calculus reasoning rule[END_REF].

Examples of the treatment of quantifiers in school textbooks

The reference I have developed allows an analysis of resources available to teachers highlighting sensitive issues that need to be paid attention to. I will then conclude with an analysis of two extracts from school textbooks.

The reintroduction of logical concepts in the syllabuses had an effect on the textbooks: those published in 2010 for the first class of high-school (15 years old students) all contained passages identified as speaking of these concepts. Nine textbooks out of ten have chosen to dedicate a few pages (between one page and nine pages) to notions of logic, usually located at the beginning or at the end of the textbook (only one textbook does it in a disseminated way). Moreover, they all contain exercises with a stamp "logic" (from ten to fifty-four exercises in the studied textbooks).

In the eight textbooks that deal with quantifiers, the letters are introduced by examples. Seven of the eight textbooks give only examples of true quantified proposition, and this choice eliminate the syntactic aspect of quantifiers: quantifiers are used only to affirm something, there isn't the idea of a proposition built with a quantifier that one could wonder whether it is true or false.

The textbook Indice gives the example of the proposition "le carré d'un réel est positif" ("the square of a real number is positive") and states that "cette proposition est vraie quel que soit le nombre réel" ("this statement is true for all real numbers"). The authors of this textbook probably want to emphasize on the various possible meanings of the word un, but do not offer at the same time an example of the meaning as existential quantification. Moreover, to know that the meaning of un in this proposition is a universal quantification, it is necessary… to know that the universal proposition is true! Mathematical knowledge is therefore needed to decide between the two possible meanings of the word un, which calls for caution when using this word in a context where student knowledge is potentially fragile. Let's go back now on the comment "this statement is true for all real numbers". It makes no sense to say that the proposition "the square of a real number is positive", which is equivalent to "for all real x, the square of x is positive" is true for all real numbers, since the variable x is dummy in this proposition. The proposition referred to in this commentary is not the quantified one, but the not quantified proposition "the square of x is positive". Finally, there is a confusion between the use of "quel que soit" ("for all") to simply mark the universal quantification, and its use to assure that this universal proposition is true. Now let's look at an example of exercise, taken from the textbook Repères, but it is an exercise that is found in many textbooks. Students must "complete the sentences (for example "… real number x… f(x)>0") using either for all… we have… or there exists… such that…" from the graphical representation of the function f. Note first that the application is not explicitly to complete so that sentences are true! Furthermore, the instruction "complete using either… or…" suggests that each time only one of the both quantifiers is correct. Yet, when the proposition "for all x P[x]" is true, the proposition "there exists x such that P[x]" is also true, so when it is possible to complete with the universal quantifier, it is also possible to complete with the existential quantifier. In everyday language, we respect the principle of maximum information, according to which we give to our interlocutor all information in our possession. So, if I say "on my holidays, it rained some days", I say in the same time that it did not rain every day. The practice of this principle leads us in this exercise to complete naturally with the universal quantifier when possible. However, the notion of truth of a proposition will be contradicted by saying that using the existential quantifier is a mistake, because in mathematics, when the proposition "for all x P[x]" is true, it is not "more true" than the proposition "there exists x such that P[x]". Some students, however, adopt this position, and we can doubt position of the authors of the teacher's textbook who offers as a correction only the universal quantifier when it is possible.

Conclusion

I presented in this paper a methodological tool, a reference to study the teaching of logic. It is of course to be completed, to be improved, both from an epistemological and from a didactic point of view. Important work remains in particular on the concept of proposition, generally not made explicit in teaching, and on the notion of variable (didactic of algebra is very concerned about the status of the letters, but it seems to me that a logical point of view on the concept of variable, such as taking in [START_REF] Epp | Variables in mathematics education[END_REF], or as I suggested in Mesnil, 2014, is more unusual), especially to identify students difficulties with these concepts that can be related to their epistemological complexity, or their use in the classroom.

Moreover, I have used for the moment this reference to analyze the syllabuses and textbooks, and a training for teachers who offers a similar approach of notions of logic. But it could also be used to study the practices of teachers, students' activity and conceptions. It would be particularly interesting to compare the effect of knowledge in mathematical logic that teachers have or have not.

Proceedings of CERME10

Name of a famous french treatise, which original title is « La logique ou l'art de penser », written in 1662 by A. Arnauld and P. Nicole, who were very influenced by Descarte's method.Thematic Working Group 01Proceedings of CERME10

For each of these themes I have summarized the mathematical content, the language related issues, and research findings on student difficulties.

This semantic characterization can be described as "naive" because I do not strictly define what "being true" means.But of course, this characterization may be more rigorous with the notion of satisfaction of a formula in a model introduced by A. Tarski.

I give the examples in French first because I will explain some difficulties linked to the word un which is used in this language in different meanings, and which is translated in English with one, with a, with the… Commentaries following the examples refer to the French expression.


Philippe Lemoisson
email: philippe.lemoisson@cirad.fr

Stefano A Cerri
email: cerri@lirmm.fr

ViewpointS: towards a Collective Brain

Keywords: collective brain, collective intelligence, knowledge graph, human learning, knowledge acquisition, semantic web, social web

Tracing knowledge acquisition and linking learning events to interaction between peers is a major challenge of our times. We have conceived, designed and evaluated a new paradigm for constructing and using collective knowledge by Web interactions that we called ViewpointS. By exploiting the similarity with Edelman's Theory of Neuronal Group Selection (TNGS), we conjecture that it may be metaphorically considered a Collective Brain, especially effective in the case of trans-disciplinary representations. Far from being without doubts, in the paper we present the reasons (and the limits) of our proposal that aims to become a useful integrating tool for future quantitative explorations of individual as well as collective learning at different degrees of granularity. We are therefore challenging each of the current approaches: the logical one in the semantic Web, the statistical one in mining and deep learning, the social one in recommender systems based on authority and trust; not in each of their own preferred field of operation, rather in their integration weaknesses far from the holistic and dynamic behavior of the human brain.

Introduction

On one side, today's research on the human brain allows us to visualize and trace the activity along the beams connecting the neural maps. When publishing the Theory of Neuronal Group Selection (TNGS) more than thirty years ago, G. M. Edelman emphasized the observation/action loop and the social interactions loop. Both loops continuously evolve the beams connecting the neural maps under the supervision of our homeostatic internal systems also called system of values, and generate learning. On the other side, we live a digital revolution where the Web plays an increasing role in the collective construction of knowledge; this happens through the semantic Web and its ontologies, via the indexing and mining techniques of the search engines and via the social Web and its recommender systems based on authority and trust. The goal of our approach is twofold: i) to exploit the metaphor of the brain for improving the collective construction of knowledge and ii) to better exploit our digital traces in order to refine the understanding of our learning processes. We have de-signed and prototyped a Knowledge Graph where resources (human or artificial agents, documents and descriptors) are dynamically interlinked by beams of digital connections called viewpoints (human viewpoints or artificial viewpoints issued from algorithms). We-as-agents endlessly exploit and update this graph, so that by similarity with the TNGS, we conjecture that it may be metaphorically considered a Collective Brain evolving under the supervision of all our individual systems of values.

In section 2, we present a schematic view of the biological bases of cognition, starting by the "three worlds" of K. [START_REF] Popper | The Tanner Lectures on Human Values[END_REF] where the interaction between minds can be studied. We then re-visit the TNGS and the role played by our system of values (internal drives, instinct, intentionality…). We finally illustrate "learning through interaction" as exposed by D. Laurillard and J. Piaget. In section 3, we explore the collective construction of knowledge in the Web paradigm, assuming that a large proportion of the traces we produce and consume today are digital ones. We distinguish three paradigms, respectively governed by logics, by statistics and by authority and trust. Thus our challenge is to integrate these paradigms and describe how individual systems of values participate to learning events. Section 4 is dedicated to the ViewpointS approach, as a candidate for answering the challenge. The metaphor of "neural maps interconnected by beams of neurons" led to the design of a graph of "knowledge resources interconnected by beams of viewpoints", where each agent can exploit the traces of others and react to them by adding new traces. As a result, the combination of all individual "system of values" regulates the evolution of knowledge. We conjecture that it may be metaphorically considered a Collective Brain.

We conclude by recapitulating our proposal which has the limits inherent to any integrator: we are not yet sure if the collective knowledge emerging from our proposed Collective Brain will perform competitively with the existing separate paradigms respectively governed by logics, by statistics and by authority and trust. Nevertheless, if our proposal does not ensure scientific discovery about learning, we hope it represents a progress toward its comprehension.

A Schematic View of the Biological Bases of Cognition

In this section, we start by adopting a well-known philosophical position where the questions of cognition and interaction can be addressed. Then we draft a schematic view of the lessons learned from Edelman about the biological mechanisms supporting cognition, and finally we use this representation within D. Laurillard's conversational learning scenario in order to test it against the question of knowledge acquisition through interaction.

The Three Worlds

To start with our analysis about minds in interaction, we need some philosophical default position; "the three worlds" of K. Popper [START_REF] Popper | The Tanner Lectures on Human Values[END_REF] provides a relevant framework. Such a framework had already found an expression in the semantic triangle of Odgen and Richards [START_REF] Hampton | Concepts in the Semantic Triangle[END_REF]. The strong interconnection of the three worlds is developed in [START_REF] Searle | Speech acts: an essay in the philosophy of language[END_REF] where J. Searle explains how the interpretation of repeated collective experiences by individuals bears the emergence of an institutional reality founded on the use of language. In the following, we shall refer to the three worlds as W 1 , W 2 and W 3 , with the following definitions: W 1 is the bio-physical world where objects and events exist independently from us, from our perceptions, our thoughts and our languages. Causal relations, insofar we are not directly implied by some event, are also considered independent from us.

W 2 is the internal world of subjectivity, where the perception of objects and events of W 1 leave traces in memory that are combined in order to participate to the construction of our own knowledge, our consciousness about the world, where intentions appear and the emotions that will be the trigger for our actions.

W 3 is the world of the cultures and languages, made of interpretable traces: signs, symbols, rules of behavior and rules for representing objects and events of W 1 . W 3 is the support of communication among individuals. Within W 3 , we find all specialized languages of the scientific disciplines, as well as the language of emotions and feelings, for instance represented by smileys. Digital images such as satellite images or scanned documents are also part of W 3 .

W 1 is where it happens, W 3 is where we can communicate about what happens, and W 2 is where the links and the learning events are. For this reason, we are going to pay special attention to the internal world W 2 .

The Internal World of the Mind

This section pays a heavy tribute to the work of G. M. Edelman [START_REF] Edelman | Neural Darwinism: The theory of neuronal group selection[END_REF], [START_REF] Edelman | Comment la matière devient conscience[END_REF], founder of the Theory of Neuronal Group Selection (TNGS), and one the firsts to emphasize that the brain is not a computer, but a highly dynamic, distributed and complex system, maybe the most complex "object" of the known universe. There is neither correlation between our personality and the shape of our skull (despite the teachings of phrenology), nor localized coding of information; no autopsy will ever reveal any single chunk of knowledge available in the brain.

According to the TNGS, every brain is twice unique: first because its cellular organization results from the laws of morphogenesis. Most important, however, is Edelman's second reason for the brain uniqueness: the brain is a set of "neural maps" continuously selected according to the individual's experiences. These cards, or adaptive functional units, are bi-directionally linked one-another by a fundamental integrating mechanism: the "re-entry". This crucial hypothesis allows a functional integration requiring neither any "super-card" nor any "supervising program": the neural maps are like "musicians of an orchestra linked one-another by wires in the absence of a unique conductor". The bi-directional re-entry links are the result of a selective synaptic reinforcement among groups of neurons; similarly: the cards result from a synaptic reinforcement internal to each group of neurons composing them. These reinforcements are triggered and managed by the homeostatic internal systems, also called "system of values" of each individual.

Fig. 1 (/left part of the figure) shows an observation-action loop that highlights several brain cards re-entering when grasping an apple. This type of loops originates the perceptual categorization event, common to all highly evolved organisms, decisive for adapting the behavior to the likelihood of benefits or dangers.

In humans as well as in some higher mammals, there is a second level of categorization, supported by cards situated in the temporal, frontal and parietal areas. Beyond the immediate cartography of the world, humans may shape some durable concepts (conceptual categorization) that consider the past and/or the future.

Finally, the human brain parts specialized in language (the Wernicke and Broca areas) play a major role in the emergence of a consciousness of a higher level, enabling the human subject to "map" his-her own experience and study him-herself.

The basic principles of the TNGS (selective reinforcement and re-entry according to the advantages offered to the subject) potentially explain any learning process, from simple memorization to skill acquisition and knowledge acquisition. All these processes are regulated by our system of values. Fig. 1. The brain according to the TNGS of G.M. Edelman: a complex network of re-entrant maps in interaction loops with the world A kernel element of the TNGS is quite relevant for us: knowledge is supported by a "physiological complex and adaptive network of neural maps"; the metaphor of "knowledge graph" is therefore justified. It induces to search for a topology allowing to define distances and proximity, like it was conjectured by the "zone of proximal development" of Vygotsky [START_REF] Vygotski | Apprentissage et développement: tensions dans la zone proximale[END_REF], [7]; such a topology will be presented in chapter 4.

Minds in Social Interaction

According to the two loops at the right of Fig. 1, we always learn through interaction: observation/action versus social interaction. These two loops clearly appear in D. Laurillard's work [START_REF] Laurillard | A conversational framework for individual learning applied to the 'learning organisation' and the 'learning society[END_REF] when analyzing the acquisition of knowledge in higher education. In her scenario, a student and his teacher simultaneously experiment and discuss.

Social interactions

Observation / action

In [START_REF] Lemoisson | Un cadre pour la construction collaborative de connaissances lors de la conception d'un observatoire des pratiques territoriales[END_REF] we extended this scenario to interactions within a group of peers coconstructing a representation of a shared territory.

In this multi-peers scenario, interactions occur at two levels: i) peers act in the shared territory (in W 1 / objects and events) and ii) peers exchange personal views of the shared territory (in W 3 / language). Doing so, the assimilation / adaptation processes described by J. Piaget in [START_REF] Piaget | La construction du réel chez l'enfant[END_REF] are activated (in W 2 / mind), which can be interpreted in terms of series of re-entry loops according to the TNGS. In [START_REF] Cerri | Tracing and enhancing serendipitous learning with viewpointS[END_REF] we proposed a roadmap for understanding and forecasting cognitive and emotional events linked to serendipitous learning.

As a consequence of all above processes, inner views tend to synchronize and yield a shared representation. We propose that what happens on the Web is a generalization of this prototypical scenario, and can be called collective knowledge acquisition. Our approach aims at tracing it; this will be exemplified in chapter 4.

Humans in Web Interaction

The change in our lives that we have been experiencing since when Internet has gained a significant place, often called the digital revolution, has been theoretically addressed by several authors, among which S. Vial [START_REF] Vial | L'être et l'écran -Comment le numérique change la perception[END_REF] and D. Cardon [START_REF] Cardon | La démocratie Internet[END_REF] who respectively provide a philosophical and a sociological approach. This revolution has suggested a significant hope: Internet as a space of shared knowledge able to bring in new levels of understanding in the sense given by Gruber in [START_REF] Gruber | Collective knowledge systems: Where the Social Web meets the Semantic Web[END_REF]. Internet is a support for a huge set of digital traces interpretable by humans but also by machines; if we refer to the conceptual framework above, it is part of W 3 /language. This space is far from being homogeneous however, and the approaches to cobuild shared knowledge are multiples; hereafter, we consider three paradigms.

The first paradigm is governed by the logical evidence: we usually call semantic Web this logically structured part of Internet where humans interact with databases encoding the knowledge of experts according to consensual conceptual schemes such as ontologies. This allows logical responses (provided by reliable algorithms) to correctly formulated questions (and only to questions with such a property). But there are problems and limits. Firstly, ontologies only represent a fragment of the reality, and the consensus they reflect is necessarily local and temporary. Secondly, formal query languages assume a closed worldwhat is rarely the case. Thirdly, formal query languages require a learning effort in order to be used properly. Finally, interconnecting ontologies and supporting their evolution with time in a rapidly changing world are very heavy and costly processes. Various approaches based on automatic alignment [START_REF] Jain | Contextual Ontology Alignment of LOD with an Upper Ontology: A Case Study with Proton[END_REF] or machine learning [START_REF] Zhao | Ontology Integration for Linked Data[END_REF] exist, but the task is endless since each ontology's evolution is domain-dependent.

The second paradigm is governed by the statistical evidence. The issue is to exploit techniques of data mining, i.e.: scan without too many assumptions a corpus, also called data set, of tweets, sequences, clicks, documents, … and detect regularities, frequencies, co-occurrences of items or terms. In other words: to feed suitable algorithms with the big data in order to reveal regularities. By reducing the complexity of the digital world W 3 , the mining algorithms make it readable for humans. However, the simplicity of these descriptions must pay a price to the expressiveness or even to the effectiveness: we are just shown the surface and not the depth, the "meaning". Today, a simple question with three independent keywords on Google may give very disappointing results. What is even worse, inferential statisticsthe only one allowing us to take significant decisions -require selecting the data according to a predefined goal, and this remains hidden from the user. Interpreting the results in order to build chunks of science is therefore heavily biased.

The third paradigm is based on authority and trust, which rely on emotions and feelings. The algorithms of the social Web provide information search and recommendations by graph analysis of the various personal, subjective and spontaneous light traces such as 'likes', 'bookmarks' and 'tweets'. They clearly operate in an open world; the limits are firstly the impossibility to logically assess the quality of responses and secondly the absence of guarantee concerning their stability along time.

Coming back to the dream of Gruber and many others to fuse the three paradigms, a first attempt is the semantic Web project [START_REF] Berners-Lee | The Semantic Web[END_REF] which somehow aims at subsuming them within the logical one; after a first wave of enthusiasm it seems that the limits listed above resist, even if they are daily pushed forward. The ViewpointS approach discussed in chapter 4 aims to offer a potential step forward in the direction of subsuming the three paradigms within the third one i.e., building up upon trust towards 'peers', would they be humans, databases or mining algorithms.

The ViewpointS Approach Discussed and Exemplified

This section first briefly recalls the ViewpointS framework and formalism for building collective knowledge in the metaphor of the brain -a detailed description can be found in [START_REF] Lemoisson | ViewpointS: When Social Ranking Meets the Semantic Web[END_REF], [START_REF] Lemoisson | ViewPointS : capturing formal data and informal contributions into an evolutionary knowledge graph[END_REF] -and then illustrates them through an imaginary case.

In the ViewpointS approach, the "neural maps interconnected by beams of neurons" are transposed into a graph of "knowledge resources (agents, documents, topics) interconnected by beams of viewpoints". The "systems of values" of the agents influence not only the viewpoints they emit, but also the way they interpret the graph.

We call knowledge resources all the resources contributing to knowledge: agents, documents and topics. We call viewpoints the links between knowledge resources. Each viewpoint is a subjective connection established by an agent (Human or Artificial) between two knowledge resources; the viewpoint (a 1 , {r 2 , r 3 },, τ) stands for: the agent a 1 believes at time τ that r 2 and r 3 are related according to the emotion carried by . We call Knowledge Graph the bipartite graph consisting of knowledge resources and viewpoints. Given two knowledge resources, the aggregation of the beam of all connections (viewpoints) linking them can be quantified and interpreted as a proximity. We call perspective the set of rules implementing this quantification by evaluating each viewpoint and then aggregating all these evaluations into a single value. The perspective is tuned by the "consumer" of the information, not by third a part "producer" such as Google or Amazon algorithms; each time an agent wishes to exploit the knowledge of the community, he does so through his own subjective perspective which acts as an interpreter.

Tuning a perspective may for instance consist in giving priority to trustworthy agents, or to the most recent viewpoints, or to the viewpoints issued from the logical paradigm. This clear separation between the storing of the traces (the viewpoints) and their subjective interpretation (through a perspective) protects the human agents involved in sharing knowledge against the intrusion of third-part algorithms reifying external system of values, such as those aiming at invading our psyche, influencing our actions [START_REF] Duchatelle | Hacker son auto-prophétie[END_REF], or even computing bankable profiles exploitable by brands or opinion-makers [START_REF] Kosinski | Manifestations of user personality in website choice and behaviour on online social networks[END_REF]. Adopting a perspective yields a tailored knowledge map where distances can be computed between knowledge resources, i.e. where the semantics emerge both from the topology of the knowledge graph and from our own system of values expressed by the tuned perspective.

The shared semantics emerge from the dynamics of the observation/action loops. Agents browse the shared knowledge through the perspectives they adopt (observation), and reversely update the graph by adding new viewpoints expressing their feedback (action). Along these exploitation/feedback cycles, shared knowledge is continuously elicited against the systems of values of the agents in a selection process.

To illustrate this, we develop below an imaginary case where learners have to select resources inside an Intelligent Tutoring System (ITS) to which a Knowledge Graph is associated. They wish to learn about the topic 'apple' and from step1 to step4 the learners adopt a 'neutral' perspective which puts in balance all types of viewpoints (issued from the logical or mining paradigms, or from the emotions and feelings of the learners). However at step5 where B chooses a perspective discarding his own viewpoints in order to discover new sources of knowledge.

Step1 illustrates the initial state of the knowledge. A, B and C are co-learners in the ITS (linked as such within the logical paradigm); the big arrows within the icons represent their respective systems of values, which play a key role both in the choice of perspectives and in the emission of viewpoints. D 1 , D 2 and D 3 are documents that a mining algorithm has indexed by the topic/tag 'apple'.

Step2: A is a calm person who has time; she browses through D 1 , D 2 and D 3 and has a positive feeling about D 1 and D 2 (she likes both and finds them relevant with respect to 'apple'); the capture of her feedbacks results in linking D 1 and D 2 to her and reinforcing the links between the documents and the topic 'apple'. B is always in a hurry; he asks the Knowledge Graph the question "which is the shortest path between me and the topic 'apple'?" According to the paths in the diagram, he gets a double answer: B-A-D 1 -'apple'and B-A-D 2 -'apple'.

Step3: B's feedback to D 1 is positive; this results in reinforcing the path B-A-D 1 -'apple'. If he would re-ask his question, he would now get only D 1 .

Step4: C likes to explore; rather than taking a short path she browses through D 1 , D 2 and D 3 and has a positive emotion about D 3 (she likes it and finds it relevant with respect to 'apple'); this results in linking D 3 to her and reinforcing the linking between D 3 and the topic 'apple'. At this stage, if A, B and C would ask for the shortest path to 'apple', they would respectively get D 1 , D 1 and D 3 .

Step5: B is not fully satisfied by D 1 . He asks for a novel short path between him and the topic 'apple' using a new perspective: he discards the viewpoints expressing his own feelings in order to discover new sources of knowledge. According to this new perspective, B-A-D 1 -'apple', B-A-D 2 -'apple'and B-C-D 3 -'apple' have the same length i.e., D 1 , D 2 and D 3 are equidistant from him. B may now discard D 1 (already visited) and D 2 (already rejected) and read D 3 . Fig. 2. The network of interlinked resources evolves along the attempts of the learners A, B and C to "catch" the topic 'apple' through performing the modules D1, D2 or D. What is figured in the schemas is not the Knowledge Graph itself, but the views (also called Knowledge Maps) resulting from the perspectives; in these maps, the more links between two resources, the closer they are.

Along the five steps of this imaginary case, the evolution of "knowledge paths" follows the metaphor of the selective reinforcement of neural beams, except that this reinforcement is not regulated by a single system of values, rather by a collaboration/competition between the three systems of values of A, B and C. The three colearners learn as a whole, in a trans-disciplinary way: the dynamics are governed by a topology mixing information and emotions, not by pure logics.

Conclusion

Starting from the three worlds proposed by K. Popper (the external world of objects and events, the internal world of mind and the world of language), we have browsed through the TNGS of G. M. Edelman and learnt how the learning events are supported by an adaptive neural network, are regulated by our systems of values and occur mainly through social interactions. We have then focused on Web interactions The ViewpointS approach and formalism offer to integrate most if not all these elements in the metaphor of a collective brain; we illustrate through an imaginary case how to trace and enhance collective knowledge acquisition. Within ViewpointS, the three paradigms for knowledge acquisition (logical inferences of the semantic Web, statistical recommendations of the mining community, authority and trust of the social Web) are merged into a knowledge graph of digital traces interpretable by human and artificial agents. Within this graph, the beams of connections are regulated by the individual systems of values that support affect i.e., culture, personality traits, as defined in [START_REF] Bamidis | Affective Learning: Principles, Technologies, Practice[END_REF].

What we gain in the integration may be lost with respect to the advantages of each of the three knowledge acquisition paradigms taken individually. For this reason, we are not yet sure if the collective knowledge emerging from ViewpointS graphs and maps (our proposed Collective Brain) will perform competitively with a similar wisdom emerging from each of the three crowds.

Nevertheless: as it has been always the case in the synergies between technological developments and scientific progress, the developments do not ensure scientific discovery, rather may facilitate the process. For instance: Galileo's telescopes did not directly produce the results of modern astronomy, but enabled a significant progress. We hope and believe that our proposed Collective Brain will have a positive impact in understanding and enhancing some aspects of human cognition.

  question of the emergence of collective knowledge partially supported by algorithms.


Monika Ewa Rakoczy
email: monika.rakoczy@telecom-sudparis.eu

Amel Bouzeghoub
email: amel.bouzeghoub@telecom-sudparis.eu

Alda Lopes

Katarzyna Wegrzyn-Wolska
email: katarzyna.wegrzyn@efrei.fr

Technical Report: Influence Model Based on Actions and Reactions in Social Networks

Keywords: influence, social scoring, social network analysis

Today's social networks allow users to react to new contents such as images, posts and messages in numerous ways. For example, a user, impressed by another user's post, might react to it by liking it and then sharing it forward to her friends. Therefore, a successful estimation of the influence between users requires models to be expressive enough to fully describe various reactions. In this article, we aim to utilize those direct reactive activities, in order to calculate users impact on others. Hence, we propose a flexible method that considers type, quality, quantity and time of reactions and, as a result, the method assesses the influence dependencies within the social network. The experiments conducted using two different realworld datasets of Facebook and Pinterest show the adequacy and flexibility of the proposed model, that is adaptive to data having different features.

I. INTRODUCTION

Nowadays, due to the increasing number of people using social network sites, the number of careers created using networking sites such as Youtube has risen as well. The majority of such businesses made via social networking are involved with creators generating content for their audiences, i.e. youtubers, bloggers, instagrammers, earning fame and influence, as well as revenue from advertisement. However, one of the concerns of both creators and advertising companies that collaborate with Internet creators is how to measure, prove and sustain their influence on audiences. To illustrate the problem, let us consider the network that consists of different users -proactive and reactive ones, having different interests, and having various numbers of connections to others. Now, we want to target the social campaign to people, using limited resources. The question is how to select creators to share the event, which not only have (possibly large) audiences, but also the audience that would be involved in the campaign?

The topic of Influence Maximization [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF] is not new in the literature, however, the proposed methods focus more on the spread of the information in the network already assumed to be indicating influence. Hence, such works suppose to have users influencing others, basing on "friend" relations on Facebook, sharing posts or even calling each other. While these approaches focus on expanding the reach of information within the network (i.e. the possibility of user seeing some content), none of them actually guarantees that we have actual influence between users (that is engaging with the message). However, while only the pairwise relation between two users seems to be insufficient to immediately implicate the impact, analyzing and evaluating the engagement and reactions concerning a particular user seem to be good basis for evaluating the influence.

In this paper, we present a simple, general model, which we called ARIM (Action-Reaction Influence Model), for evaluating influence between users using one (any) on-line social network platform. In the model, we base on the users' proactive and reactive behaviors, that can be found on basically any social networking site. We concentrate on data flexibility aspect of the model, so that the model can be used with different datasets. Our focus is to create a model which uses data features that are possible to obtain for research, and in which each of the properties is connected to its overall expressiveness. Our contributions are as follows:

• We analyze characteristics of different social network sites, establish the key influence-related terms and present the simple social network sites interactions schema (Section II), • We propose a simple, general influence model ARIM, that focuses on three influence aspects important for model expressiveness namely intensity of users' reactions, spread and audience engagement differentiation and time dependency (Sections II and III), • We additionally propose a submodel based on ARIM, that is focused on trends in time intervals, called Inic -Interval Specific SubModel (Section IV),

• We built a framework based on the proposed models and test it using two real-world datasets, that allow us to validate the two models (Section V).

II. DESCRIPTION OF ACTION-REACTION INFLUENCE MODEL

In this section, we firstly define the basic notions and the core of our idea. Then, we describe three components of ARIM model and explain their importance for the model expressiveness.

A. Definitions

The aim of this work is to create a general model for evaluating influence, that is independent of the data platform on which it is applied on. On top of that, we focus on the simplicity of the model on both abstraction and data levels, for two reasons: to be compatible with the intuitive understanding of influence, and to be able to operate on the minimum features expressive enough to evaluate influence.

We define the influence from the perspective of Actions and Reactions on social network sites, which goes with intuitive understanding of influence, and the way we tend to evaluate influence in real world. The core of the Action-Reaction Influence Model (ARIM) consists in the simple schema of information exchange on social network sites, depicted in Fig. 1. We base on the situation in which one user is performing an Action by generating the content, i.e positing photo or post. The second user, being (a part of) audience, is performing Reaction1 ) to the content. We follow the categorization of the relation from social networks that distinguishes upvotes, comments, etc. (see Section II-B). We aim to measure the influence between Action generating user -the Subject of Influence -and the reacting user -the Object of Influence. In the following, we formally define the notions of influencer, reaction and the influence starting with a social network:

Definition 1: Action: An entity e 2 E such as a person (user), a group of people (e.g. company) or an object (content, conference, etc.) performs an action a 2 A such as posting, photo-sharing etc.

Definition 2: Reaction: An entity e 1 2 E can react to an action a 2 A performed by other entity e 2 2 E Definition 3: Social Network: A social network SN is a triple SN < E, A, R > where E is a set of entities, A is a set of actions performed by entities from E, and R is a set of reactions of entities from E Definition 4: Influencer: We say that e 1 2 E is an Influencer iff:

1) e 1 performs actions A 0 ✓ A observable in SN 2) Actions from A 0 are recognized by other entities e i 2 using reactions from R 3) Actions from A 0 have an impact on the behavior or structure of SN . Definition 5: Influence: Let e 1 , e 2 2 E from SN < E, A, R >. We say that e 1 influences e 2 iff actions A 0 ⇢ A performed by e 1 affects the behavior of e 2 resulting in the influenced entity e 2 engagement demonstrated in reciprocal actions-reactions R 0 ⇢ R. e 1 is then called the subject of influence and e 2 the object of influence.

Proposition: The relation of influence is asymmetric, timedependent and oriented. It involves actions generating content (proactive) and reactions (reacting on content).

Remark: Importantly, we do not assume that the Subject and Object of Influence have a relation between each other, i.e. follow, friendship, etc. Obviously, the actions of the Subject of Influence must be visible by the Object of Influence in order to the Object to be able to react. However, this assumption

• Subject of influence (u 1 ) • Object of influence (u 2 ) Action Reaction 1 . . .

Reaction n

Fig. 1. The Action-Reaction schema. Subject generates content visible to the Object who reacts to the content, possibly by using multiple types of reactions several times. implies only the visibility of the actions performed by the Subject, and represents more flexible approach in terms of the data requirements and applicability to various social network platforms without a notion of direct relation between users.

The presented general influence definitions and schema aim to cover three degrees of expressiveness, namely: intensity of influence depending on the reactions, influence spread and audience engagement and influence time dependency, which we discuss in the next section.

B. Intensity of influence depending on reactions

The fact that our model is based on the Action-Reaction schema (Fig. 1) implies the flexibility in terms of the number of features the data can include. Obviously, by minimum, the data is required to have the users and reactions of one kind (e.g. comments). Considering the existing to-date potential sources of data, we enlisted most of major social network sites in order to gather all the similar features categorized to Action and Reaction. The results are presented in Table I. From this comparison, it can be seen that most omnipresent reactions available to the users to perform upon the generated content are upvotes (also named likes, claps, hearts, etc. -in this work, all one-click reactions that imply appreciation of the content will be called by the general name upvote), and comments, and then shares (the action of sharing the content originally posted by other user). In this article, without loss of generality, we concentrate on these three reactions due to their ubiquity.

In particular, we want to focus on the implications that different types of reactions have on the influence strength. Indeed, upvotes, comments and shares have various functions and meanings within social networks. According to [START_REF] Scissors | What's in a like?: Attitudes and behaviors around receiving likes on facebook[END_REF], the upvote is treated as a lightweight reaction, easy to perform to acknowledge the posted content, similar to "wordless nod". In comparison, comments are regarded as "more satisfying to receivers" [START_REF] Scissors | What's in a like?: Attitudes and behaviors around receiving likes on facebook[END_REF]. Moreover, because of the quickness and easiness of upvoting, it may also be regarded as less meaningful than other reactions. Clearly, reacting using comment involves both effort (writing comment content in comparison to "default" value of upvote) and time (writing and answering versus just one-click). Furthermore, a study [START_REF]The psychology of sharing. why do people share online?[END_REF] has shown that the majority of people share content due to its value -94% of subjects share "valuable or entertaining content with others", 84% share to support causes and issues they care about. This means that when we see content that is highly impacting we are more willing to share it. This phenomenon is much different to the "casual" upvoting. It was also observed that receiving more complex reactions from acquaintances is corresponding to increase of relationship strength and closeness, as opposed to getting upvotes where no such association was noticed [START_REF] Burke | Growing closer on facebook: changes in tie strength through social network site use[END_REF]. Taking all the above into consideration, we assumed the hierarchy of the reactions, in which the reaction of sharing is regarded as better descriptor of influence than comments and comments implying higher influence than upvotes. To illustrate, let us consider the following example: user u 1 created the post. User u 2 liked the post, while user u 3 commented on the post. The hierarchy aims to evaluate influence according to the importance of the reactions, in this case evaluating the influence of user u 3 higher than user u 2 . Moreover, for each generated influencer content, we can specify several combinations of reactions of the influencee -user can only upvote, comment or share the content, can upvote and comment, upvote and share, comment and share, and obviously, all three at the same time, meaning user is upvoting, commenting and sharing the same content.

We do notice that one might argue that the importance of combinations could be modeled by using the linear combination of weights and vector of reactions. However, we aim to put different importance to each of the reactions (and their possible combinations) by not only using weights vector but also by utilizing non-linear multiplications corresponding to reaction combination, so that the importance of existence of two or three reactions at the same time can be stressed, and have greater value on final score. In order to achieve this, we need to model reactions with the use of a non-linear function.

C. Spread and engagement

The reaction schema presented in Fig. 1 is a simple, but important generalization of interactions on social networks sites. However, further consideration involves the case of multiple reactions to the same user's content.

Let us imagine a user u 1 performs an action -creates a post. The users u 2 , u 3 can react differently, as pictured in Figure 2. In two cases (Fig. 2 A and Fig. 2 B), only user u 2 reacted to the post. Intuitively, the comparison of these two situations leads to the conclusion that the influence of user u 1 should be higher in the Fig. 2 B, as the user u 2 has stronger response in this case. In other two cases (Fig. 2 C and Fig. 2 D), both users u 2 , u 3 reacted. We can say that the overall influence of user u 1 in Fig. 2 D should be higher, because of more reactions obtained. While it is fairly easy to compare these pairs of situations, the a and b or c and d, the issue starts to be complicated when we want to order influence from highest to lowest in all the presented situations. Intuitively, without any particular model we can state that influence in case a will be the lowest, while influence in case d the highest. However, it is not that obvious in case of b and c. Is the fact that user u gets reaction from two separate users important enough to evaluate influence of u in situation b higher? Or is the strength of particular reaction more important than the spread of the audience?

In order to tackle this problem, we propose to differentiate two components of influence, namely influence spread and audience engagement. Intuitively, in the example above, the user u 1 in situation c would have higher spread, while in situation b would obtain higher audience engagement value.

Definition 6: Spread: property of influence which determines how many users were actually affected with the generated content. The idea behind the spread is to calculate the active audience (meaning the one that is reacting to the initial activity), in order to determine the actual overall broadcast range. We define spread as a number of users that made reaction at least once, i.e. general audience cardinality.

Definition 7: Audience engagement: property of influence determining the strength of the reactions obtained from the reached audience. The audience engagement notion aims to conceptualize how powerful is the user's influence, therefore evaluates the overall involvement of already active users reacting to the content.

D. Time dependency

The third crucial factor we consider as an influence component is time. As it was mentioned in the influence definition, we acknowledge that influence is occurring in time, hence

• u 1 • u 2 • u 1 • u 2 • u 1 • u 2 • u 3 • u 1 • u 2 • u 3 A B C D 1 2 1 1 1 2 Fig. 2.
Examples of different scenarios of users (u 2 , u 3 ) reacting of the content generated by user u 1 . The arrow symbolizes the reaction (e.g. comment) and the number signifies the number of reactions it is necessary to examine and include different aspects of time in the influence estimation process. Here, we present the approach to include time dependency within ARIM.

Gaining the influence is a long-term and continuous process. Therefore, in many cases, considering the whole available dataset gives the best estimation of users influence to-date. However, while considering the whole data time period for calculation, including time aspects connected to actions occurring within this time is still valid.

To illustrate, let us consider two situations, in which users u 1 and u 2 are posting content for time period (t 1 , t 10 ), and content is visible to the same number of users (audience). In the first situation, user u 1 is posting in each of the time points t 1 , ..., t 10 . In the second, user u 2 has posted only two times, in t 1 and t 10 . Additionally, both users u 1 , u 2 received during this time the same amount and type of reactions. While evaluating influence, the question is whether user u 1 posting much more ("constantly") should have equal value of the influence (as we count only the reactions that both users received), or should user u 2 indeed have higher influence? We propose to tackle this issue by introducing the component that favors lower frequency of performing actions. This is in accordance with the fact that we tend to appreciate more users with less posts but with higher quality and possible maximum gain for reactions.

The presented idea aims to incorporate time-dependency aspect into ARIM without the need to divide the data according to the time. However, we do acknowledge that temporal changes are also of importance. Therefore, in Section IV, we additionally show the second proposed model, basing on ARIM, that is especially dedicated only to examine the shortterm influence trends.

III. SPECIFICATION OF ARIM MODEL

In this section, we present the formulas used for the model that are combined for the final calculation of the influence. All the notions used in this section were previously described in Section II.

First, in order to deal with distinction of different types of reactions (see Section II-B), we propose a Reaction Strength Per Reacting User function that combines all reactions done by particular reacting user ru on particular action a for user u using the following formula:

Strength(u, a, ru) = w1 ⇤ upvotes + w2 ⇤ comments+ w3 ⇤ shares + w4 ⇤ upvotes ⇤ comments+ w5 ⇤ upvotes ⇤ shares + w6 ⇤ comments ⇤ shares+ w7 ⇤ upvotes ⇤ comments ⇤ shares (1)
where upvotes, comments, shares are the number of each of the reactions for each actor respectively and w 1 , ..., w 6 are the weights that introduce additional hierarchy of the reactions. The weights should be tuned to the emphasize each reaction combination (lower weight -less emphasis). The number of possible reactions can obviously be higher or lower depending on the types of reactions the dataset includes. The function includes the multiplication of each pair of the reactions, so that in the case in which there is a combination of reactions the overall value of the measure increases. Inclusion of these combinations -cases when two or three reactions where done by one reacting user concurrently -reflects the fact, that Reaction Strength is considered per Reacting user. Then, we measure the Post Reaction Intensity (PRI) metric, which specifies how much, on average, appraisal (in the form of reactions) a post received, formally defined as:

P RI(u, a) = P |RU (a)| ru=1 Strength(u, a, ru) |RU (a)| (2)
where Strength(u, a, ru) is the defined above (Reaction Strength Per Reacting User)e, and RU (a) is the set of Reacting Users on the action a.

Next, the two influence components, Engagement and Spread, are defined (in accordance to their definition presented in Section II-C) as follows:

Engagement(u) = P |A| i=1 P RI(u, i) |A| ( 3
)
where A is the set of actions performed by user u.

Spread(u) = |RU | 6 = ( 4
)
where RU is the set of Reacting Users from the whole considered data, and index 6 = implicates the distinct set.

The component that favors lower frequency of performing actions (Section II-D), called fc (frequency component), is defined as:

fc(u) = e 1 |A| ( 5
)
where A is the set of actions performed by user u. Finally, the influence for user u can be described using the formula:

Inf luence(u) = Engagement(u) ⇥ Spread(u) ⇥ fc(u) (6)
The formula calculating the final value of influence is using the whole available data, in order to achieve the best estimate. Formula 6 binds all the measures presented before using the multiplication. Thanks to this, the contribution of spread and engagement to the final influence score depends on their value. Therefore, the user with high engagement and low spread will have the same influence as the user with low engagement and high spread, assuming that they have equal amount of posts. This is important, as being the better spreader does not straightforwardly implies being more influential (as shown in e.g. [START_REF] Kitsak | Identification of influential spreaders in complex networks[END_REF]). Additionally, the fc component penalizes the users with bigger number of posts, which means that for two users with equal spread and engagement, we prefer user that posts less.

IV. INIC: INTERVAL SPECIFIC MODEL

As mentioned before, time is an important aspect of influence. ARIM presented in the previous section is focused on the frequency problem for the whole dataset. However, we also want to examine the short-term, temporal changes in reactions, that would give us the possibility of observing the trends of users' reactions in terms of potential influence. In order to be able to observe these trends, we propose a second sub-model, Interval Specific Model (Inic), that bases on ARIM model, however it is dedicated only to dealing with the influence trends within time windows. Inic uses constant time windows, i.e. it will consider each time period resulting from time window separately. As we utilize the same concepts as presented in previous sections, here we will only precisely present the way is dealing with time.

First, it is important to notice that in many cases the length of the time window should be tuned to the specific portal characteristic, from which the data come. This is due to the fact that there are sites with a very rapid information exchange patterns, e.g. Twitter, where posts and reactions can be done in minutes and even couple of seconds, and there are those that have instant but more steady exchanges, e.g. Instagram. Moreover, the time window will also be highly dependent on the data granularity -data collected daily and aggregated into one snapshot will be by default unable to capture more rapid changes in the network.

It is crucial to notice that there are two time-lines of the data to consider: the action time-line and the reactions timeline. For instance, presented before ARIM model is basing on the Action-Reaction schema, hence it uses the information about the time of occurring reactions to estimate influence. Any influence model focused on time that utilizes quantities such as post engagement, spread or target audience has to determine a time period in which reactions targeted towards the post. For example, if a user posts some news (performs action) on social network portal, the impact of this post on the user's audience is not immediate. Instead, the influence on the user's audience is stretched in time, over the period from the post creation until the last audience reaction, e.g. last comment. Therefore, the question would be: is the post influential at the creation moment or when the reaction occurs? We propose to assign the influence value to time window, considering the reactions time-line (and not the time-line of action occurrence). In particular, we propose two alternative ways of assigning influence value to time window:

• Using only reaction time-line -We assess the influence value for each window in which the reactions to particular action occurred. In that way, although we do ignore the moment when the post was created, we correctly find the particular time window for which the influence (coming from reactions) have surfaced. For instance, let us assume that the action (e.g. post creation) was performed in time window 1, and reactions (e.g. comments and share) occurred in respectively time window 2 and 3. In this case, Inic would associate time windows 2 and 3 with respective influence values (and none in time window 1).

• Using point in time resulting from median of Action and

Reaction times -The idea is to chose the time window according to both action time, and the time of receiving the reactions. In particular, we determine the time window according to the median of time series consisting of delay between publishing the post and publishing the reaction. The calculated influence value for this action and reactions is placed in the time window which includes this "median" time. In that way, we bind influence value to the time window in which already half of all reactions towards the action have occurred. The choice of one of these methods will depend on the time window length and the social network site characteristics2 .

V. EVALUATION

This section describes experiments conducted in order to validate and check the proposed models, namely general ARIM and interval specific Inic showed in previous sections.

A. Datasets

In order to evaluate our proposed approach, we performed the experiments using two real datasets. The first consists of data from Facebook [START_REF] Spasojevic | When-to-post on social networks[END_REF] containing information about posts and their comments (without the text content) with precise information about the time of each of action/reaction. Second dataset includes data from Pinterest [START_REF] Zhong | Sharing the loves: Understanding the how and why of online content curation[END_REF], that contains repins (shares) and likes (upvotes). Table II presents the basic statistics about the both used datasets. We conducted three sets of experiments: two resulting in general influence score using ARIM model, utilizing datasets of Facebook and Pinterest, and one involving trend examination within time using Inic model, utilizing Facebook dataset, due to the fact that only this dataset includes precise time stamps for each action and reaction. The implementation was done using PostgreSQL ver 9.6 and R language ver 3.3.1.

B. Results of ARIM 1) Facebook:

The experiments performed on Facebook dataset, due to the nature of the data that contained one relation -comments, were done using all weights equal to 1. Table III shows the Top 10 ranking of the most influential users. It can be observed that the user in the first place, while having low (relatively to other top 10 users) engagement rate, is having exceptionally high spread. At the same time, this person has created 96 posts, which is also relatively low. Despite the fact that the audience is not very reactive, i.e. commenting only once, and not entering into discussions, the user is considered very influential due to the user's huge spread for very few actions. Complementary to Table III, the Figure 3 presents a comparison of Top 3 users in terms of influence score, engagement and spread rate, and number of posts. The high spread rate of top 1 user relative to two other top users can be easily noticed.

Another interesting case can be observed on the 8 th position in the ranking, with the person having the lowest spread rate in the ranking, significantly lower than both users on 9 th and 10 th position. Interestingly though, this person engagement rate is very high (highest value in the ranking), with additionally small number of posts. Therefore, this user higher place can be explained with the fact that ARIM is not only focused on both engagement and spread equally, but also it favors the low post number (see Section III). Hence, the user on the 8 th position in the ranking surpasses the next user (9 th position) that although having high spread, he/she has also lower engagement and very high (998) number of posts (the biggest number of posts in the whole ranking).

2) Pinterest: The second set of experiments was conducted on Pinterest database, containing two types of reactions, namely shares (called on the site "repins") and upvotes ("likes"). In order to show the results difference when considering shares to be of more significant value than upvotes, we used ARIM model with: (1) equal weights for all reactions, w 1 = w 2 = w 3 = 1 (see Equation 1in Section V-B), and (2) higher weight for shares, w 1 = w 3 = 1, w 2 = 2. Tables IV and V present the obtained results from both runs. Additionally, the complementary Table VI presents the detailed information about the users, containing the aggregated sums of upvotes and shares (2 nd and 3 rd column) and the number of times that both of the reactions occurred simultaneously (for the same post and from the same user). The information about the latter number (4 th column) is important, as the Equation 1 (Section III) in ARIM model includes an addition component in the formula regarding cases of simultaneous occurrence of the reactions (as was described in Section II-B).

Going back to the Top 10 rankings (Tables IV andV), it can be seen that for both experiment executions, the first three positions are unchanged. This can be explained by a very high spread value of each of the users, which is the predominant component for their high influence. Figure 4 shows in detail the ratios of spread, engagement and post number for each of the top 3 users. Moreover, from Table VI we can see that all of the top 3 users have a very high "combination" number, which means that they were apprised by other users simultaneously using upvotes and shares.

The use of the emphasis on the shares can be clearly seen by the example of user 820. In the first rank with equal weights, this person position is low (8 th ). However, the stress by using the higher weight for share reactions results in increase of the position of user 820 to 4 th , jumping ahead of users like 5656 who, while having higher spread value, has less shares in total. Similarly, user 4968, who also has a high number of shares, is promoted from 9 th to 7 th position. Interestingly, in the case of second rank (Table V), the emphasis on the shares also resulted in the top user 2777 having much higher influence score in comparison to all other users. Indeed, the gap between users 2777 (1 st ) and 20703 (2 nd ) significantly rose (two fold), while the gap between users on second and third position (20703 and 2367) stayed similar. This is due to the fact that user 2777 has significantly higher share sum (28920 versus 15944 and 19404 for users 20703 and 2367 respectively), along with the high spread rate.

C. Results of Inic

Experiments involving the trend analysis using Inic were done using Facebook dataset, due to the fact that only this data included time information for actions and reactions. Importantly, in Section IV, we presented two methods for assigning influence, using reaction time-line and using time resulted from median of time series of action and reaction. However, after initial analysis of the data, we noticed that for the majority of users, the time to obtain comment after posting is less than 200 minutes. Such a short time is not sufficient for observing the trends and obtaining conclusions about one's influence. Therefore, in the experiments, we used only the first method, while using arbitrary chosen time window of 7 days. Due to the space limitation, we present the trend analysis for only two, interesting users, selected from the list of top 100 influentials, obtained from previous experiments with the dataset (see Section V-B1). The information about the two selected users can be seen in Table VII. The results in the form of trends are presented in Figure 5 for user u 6 and Figure 6 for user u 89 , and include the trend of post count, engagement and spread ratio trends and influence trend. The dotted line in an approximation of the trend done using Linear Regression.

It can be seen that, in case of user u 6 , while the amount of created posts varies, both spread and the engagement decrease in time. On the contrary, user u 89 gains not only spread for his/her posts, but also the users that are commenting are highly active -the engagement ratio equals at the beginning 4 and is rising with time. In the case of user u 6 , the possible explanation for such trend is that independently from the number of posts that the user is generating, the audience is gradually loosing interest. Additionally, we can observe one significant peak in the spread and one time window, which suggests that the person was very popular (maybe controversial?) at one particular moment, which spiked the number of users reacting to the content. However, after that the user was not able to maintain such an audience. The case of user u 89 shows how the user is systematically gaining popularity (increasing the spread value). Moreover, observed increase of spread is simultaneous with the increase of audience engagement. This shows that Inic was able to determine influence evolution would be particularly helpful for campaigns in which the audience should be an active collaborator, for example giving feedbacks about the product.

VI. RELATED WORK

Much of the current research is focused on the problem of Influence Maximization, in which the aim is to find the group of nodes in the social network for which the information spread will be maximal. The systems, e.g. [START_REF] Chen | Robust influence maximization[END_REF], [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF], targeting influence maximization problem are focused on selecting a subset of users basing on the graph, where users are already connected via influence relations. In comparison, in this work we target evaluating the influence between the users and establishing a rank for them. There are several works proposing various influence metrics for ranking influential users in social networks. Many of them are platform specific, for example operating on Twitter [START_REF] Laflin | Discovering and validating influence in a dynamic online social network[END_REF], or job portals [START_REF] Rames | Multi-relational influence models for online professional networks[END_REF], as opposed to this work, which aims to propose the general approach that can be tuned to any social network site. Few of the well-known, well-established influence metrics base their model only on the typology of the network, i.e. PageRank [START_REF] Page | The pagerank citation ranking: Bringing order to the web[END_REF], or centrality measures [START_REF] Zafarani | Social media mining: an introduction[END_REF]. However, these methods do not consider deeper network characteristics, such as differences between types of reactions between the users, or the aspect of time dependency.

To the best of our knowledge, the work of Rao et al. [START_REF] Rao | Klout score: Measuring influence across multiple social networks[END_REF], [START_REF] Spasojevic | When-to-post on social networks[END_REF] is the closest to our approach. However, there are several key differences between the methods. Rao et al. system called Klout focuses on being multi-platform, while our ARIM is build for single platform, targeting a platformspecific influencers' analysis. This means that ARIM not only needs less data, but also it can be more helpful to use it in practice, i.e. for small advertiser companies. On top of that, probably due to the company privacy, none of the articles published by Klout reveals enough details about the metric to really have inside into their work, e.g. in their article they mention using both weights and more than 3 thousand features to calculate the final score, however, they do not include any details about them.

VII. CONCLUSION

In this paper, we presented the Action Reaction Influence Model that evaluates influence for each user basing on the proactive (actions) and reactive (reactions) behaviors of social network participants. The model targets three aspects of influence, namely intensity of influence, spread and audience engagement in the context of influence value, and time. Importantly, the model is flexible in terms of features that may or may not be available for particular social network site. Moreover, it also embraces the fact that different reactions types should be differently considered and should have various significance for the final influence score of a particular user. Furthermore, we have proposed an additional model, Inic, that focuses solely on influence trends in time. We have performed experiments on two real-world datasets including data from two well-known social network sites, namely Facebook and Pinterest. The results present interesting discoveries about the users influence and indicate the adequacy of the proposed ARIM model. They also show how different emphasis on various types of reactions can change overall influence rank.

As future work, we intend to deepen further analysis of influence trends. In particular, we want to focus on users that, while being not at the highest positions in influence ranks, show the potential to gain the influence with time.

Fig. 3 .

 3 Fig. 3. Detailed comparison of top three users from influence score using Facebook dataset

Fig. 4 .

 4 Fig. 4. Detailed comparison of top three users: 2777 (u1 -red), 20703 (u2 -green), 2367 (u3 -blue) from both influence scores using Pinterest dataset

Fig. 5 .Fig. 6 .

 56 Fig. 5. Falling trend of user u 6

TABLE II STATISTICS

 II ABOUT USED FACEBOOK[START_REF] Spasojevic | When-to-post on social networks[END_REF] AND PINTEREST[START_REF] Zhong | Sharing the loves: Understanding the how and why of online content curation[END_REF] DATASETS

		Parameter	Number
		Number of users	1 067 026
	Facebook	Number of users that reacted Number of posts Number of comments Time span of data	23 426 682 25 937 525 104 364 591 15/10/14 -11/02/15
		Number of users	1 307 527
	Pinterest	Number of users that reacted Number of posts Number of shares Number of upvotes	8 314 067 2 362 006 37 087 685 19 332 254
		Time span of data	03/01/13 -21/01/13

TABLE III TOP

 III 10 INFLUENTIAL USERS FROM FACEBOOK[START_REF] Spasojevic | When-to-post on social networks[END_REF]

	#	Engagement	Spread	#Actions	Influence Score
	1	1.039	66181	96	69478
	2	1.216	19793	549	24116
	3	1.208	18093	148	22012
	4	1.204	17030	103	20701
	5	1.071	17817	200	19183
	6	1.097	17040	941	18717
	7	1.092	16087	263	17637
	8	1.413	11086	67	15899
	9	1.053	14185	998	14953
	10	1.066	12678	34	13924

TABLE IV TOP

 IV 10 INFLUENTIAL USERS FROM PINTEREST DATASET OBTAINED USING ARIM WITH EQUAL WEIGHTS (ALL REACTIONS CONSIDERED EQUAL)

	#	UID	Engagement	Spread	#Actions	Influence Score
	1	2777	1.314	23386	1282	30743
	2	20703	1.249	19777	566	24747
	3	2367	1.367	13512	1025	18487
	4	5656	1.314	9843	535	12958
	5	4000	1.286	9908	360	12778
	6	1731	1.442	8553	328	12372
	7	5074	1.389	8876	465	12358
	8	820	1.262	9735	615	12304
	9	4968	1.301	9013	569	11742
	10	993	1.344	8580	387	11559

TABLE VII INFORMATION

 VII ABOUT TWO SELECTED USERS FOR TREND ANALYSIS

In the Action-Reaction schema, we refer to an Action as a self-activity of user u, while Reactions symbolize activities overtaken by other users in response to the user u Action.

For example, the median method will be insufficient in the case in which the average reaction time for the action will be short (minutes, some hours), and the time window will be longer (as we want to consider a big enough sample for the trends, e.g. couple of days), in which case the median will be included in the time window.


Adeline Morez

Nicolas Brucato

Kylie Susaki

Roxanne Tsang

Jason Kariwiga

Lauri Saag

John Muke

Kenneth Miampa

Alois Kuaso

Toomas Kivisild

la diffusion de documents scientifiques de niveau

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

POLYGENICITY OF SKIN COLOR Population Adaptation in Papua New Guinea A UNIQUE GENETIC PATTERN

Sampling

Biostatistical analysis

The uniqueness of Papuan skin pigmentation diversity is directly inherited by a set of genes with population-specific variants.

Adeline Morez 1 , Nicolas Brucato 1 , Kylie Susaki 2 , Roxanne Tsang 2 , Jason Kariwiga 2 , Lauri Saag 3 , John Muke 4 , Kenneth Miampa 5 , Alois Kuaso 5 , Toomas Kivisild 3,6 , Mait Metspalu 3 , William Pomat 7 , Matthew Leavesley 8,9 , Francois-Xavier Ricaut 1

Contacts: adelinemorez@gmail.com; nicolasbrucato@gmail.com and francois-xavier.ricaut@univ-tlse3.fr

More information on https://papuanpast.hypotheses.org/ To identify loci significantly associated with skin pigmentation, we performed a genomewide multivariate association test on pigmentation variables (Figure 2). 8 genes were found significantty associated with skin pigmentation (p<10 -8 ). None of these genes were previously associated in humans. However, GO term enrichment analysis revealed their biological role in skin cells and inflammatory response. Indeed, based on bibliography, these genes are known actors of the epidermis physiology, some even being influenced by UV-induction (like AKR1C1). Our results show that these 8 genes have specific variations in Papuan groups that significantly explain the large diveristy of skin pigmentation.

A more in-depth analysis is now necessary to determine if some of these SNPs are population-specific, which would explain the striking difference of skin pigmentation between Papuan groups.

106 participants for which four skin pigmentation variables were collected (using DSM II DermaSpectrometer,Cortex Technology) (Figure1):

-on the inner arm (baseline constitutive skin color) and on the top of the hand (most tanned region) -melanin (main skin color pigment) and erythema (vascularisation in response to sunburn) index.

Anthropological questionnaires (language, genealogy) to access individual origins. 58 whole genome sequences (30X, Illumina X5).

Clustering analysis of skin pigmentation variables.

Pigmentation variables corrected for gender and East Asian ancestry (inferred from ADMIXTURE analysis).

Multivariate

genome-wide association with Plink.multivariate (10 8 pernutations).

GO term enrichment analysis with Enrichr.

Candidate gene univariate associations with Plink:

-new associations found -previously associated genes.

Papua New Guineans are one of the most biologically and culturally diverse people of the world, resulting from complex demographic processes and the wide spectrum of geographical contexts. They settled all territories, from the high-altitude mountains to the coastal swamps on New Guinea Island as well as in the neighboring islands. Each of these populations adapted their culture and technologies to these environment, as did their biology. A remarkable feature of the Papuan biological diversity resides in the large panel of skin pigmentation. Not only groups of Papuan ancestry show the darkest skin color worldwide, as in Bougainville, the variability between group is unmatched. This observation has long been made but so far no study focused on the genetic information that drives it in order to understand the anthropological mechanisms that led to this exceptional biological patrimony. Using whole genome sequences and multiple standardised skin pigmentation measurements, we conducted the first genome-wide analysis of Papuan skin pigmentation. Based on our multivariate association test and on bibliography, we selected 30 candidate genes related to skin pigmentation in diverse human groups. We ran univariate association tests on each candidate genes.

CONTEXT

Each of the genes found in our multivariate analysis show more significant signal of association with the variable related to the melanin index of the inner part of the arm. None of the genes previoulsy identified to be associated to skin pigmentation in Europeans and East Asians are implicated in skin color in Papuans. Only SMARCA2-VLDLR, previously found to be responsible for skin pigmentation in an African population, show a significant association (Figure 3). This could be due to the shared ancient ancestry or to an evolutive convergence caused by a similar environment.

We thanks the members of the Archaeological Laboratory of the University of Papua New Guinea for their help and support, and the all the participants to this study from the University of PNG, the NMAG. We also acknowledge support from various Papua New Guinea partners: National Research Institute, National Museum and Art Gallery, Institute of Medical Research, University of Papua New Guinea. This research is supported by the French National Research Agency, the French Ministry for Europe and Foreign Affairs (France), the French Embassy in Papua New Guinea and the University of Papua New Guinea (PNG). All picture copyright MPF-PNG.

Figure 1 :

 1 Figure 1: Skin color variation within Papua New Guinea. Melanin index was took from the upper inner left arm. black dots representindividuals's location. x and y axis represent respectively longitude and latitude

  Laboratoire d´Anthropobiologie Moléculaire et Imagerie de Synthèse (AMIS), Université de Toulouse (Paul Sabatier), Toulouse, France; 2 Archaeology, School of Humanities &Social Sciences, University of Papua New Guinea; 3 Estonian Biocentre, Tartu, Estonia; 4 Social Research Institute, Papua New Guinea; 5 National Museum and Art Gallery, Papua New Guinea; 6 Division of Biological Anthropology, University of Cambridge, United Kingdom; 7 Papua New Guinea Institute of Medical Research, Goroka, Papua New Guinea; 8 Archaeology, School of Humanities &Social Sciences, University of Papua New Guinea; 9 College of Arts, Society & Education, James Cook University, Cairns, Australia

	1 CNRS UMR 5288,
	Figure 3: Univariate association of SMARCA2-VLDLR to each of the four skin pigmentation variables.
	Red line represents p-value = 4,2E-4. Significantly associated SNPs are represent by blue dots. ArmM: arm melanin,
	HandM: hand melanin , ArmE: arm erythema, HandE: hand erythema.Associations were runon 1.0E-8 permutation using Plink.