P Marquet

& B Coulibaly

The concept of instrumental conflict: an application of the theory of activity to computer-supported teaching-learning situations

Keywords: theory of instruments, theory of activity, instrumental conflict, didactical artefact, pedagogical artefact, technical artefact

The purpose of this chapter is to define and to illustrate the concept of instrumental conflict, as a tool to a better understanding of the difficulties that learners encounter while using ICT (Information and Communication Technologies). An instrumental conflict takes its meaning within the framework of Rabardel's theory of instruments, which in itself is a part of the more general theory of activity.

The main idea is that VLE (Virtual Learning Environments) associate three different types of artefacts: didactical, pedagogical and technical. These artefacts must be correctly combined, so that learners can have access to knowledge, which is embedded in the device. We point out the usefulness of such a concept. We thus describe how scientific communities, which focus on didactics of Mathematics and computer science, and on ICT in education and training in particular, deal with difficulties of implementation and exploitation of ICT.

It appears that some objects called either didactical objects or pedagogical objects represent a reality, which is similar and extremely wide at the same time. In fact, this ambiguity justifies the distinction between didactical artefacts, pedagogical artefacts and technical artefacts. We finally give some examples of obstacles that can be considered as instrumental conflicts as well as some possible developments offered by the concept of instrumental conflict.

Introduction

Over several years, a unanimity has grown up around interest in an instrumental approach as a particularly rich heuristic perspective upon understanding human activity, particularly those in which computers are used [START_REF] Bruillard | Apprentissage coopératif à distance : quelques repères sur les questions de recherche[END_REF][START_REF] Linard | Concevoir des environnements pour apprendre : l'activité humaine, cadre organisateur de l'activité technique[END_REF]. It must be remembered, however, that this instrumental approach takes as its conceptual base theories of activity developed by Vygotski since the 1930s, the period in which he advanced the first attempt at a theory conceptualising activity mediated by tools and signs, which Rabardel [START_REF] Rabardel | Les hommes et les technologies, Approche cognitive des instruments contemporains[END_REF] reframed and extended to contemporary technologies.

The aim of this article is to show how Rabardel's approach can be applied to school teaching situations or to training involving ICT (Information and Communication Technologies). Such situations constructed by design, are often complex and demand that their subjects interact with objects of different natures: didactical, pedagogical, technical. Virtual Learning Environments (VLEs) may be considered in the same way in that they bring together on platform programmes different objects, posing the question of their optimal organisation, in the sense that the user has to some degree acquire the knowledge presented and made accessible by the system.

Initially this article will revisit the principal concepts of instrumental theory and will show how the matter of instrumental conflict arose in the body of theory which it could complete. The main idea is to make a distinction between the three components of a technical system dedicated to teaching, to know, firstly, the content of the discipline being taught, secondly the possible forms for their representation and the scenario for their presentation and, thirdly, the functionalities of the software programme. These three types of components are each the object of a process of adaptation each of which can interfere with the other and produce a conflict.

The second element of this paper will show that the simultaneous presence of didactical, pedagogical and technical objects presents difficulties in terms of theoretical approaches which relate to situations of computer-supported teaching and learning. This is the case particularly with attempts to analyse pupils' activities in Mathematics using certain software programmes, and also with models which depend upon descriptive norms of pedagogical objects such as can be used in computing. It appears that teachers of Mathematics and computer scientists are both confronted with a similar problem, the fact that the concepts that they each propose tend to cover too diverse a set of realities. In this respect, the concept of instrumental conflict offers the opportunity to describe situations and in a more precise manner than other approaches that might be mentioned.

The subsequent element presents the articulation of an extension to the instrumental approach which is proposed to run alongside the instrumental approaches of teachers of mathematicians and computer specialists. It is considered that, without presuming it to constitute a major epistemological leap forward, the concept of instrumental conflict offers greater precision in matters of computer-supported teaching and learning to the extent that it identifies where there may exist a discrepancy between a proposed element of content, a structural format or a presentational scenario and the possible system options for action. In this regard, the developments in pedagogical engineering put into operation for distance learning will gain from this research.

Finally the article concludes by suggesting how and in what circumstances instrumental conflict is likely to arise, such that readers interested in this approach can use the concept to their advantage.

Principal concepts in instrumental theory

The Rabardel approach

One of the Rabardel central concepts concerns the instrument, tool or technical object, this being long the subject of considerable interest on the part of the scientific community. According to Rabardel, the concept generally involves two different facets: the one centred upon the intended function of the instrument in the activity and the other centred upon the activity itself.

Simondon [START_REF] Simondon | Du mode d'existence des objets techniques[END_REF] illustrates this first perspective. Simondon established a distinction between instrument and tool. The instrument serves to draw upon information while the tool serves to carry out an action (p. 80). These two dimensions correspond with what he called the technical object. Although taking it in a broader perspective intending to address the relationship between man and his socio-technical environment, he considered that the technical object is always oriented towards the accomplishment of a particular function. He notes: "the beginning of the mediation between the organism and its environment", which heralds the arrival of biological metaphor to explain the concept of instrument. Running counter to this, Guillaume and Meyerson [START_REF] Guillaume | Recherches sur l'usage de l'instrument chez les singes : choix, correction, inventions[END_REF], offered a concept of the instrument centred upon activity. They illustrated their conception of the instrument by analysing the use that monkeys make of instruments in their activities. What was important in this latter context was what the subject did with the instrument and the active power that it conferred upon the monkey concerned. For them, the instrument was an intermediary between the subject and the world whereas for Simondon this intermediation existed between the organism and its environment.

Wallon [START_REF] Wallon | L'évolution psychologique de l'enfant[END_REF] compared the human instrument with that of the monkeys. He wrote taking the perspective of an instrument centred upon an activity, with, in addition, the idea that the instrument enables the accumulated experience to be capitalised upon. That renders the instrument not only an intermediary, but also an experience and a capitalised knowledge. This perspective is also developed by Vygotski and above all Leontiev in their theory of activity. The fundamental aspect to draw from this conceptualisation of the instrument is that it (the instrument) has no significance except in relation to the subject, a subject engaged in the process of producing an activity.

That is the reason why Rabardel proposed a distinction between artefact and instrument. The artefact is a man-made, material object: "in anthropology, the notion of artefact designates anything that has undergone a transformation, however minimal, of human origin" (p. 39 of the translation), whereas an instrument designates "the artefact in situation, inscribed in usage, in an instrumental relation of action to subject as a means of the action" (p. 39 of translation). The instrument is thus the result of the utilisation of a tool [START_REF] Contamines | Approche instrumentale des banques de ressources éducatives[END_REF]. One can thus say that the tool is itself an artefact, and that the term instrument can be used to establish the artefact as a means of realising the activity of the subject. Hence it is the subject who confers upon the artefact the status of instrument.

Another fundamental dimension in Rabardel's conceptualisation of the instrument is that it cannot be reduced solely to its material nature: the technical object in the Simondon [START_REF] Wallon | L'évolution psychologique de l'enfant[END_REF] sense of the term. Thus the symbols, the signs, the language and all the intellectual constructions are also instruments, from whence comes a second, symbolic nature which it may take on. In effect, in his conceptualisation of an extended instrumental theory, Rabardel [START_REF] Rabardel | Le langage comme instrument ? Élément pour une théorie instrumentale élargie[END_REF] considered only that the representation of the instrument should not be limited to a particular type of instrument like technical or psychological tools in the sense employed by Vygotski. In this way all constructed material or symbolic objects are artefacts and become instruments in interaction with a subject.

Thus the instrument is considered as combination of attributes in one entity, with two facets relating to subject and artefact. And yet more important, this entity relates also to the subject and society because every instrument has an eminently social dimension. The mode of operation or usage employed by the subject or community refers back to another critical component of Rabardel's theory, that of instrumental genesis. This results from the dynamics of the subject's activity and incorporates two processes:

-On the one hand, a process of transformation of the artefact in an activity situation is as much a matter of its structure as its functioning -it is instrumentalisation which connects the attribution of a function to the artefact on the part of the subject. -On the other, there is the transformation of the subject himself at the cognitive level: this is instrumentation, which consists of the adaptation of the subject's knowledge to the artefact or in the creation of new knowledge. These two processes are jointly involved in a reciprocal, two-way relationship. They represent two inseparable dimensions of instrumental genesis.

To synthesise, it is held that constructed cultural objects, be they material or symbolic, possess identical characteristics of which artefacts are made. Whenever a subject interacts with an artefact, an instrument emerges following a phenomenon of instrumental genesis, composed of a process running from the artefact towards the subject (which is instrumentation) and a process running from the subject towards the artefact (which is instrumentalisation) (cf. fig. 1).

Fig. 1 The concepts of artefact, instrument and instrumental genesis.

Application to learning and teaching situations

The extension of instrumental theory to symbolic objects as proposed by Rabardel provides the opportunity to consider knowledge in the process of being acquired in learning and teaching situations as much as the artefacts themselves [START_REF] Marquet | Informatique et enseignement : progrès ou évolution[END_REF]. In all cases, these artefacts need to be sub-divided into didactical artefact and pedagogical artefact. Everyone writing in the field of instrumental theory has to specify that the term didactical is used in a sense far removed from its everyday sense. Here, didactical is neither to be understood in the sense of teaching method, a technical means or a particular pedagogy, nor as the art of teaching…. but rather in the sense of content [START_REF] Portugais | Didactique des mathématiques et formation des enseignants[END_REF]. In fact for Brun [START_REF] Brun | A propos de la didactique des mathématiques[END_REF] the redefinition of the term didactical contains a willingness to refocus upon the importance of teaching content. It is felt that this brief clarification is warranted to resolve any misunderstanding between the terms pedagogy and didactics as the distinction is not always immediately obvious.

Didactical artefact is thus the term used here for all the disciplinary content which needs to be learned in order to become an instrument of the one mastering it. A pedagogical artefact is defined here as being like the formalism for representation and/or the disciplinary content presentation scenario appropriate to its teaching. Formalism for representation is taken here to mean the semiotic processes of the designation of the didactical artefact, and presentation scenario is taken to mean the presentation of the didactical artefact, also called pedagogical scenario [START_REF] Paquette | L'ingénierie pédagogique, pour construire l'apprentissage en réseau[END_REF].

In distinguishing a didactical and a pedagogical artefactual dimension within knowledge taught, this paper makes the same sort of distinction as does Peraya [START_REF] Peraya | Le cyberespace : un dispositif de communication et de formation médiatisée[END_REF], for whom, each time something is taught, he sees it as being taught within a semio-pragmatic context, which understands the signifiers essential to its formulation, and which are situated in a discourse oriented towards the appropriation by the learner of the particular object being taught.

Duval [START_REF] Duval | Sémiosis et pensée humaine, Registres sémiotiques des apprentissages intellectuels[END_REF] also himself makes a distinction between two aspects of the same element of knowledge. He calls it noesis, a term which he borrowed from Plato and Aristotle, taken to mean the cognitive acts like the conceptual understanding of an object, discrimination of a difference or the understanding of an inference, and he indicates by semiosis the production of a semiotic representation. In the field of linguistics, this conception corresponds respectively to the signified and the signifier. The nature of expression is built upon that raft of signifiers, i.e. that of the form and nature of that which is signified as content [15].

Two families of artefacts are thus present, one being the knowledge ordained by the situation, and the other the discursive setting for learning. In order for one element of knowledge to be effectively acquired by the learner so that, together, they interact as an instrument, it is essential that the learner takes on board both the didactical understanding (content) and the pedagogical (the formalism for representation and/or the presentation scenario). Were one to risk an analogy here: where some would see water, others in fact see one atom of Oxygen and two atoms of Hydrogen which together constitute a molecule of water.

A fundamental principle is thus proposed here that every didactical object is associated to a pedagogical object in a teaching situation. Each of these objects, considered like an artefact, must be instrumentalised and instrumented suitably by the subject, i.e. that two concomitant phenomena of instrumental genesis have to be in operation. To put this another way, the learner has to attribute the right functions at times to the content and to its formalism for representation and he must also adapt his knowledge and apply it at times to the content and its formalism for representation.

In order to illustrate this didactical and pedagogical duality relating to associated artefacts the example of the multiplication may be used. To help the instrumentalisation of this didactical object one has to typically resort to two formalisms. The first consists in writing in lines and columns the list of results to be recorded (cf. fig. 2a). The second is presented in the form of a double-entry table (cf. fig. 2b). Thus there are the two objects: the one didactical -multiplication, the other pedagogical -be it writing in lines or the double-entry table, which have to be mastered by the pupil at some point in his schooling. The didactical artefact can function very well with one or other of the pedagogical artefacts, and reciprocally, each of the pedagogical artefacts can function very well without the didactical artefact, as is the case when writing on lines is used to represent the results of other calculations, like addition, for example, or when this type of table serves to represent a calendar with which nursery school children are familiar.

a) […] b) 3 x 1 = 3 4 x 1 = 4 x Ú 1 2
In order to complete the matter of the distinction between didactical and pedagogical artefact, it is interesting to reflect upon the follow-up to Duval's work which was considered above, which describes the existence of two postures in the learning of Mathematics: one being of dissociation and the other of non-dissociation. The first posture, corresponding to the distinction already made between didactical and pedagogical artefact, makes the distinction between mathematical objects, being numbers, functions, straight lines, graphs etc, and their representation, an indispensable condition for better learning. What matters is more the object represented rather than the style or form for its representation, the latter being only ever a means of communication of the object. This point of view postulates that the fact of not distinguishing the content from form leads in the long term to a loss of understanding. The representation of mathematical objects here is then secondary and constitutes a property extrinsic to cognition and the conceptual appreciation of these objects.

The second posture is that of non-dissociation between noesis and semiosis. The field of Mathematics could constitute the primary domain where semiotic representations are indivisible from the objects represented because the possibility of carrying out applications upon mathematical objects could depend directly upon the semiotic system of representation utilised. The analysis of problems in the learning of Mathematics and of the stumbling blocks students regularly come up against suggests that non-dissociation is a fundamental law in cognitive functioning. Not only does semiotic representation serve as a means of communication, but it is also indispensable to activity in Mathematics. It is, from this point of view, an intrinsic property of mathematical objects.

Towards a theory of instrumental conflict

The didactical and pedagogical traditions which have arisen owing to the laws and regulations of teaching and consequently since the industrialisation of teaching [START_REF] Rabardel | De l'apprentissage par les artefacts à l'apprentissage médiatisé par les instruments[END_REF] have progressively determined how the majority of content should be presented in order for it to be assimilated by the greatest number of learners. It is a fact that these traditions are not always a great help when one wishes to introduce ICT into a learning and teaching situation.

In fact, things get even more complicated when didactical and pedagogical artefacts are associated with technical artefacts. For example, a software which teaches multiplication in elementary school is a technical artefact, which, to become a technical instrument has to become a learning object which, in turn, depends upon the instrumentalisation and instrumentation of the user. But as much as it may be a technical artefact, this VLE also brings into play the aforementioned didactical and pedagogical artefacts, which, in their turn, have to be suitably instrumentalised and instrumented in order to become real instruments. That which Peraya [START_REF] Peraya | Le cyberespace : un dispositif de communication et de formation médiatisée[END_REF] terms techno-semio-pragmatic appears similar to what is referred to here as an overlay of three artefactual layers: didactical, pedagogical and technical. (cf. fig 3).

Thus, the introduction of a technical system may provoke a disturbance of the balance between didactical and pedagogical artefacts, to the extent that the formalisms representation and/or the representation scenarios which were pertinent beforehand are found no longer usable. These disturbances to equilibrium may be termed instrumental conflicts, suggesting that the processes of instrumentalisation and instrumentation of the various artefacts in question can interfere with each other.

In an instrumented teaching and learning situation, the learner-subject is not only a physical, cognitive or social entity in interaction with a technical system, he is equally a subject who is intentionally engaged in the undertaking of his tasks [START_REF] Rabardel | De l'apprentissage par les artefacts à l'apprentissage médiatisé par les instruments[END_REF]. In the realisation of these tasks, the learner carries out activities which can be both productive and constructive, to the extent that the subject produces a response to the situation and where the task concerned confers upon him an additional cognitive development. Each time one introduces a technical system, one takes the risk that the different levels of instrumental genesis may interfere with one another and deprive the learner at times of the possibility to respond to the situation and of constructing the didactical instrument as envisaged in the particular situation. It is in this combination and usage that learner-users make of didactical, pedagogical and technical artefacts that the optimisation and efficiency of learner activity comes about, be it productive or constructive and the hypothesis is advanced here that failure to produce the anticipated responses results from what may be called instrumental conflict, in other words, the unfortunate association between one or several artefacts which produces a failure in the instrumental genesis of at least one of the three artefacts.

IT-based didactical and pedagogical objects in the teaching of Mathematics

A significant portion of VLEs have been developed by ICT specialists with a view to autonomous learning with the benefit of pre-existing knowledge or competencies [START_REF] Pernin | LOM, SCORM et IMS-Learning Design : ressources, activités et scénarios. Babel-edit[END_REF]. VLE designers not necessarily being teachers or specialist trainers in tutoring learners, the question of the integration and adaptation of these technical artefacts to classic teaching and training practices cannot fail to be posed.

This question is as much, if not more, important than the conception of learning objects, which in the present context of ICT development covers matters which run far beyond the realm of pedagogical engineering. In effect, it is all about normalising teaching content in order to be able to organise it by function relating to the intended learning objectives, given VLE functionalities, the language in which they are presented, the area of knowledge they address, etc. This paper holds to the logic of conception and representation of didactical and pedagogical objects.

Mathematics is the school discipline area where the introduction of VLEs is the most evident, from the very inception of which one often finds a quite animated discourse [START_REF] Artigue | Une approche didactique de l'intégration des EIAO à l'enseignement[END_REF]. According to Guin and Trouche [START_REF] Guin | Analyser l'usage didactique d'un EIAH en mathématiques : une tâche nécessairement complexe[END_REF], this discourse draws its legitimacy from Piaget's constructivism and is in fact characterised by a twin illusion: -That of a naturally positive contribution to learning: the environment allows one to see and therefore to understand.

-That of a naturally positive contribution to teaching: this illusion is based upon the principle that the introduction of a VLE results in reducing the cognitive loading upon learners in the resolution of mathematical problems. In effect, the computerised artefacts would lift the technical operations off the learners' shoulders and thereby allow them to focus upon the mathematical objects.

For the IT specialists, putting in place a training structure can be achieved by the successive addition of 'digital building blocks' of different shapes and sizes, which can vary from a simple document right through to an entire training programme. This vision arises from the engineering of pedagogical objects based upon an object-centred approach, also used in the development of software. What is important in this conception is the internal coherence of the technical artefacts operating in the learning context. But in the particular context here technical artefacts constitute the third element, giving rise to the emergence of instrumental conflicts. What, then, does the didactical object actually signify for Mathematics teachers and the pedagogical object for computer specialists?

Computerised didactical objects as viewed by teachers of Mathematics

A study of the usage of Dérive1 software programme by Lagrange and Drouhard [START_REF] Lagrange | L'intégration du système de mathématiques symboliques DÉRIVE, une évaluation didactique auprès des élèves de 14 à 19 ans[END_REF] has shown that the pupils did not automatically manage the transition from the technical to the conceptual and that they did not directly access the didactical objects which could be manipulated by the software. In reality, this process did not work to solve every problem and its operation could only be technical because resorting to Dérive did not mean by its very use that a better explanation of processes would be achieved.

From this observation Artigue [START_REF] Artigue | Une approche didactique de l'intégration des EIAO à l'enseignement[END_REF] deduced the existence of two phenomena, that of double-reference and pseudotransparency, in order to explain the integration complexity of a VLE, in this case Dérive, in the context of teaching and learning in Mathematics.

Double reference

The situations observed by Lagrange and Drouhard [START_REF] Lagrange | L'intégration du système de mathématiques symboliques DÉRIVE, une évaluation didactique auprès des élèves de 14 à 19 ans[END_REF] occurred in two environments: that of the software programme and that of paper-pencil and consisted of factorisations of the polynomial X n-1 and of trigonometrical calculation. The phenomenon of double reference thus arises from a confrontation of the traditional environment, paper-pencil, and that of Dérive [START_REF] Guin | Analyser l'usage didactique d'un EIAH en mathématiques : une tâche nécessairement complexe[END_REF].

Artigue [START_REF] Artigue | Une approche didactique de l'intégration des EIAO à l'enseignement[END_REF] thus takes account of the rational factorisation of the polynomial in the penultimate year of highschool: "In the paper/pencil environment the factorisation of the polynomial is linked, at this educational level, to research into real roots […], to techniques of polynomial division […]. Dérive's algorithm in the internal workings of the machine worked by intermediary factorisations in Z/pZ. Evidently these two levels were not accessible to these 17 year old pupils, because Dérive was to function as a 'black box' producer of various results which would be valid a priori" (p. 20). In reality, the fact that the technical system was similar to a 'black box' is not unusual. For example, when pupils use a calculator, they do not have access to the way in which the machine does its calculations. The difference between the factorisation of polynomials situation and that of doing simple calculations with a basic calculator lies essentially in the difference in degree of complexity, which is determined by the teacher.

The author shows that in this case there are two possible interpretations: one that for the pupils it is a case of bringing forward ideas of factorisation in the classic paper-pencil model with the help of Dérive, and the other in which the pupils would produce results of factorisation coming from Dérive. In both cases, the pupils encounter difficulties. These difficulties result from the computer transposition as Balacheff [START_REF] Balacheff | La transposition informatique. Note sur un nouveau problème pour la didactique[END_REF], expressed it, and from constraints associated with such a transfer. This will be addressed further later.

The second observation relates to trigonometric calculus with the help of Dérive. It resulted in the same conclusions being drawn according to which pupils are confronted with simplification difficulties with the software programme. Although the simplifications that Dérive enables are based upon the formulae of classical trigonometry, not least there remains the problem that Dérive's simplifications are difficult to put into practice.

It would appear to be very clear that the integration of a VLE in learning does not make any easier or better teaching and learning situations in mathematics, as the proponents of this discourse would suggest. On the contrary, the computer transposition often comes with constraints which can constitute real handicaps to learning. These constraints thus weaken the mediation capacity of the technical artefact. Under these conditions the software programme no longer plays its role in epistemological mediation such that achieving the didactical objective (in this case the cognitive mathematical objective) is no longer possible.

In her thesis concerning the integration of spreadsheets in algebraic calculations, Haspékian [START_REF] Haspekian | Intégration d'outils informatiques dans l'enseignement des mathématiques[END_REF] cites evidence of the difficulty teachers have in moving from the traditional paper/pencil environment to the electronic spreadsheet. The difficulty lay in integrating a tool with such variable functions as this. She introduced the notion of instrumental distance which she summed up as "the stronger the degree of instrumentation when compared with the traditional reference environment (paper/pencil), i.e. the greater the distance from 'habitual scholarly practice', the more the tool will seem difficult to grasp" (p. 296). She demonstrated that in such a situation, a teacher who is not an expert user of the tool can present an additional complexity to the organisation and management of teaching, because the introduction of the spreadsheet, as in this case, implies that new teaching and learning practices be put in place which take full account of the constraints and properties of the spreadsheet.

It is crucial to assert that instrumental distance as measured by the greater or lesser degree of difficulty in integrating the spreadsheet may be translated as the term instrumental conflict used herein, as the notion of difficulty makes reference to the problematic combining of didactical, pedagogical and technical instruments. In effect the use of spreadsheets implies the introduction within the teaching and learning system of new objects, of a new representation, of new functions and significance, thus new symbolisms. The period necessary to master these new capacities is inevitably going to be one of upset and tension: one of disequilibrium in the teaching and learning process.

The double reference appears very similar to the fact that didactical objects such as defined by teachers according to the posture of non-dissociation between noesis and semiosis are transposed by the student: such as paper-pencil for the VLE. The fact even that the notion of double reference should be necessary to explain usage difficulties encountered by students demonstrates the consequences of non-dissociation whenever didactical objects are computerised.

Pseudo-transparency

In order to provide an illustration of the notion of pseudo-transparency an example drawn from the work of Guin and Trouche [START_REF] Guin | Analyser l'usage didactique d'un EIAH en mathématiques : une tâche nécessairement complexe[END_REF] will be adopted, as borrowed from Artigue [START_REF] Artigue | Une approche didactique de l'intégration des EIAO à l'enseignement[END_REF]. He defined this phenomenon as the gap between what is written by the student and that which is shown on the screen: "to enter (a+2)/5, certain pupils, having correctly added the pair of brackets around the (a+2), were astonished to find their screen showing the data without brackets and asked if what they had done was right or not. The appearance and disappearance of brackets seemed, to some of the students, to be playing a rather mysterious game which they little understood such that they could not work out what brackets were supposed to be about". (p. 64)

Artigue [START_REF] Artigue | Une approche didactique de l'intégration des EIAO à l'enseignement[END_REF] points out that the Dérive interface did not at a stroke enable students to alter the length of the line between the upper and lower elements of a fraction which they could do all too easily by hand. And yet this information is necessary as it allows students to know where the line in a fraction should go. There is in this a constraint linked to the fact that the keyboard only provides for one keystroke for division. There is unarguably a discrepancy produced by the transition to computer between the traditional didactical object and the computerised didactical object: this is the phenomenon of pseudo-transparency.

This situation represents an obstacle to the identification of mathematical symbols whose function is precisely to enable pupils to develop their capabilities in Mathematics. The lack of the facility to be able to produce these lines with the Dérive software programme is an example of a situation in which the introduction of a VLE is responsible for introducing a disequilibrium in the learning process. As symbolic representations, the lines in a fraction, which here are taken as pedagogical objects, only have one role, which is to assist in the resolution of the mathematical problem. They are also a means of more clearly identifying a mathematical object critical to conceptualisation [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF]. This aspect is very much in line with a case which in the field of conceptual theory, Vergnaud [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF] considered as a rupture of cognitive development. This research, however, considers such a disequilibrium caused by the Dérive environment as being instrumental conflict, as the failure in the implementation of the pedagogical artefact and the line between the two components of a fraction act against the pupil's way of working and thus prevent him from coming to an understanding or of appreciating significance, i.e. from the didactical artefact.

This example of pseudo-transparency provides the opportunity to confirm the existence of a semiotic non-conformity between the traditional and VLE environment. The fact that showing brackets was simply not possible on the Dérive interface or that the keyboard could not be given specific functions enabling the writing of differentiated signs of lines duly adapted to a perfect and complete representation of the mathematical contents of division served to disturb the majority of pupils. Such ambiguities could also arise without the use of a computer, but they are normally well dealt with by teachers who can most easily resolve the disequilibrium between the formalisms of representation, that is to say between the semiotic registers and cognitive objects. Dérive here creates a disequilibrium which concerns the pedagogical artefact (in this case the line in the fraction or division) and thus the formalism of representation of the didactical artefact which can also create difficulties for the teacher without good anticipation on his part.

It would seem that, beyond the perspective offered by Artigue [START_REF] Artigue | Une approche didactique de l'intégration des EIAO à l'enseignement[END_REF] in proposing this notion of pseudo-transparency as a means to study semiotic non-conformity between the traditional paper-pencil context and that of a VLE in a transposition to a computerised situation, the real problem is to take account of the possible deformation of didactical objects as it arises from the use of technology no matter how well conceived it may have been.

Computerised pedagogical objects as viewed by computer specialists

It is interesting to note that computer specialists' thinking regarding pedagogical objects emerged at the same time as the advent of VLEs. The term pedagogical object, synonym of learning object, only makes sense in relation to the latter. This object-oriented approach has gone through three successive phases, which were crystallised in norms: LOM [START_REF]Learning Object Metadata[END_REF], SCORM [START_REF]SCORM[END_REF] and EML [START_REF] Kopper | Modelling units of study from a pedagogical perspective[END_REF]. It should be recalled before moving on that these three models of pedagogical artefact correspond to three drivers (respectively economic, technical and pedagogical) which preside over object conceptualisation. Moreover, what computer specialists call objects are in reality artefacts in as much as they are not embodied in a VLE and in interaction with a user-learner, they remain symbolic constructions fixed by digital processes.

Pernin [START_REF] Pernin | Objets pédagogiques : unités d'apprentissage, activités ou ressources ? Sciences et Techniques Educatives[END_REF] highlights the lack of consensus as to the definition of a pedagogical object, and this despite the definition given to it by the work group IEEE-Learning Technology Standards Committee. In effect for the IEEE-LTSC a pedagogical/learning object is defined as "any entity, digital or otherwise, which can be used or referenced in training provided by a means of technological support". Looked at more closely, the definition which computer specialists give of the pedagogical object is not too far from this. For David [START_REF] David | Modélisation et production d'objets pédagogiques. Une approche par objet pédagogique[END_REF] a pedagogical object is a digital document allowing the learner to get engaged in an autonomous learning activity regardless of the context of object utilisation. Put another way, it has to be reusable in all learning contexts.

But in order for a digital object to stake a claim to being a pedagogical object, its conception has to integrate the recommendations of pedagogical activity. The model object to which he makes reference is that which complies with the LOM norm specifications, the structure of which is based upon four levels comprising the course, the lesson, the curriculum and the media. This latter component is supposed to enable a replication of the granular structure in all technological learning environments. What is central to the conception of this model is its characteristic of reusability. It is very much a vision which gave rise to the concept of the inter-operability of VLEs, according to which digital resources have to be able to be compatible with the technical structures where they are likely to be used. However, the LOM model has not enabled convenient and 'universal' inter-operability to be achieved.

Another very computing-based conception of the concept of the pedagogical object is provided by Contamines, George and Hotte [START_REF] Contamines | Approche instrumentale des banques de ressources éducatives[END_REF]. It must be borne in mind all the time that these authors did not use the term pedagogical object but that of educational resource, covering a great variety of learning objects. Beyond the indisputable relevance and interest that can be accorded to their work, it is no less well founded than the meaning -of the rest borrowed from Klassen [START_REF] Klassen | Architectures for learning object repositories[END_REF] focusing upon four points -which they give to pedagogical objects, which serves to increase the confusion which reigns around the definition of pedagogical objects. For them, an educational resource is an 'atomic' entity, a video clip or a web page for example. It is also of a composite nature and refers to a non-dissociable whole (didactical multimedia) or an assembly of learning objects (p. 161). It is appropriate to note that this ambiguity concerning pedagogical objects can on the part of learners themselves lead to a situation in which they have altogether different ideas of what constitutes a pedagogical object.

If the construction of the LOM model has not offered much satisfaction in respect of its own expected constituted functions, that is to say the reutilisation of pedagogical objects in all VLEs, one can nevertheless recognise that the SCORM model represents progress in the computing conception of pedagogical objects. It concerns a model by Pernin [START_REF] Pernin | Objets pédagogiques : unités d'apprentissage, activités ou ressources ? Sciences et Techniques Educatives[END_REF] composed of three well-identified levels:

-The first is that of the basic digital resource, such as an image: JPEG or GIF, a WAV or MP3 sound file, a Web page etc. -The second level is termed intermediary. It constitutes of a coherent grouping of basic digital resources capable of being shared amongst learners on a distance learning platform. At this level the system allows control of the carrying out of learning activities. It makes possible the provision of information on resources utilisation and the carrying out of activities on the platform by the key players.

-The third level is that of the bringing together of the content. This provides a coherent structuring of content at the core of an entity deemed of higher level, like a course, chapter or module. The LOM and SCORM models, let it be remembered, serve to facilitate the orientation and indexation of pedagogical objects, and precisely apply this role of the pedagogical object to very diverse entities. The principal consequence of this is that one cannot discern between a pedagogical object and a didactical object, such that this research is left to attempt to do it by separating that which relates to the disciplinary content taught from the formalism of representation or presentation for teaching purposes. This lack of discernment would appear to reside in the fact that the central aspect of the object-oriented approach relies less upon learning activity than upon computing artefacts. In effect, these models consider elementary artefacts to be just as much pedagogical objects (although they are located at different levels), such as images, web pages, content structures, courses, lessons and modules. Yet it would seem necessary to make a distinction between pedagogical artefacts which can be considered as scenarios and formalisms which serve to present the didactical artefacts which are the contents of learning.

Relevance of the concept of instrumental conflict

The concept of instrumental conflict draws its relevance from the generalisation of the use of ICT in teaching. As it has been noted earlier in this paper, the introduction of a VLE might disturb the very equilibrium of a classical teaching situation, in which the didactical artefacts can be conveniently combined with their pedagogical artefacts, so that they can be instrumentalised and instrumented by the learners, and so that they thus become socially useful instruments. But the evidence provided herein would seem to indicate clearly that the two scientific communities interested in ICT in teaching are coming up against difficulties in identifying didactical objects and pedagogic objects when they are in computerised form.

Further notions concerning related objects

For teachers of mathematics, the notions of double-reference and pseudo-transparency take account of the fact that accessible didactical objects in some software programmes do not always work for their pupils whether it be relating to their paper-pencil representation or in accommodating the constraints imposed by the user-interface. From an instrumental perspective, the difficulties encountered by pupils are an inadequacy in the combination of didactical artefacts which are the mathematical objects and pedagogical objects, i.e. their formalisation by mathematical signs in a computerised environment. Although it is always useful to represent mathematical objects by several semiotic systems, what is clear from the classical form of teaching can reveal itself to be that much more difficult, even impossible when a technical artefact is introduced.

For computer specialists, the notions of granularity and inter-operability enable the LOM and SCORM indexation norms to deal with the variety of pedagogical objects that they would seek to describe, but also to conceptualise the difficulty brought about by the absence of a distinction between the pedagogical object as such and its integration within a technical system. Everything happens as if (and this would seem both accurate and to be the norm) the mathematics teachers could not easily computerise certain of their didactical objectives, for lack of ability to conceptualise the dissociation between the taught content and its formalism of representation or its presentation for teaching purposes, and as if the computer specialists could not suitably put pedagogical objects into a teaching mode by reason of also not being able to make the same distinction.

In a way, teachers of mathematics and the computer-specialists are giving two different names to the same objects and are in need of further objects to account for the difficulties posed by their respective nomenclatures (cf. fig 4). Instrumental theory and the separation that has been introduced here between didactical objects, pedagogical objects and technical objects provides the opportunity to unify these two conceptions of the integration of ICT in teaching.

The distinction between didactical objects, pedagogical objects and technical objects is not just an exercise in rhetoric which will depend upon conceptual common ground between the teaching of mathematics and computers science applied to VLEs. In fact, if the term object has been used here for the purpose of clarity, it is important to specify that these objects, from an instrumental perspective, are, in reality, artefacts which become instruments in interaction with a subject. Moreover, it is the simultaneous instrumental genesis of these types of artefact which can cause difficulty, difficulties which here have been termed instrumental conflict.

Usefulness in distance learning and pedagogical engineering

The concept of instrumental conflict would appear to be useful in the analysis of current developments in distance education although the action modalities are different from the classic classroom teaching one, even with a VLE included. In fact, in the foregoing, and equally in the conception of mathematics teaching as in the object-oriented approach, it has been seen that the computerisation of learning and teaching is determined more by the objects of knowledge than by learning activity.

In the field of distance education the activities have a central place. This change in perspective calls indisputably for another approach to the conceptualisation of artefacts operating in various situations. For example, the EML (Educational Modelling Language) developed by Kopper [START_REF] Kopper | Modelling units of study from a pedagogical perspective[END_REF][START_REF] Kopper | Learning Design: A handbook on modelling and delivering networked education and training[END_REF] which is at the origin of the IMSLD2 [START_REF]IMS Learning Design Best Practice and Implementation Guide[END_REF] presents a real leap forward in the pedagogical realm when compared to the LOM and SCORM models already addressed above. This language for pedagogical modelling identifies several types of activity amongst which are learning activities, student support activities and instrumentation activities [START_REF] Pernin | Objets pédagogiques : unités d'apprentissage, activités ou ressources ? Sciences et Techniques Educatives[END_REF].

This refocusing upon the activity is becoming common while designing distance learning platforms. The majority of VLE platforms draw upon a representation of learner function or teaching model, the object of which is to enable isolated learners, because they are operating at distance, to get on effectively with their learning activities [START_REF] Legros | Les théories de l'apprentissage et les systèmes multimédias[END_REF]. This effectiveness naturally depends upon the support to which they are entitled, but also and, perhaps above all, upon the combination of didactical and pedagogical artefacts which, after a certain fashion, are seeking a pedagogical engineering approach. The concept of instrumental conflict can at this point be put to the service of practices in pedagogical engineering which have been developed in distance education [START_REF] Carre | Ingénierie pédagogique et formation ouvertes[END_REF] and contribute to the attainment of specific pedagogical objectives, notably in its capacity of accounting for the imperfection of associations between artefacts.

The concept of instrumental conflict can also be called upon in the evaluation of the introduction of ICT in the practice of teaching and learning. In fact, the recourse to computer-based solutions in the context of pedagogical innovation has often been accompanied by a fierce and impassioned discourse more typical of the political or economic spheres than the pedagogical. As a result, it is not at all unusual that there is a gap between the expected benefit of the introduction of these computer-based artefacts and the actual impact in teaching and learning situations. Having a concept of instrumental conflict enables the review of possible inadequacy in the articulation of didactical, pedagogical and technical artefacts where ICT is involved. In enabling the failure of instrumental genesis to be identified, this concept offers the opportunity for the adjustment of one or more of the artefacts concerned in such a way as to ensure their harmonisation with the learning outcomes desired.

Discussion

Although it is agreed with De Vries and Baillé [START_REF] De | Apprentissages : référents théoriques pour les EIAH[END_REF] that bringing together other existing theory can offer support to the concepts of VLEs and describe the mechanics of learning in action, this paper has attempted to demonstrate the value of reflecting upon the concept of instrumental conflict in order to explain certain dysfunctions in computer-based teaching and learning situations, and in so doing, has tried to help avoid user problems in a distance learning pedagogical engineering approach.

Instrumental conflict only applies in the case of instrumental theory, the principal elements thereof being addressed herein, and identifies interferences between the simultaneous processes of the instrumental genesis of didactical artefacts (disciplinary content), pedagogical artefacts (formalisms of representation and presentation scenarios) and technical artefacts (VLEs, platforms). The main argument, which is singled out from prior analyses of computer-based teaching and learning situations, lies in the explicit distinction between the three families of objects which can bring about an instrumental conflict in many possible forms.

The first of these manifestations and without doubt the most usual is what has been termed a rupture in the equilibrium between a classical situation and an instrumented one. This happens when combinations of didactical and pedagogical artefacts, tried and tested by academic tradition, find themselves no longer to be properly instrumentalised and instrumented once embodied in a technical system. This can come about each and every time that the disciplinary content is delivered within a VLE without modification being made to its presentation format or its role in learning. How many supposed e-learning solutions turn out to be barely disguised slides or photocopies reformatted for the menu systems? No matter how much care has gone into the taught content, the computerisation of such matters requires that the learning and teaching scenario be adapted, in order that it can work with the constraints imposed by the system. In the absence of such precautions, the addition of a technical artefactual layer to a relevant combination of didactical and pedagogical artefacts is bound to lead to instrumental conflict. This is what Lagrange and Drouhard [START_REF] Lagrange | L'intégration du système de mathématiques symboliques DÉRIVE, une évaluation didactique auprès des élèves de 14 à 19 ans[END_REF] identified in their research into the use of the Dérive teaching software, although they didn't call it this.

The second type of manifestation of instrumental conflict corresponds with what can be observed when disciplinary content has not been suitably adapted to the generic application which diffuses it. This is the case, and unhappily it is not altogether rare, when a distance education platform not only imposes its own functionalities, but also forces a particular pedagogical scenario. In effect, a certain number of LMS (Learning Management Systems) are built around modes of organisation of pedagogy which do not suit all academic disciplines or all professional training contexts. These modes of organisation of pedagogy cover a broad range, which extends from the downloading of files to print and be read in complete seclusion to systems of collaborative learning with tutor support and computer sessions. Neither these extremes nor the possibilities between are, a priori, good or bad. They can become one or the other depending upon the functioning of the content presented when they fail to accommodate the nature of the teaching concept and the learning vehicle provided by the platform.

The third and final manifestation of instrumental conflict comes about when a technical system dedicated to a family of didactical objects is inappropriately used by the learner or the trainer. In this instance, neither the contents involved nor the VLE functionalities are to blame, but rather the pedagogical expertise of the teacher. This happens, for example, when teachers are delivering sessions on dynamic geometry in relying upon scenarios developed for paper-pencil geometry. In doing this, pupils cannot access the properties of the geometric objects that the software emphasises, since it is precisely these properties that are not updated in the same way in relation to how the software is used or how one produces figures by hand.

From the moment that one of the didactical, pedagogical or technical artefacts is not in place, or, to put it another way, is not in harmony with the two others, the processes of instrumentalisation and instrumentation necessary for the construction of knowledge risks becoming the object of instrumental conflict.

This represents the sum total of the knowledge arising from this research which can be passed on and applied to the computer-based teaching and learning situations which have been investigated, in so far as care can be taken to bring together in an optimal arrangement, content, formalisms and functionalities and to verify them by means of studies of actual use.

Fig. 2

 2 Fig. 2 Formalisms for presenting multiplication tables.

Fig. 3

 3 Fig. 3 Relationship of artefacts as sources of instrumental conflict.

Fig. 4

 4 Fig. 4 Distinction between didactical, pedagogical and technical objects according to existing approaches.

This software programme is just part of the panoply of digital calculation programmes. The originality of Dérive lies in the fact that it was conceived as the system covering the broadest possible range of formal calculus. It did not really address a specific teaching need.

IMSLD: Instructional Management System Learning Design.


Robert E Kooij

Horsevad Sørensen

Roland Bouffanais

Tuning the Clustering Coefficient of Generalized Circulant Networks

Keywords: Circulant Graphs, Clustered Networks, Consensus Dynamics

Apart from the role the clustering coefficient plays in the definition of the smallworld phenomena, it also has great relevance for practical problems involving networked dynamical systems. To study the impact of the clustering coefficient on dynamical processes taking place on networks, some authors have focused on the construction of graphs with tunable clustering coefficients. These constructions are usually realized through a stochastic process, either by growing a network through the preferential attachment procedure, or by applying a random rewiring process. In contrast, we consider here several families of static graphs whose clustering coefficients can be determined explicitly. The basis for these families is formed by the k-regular graphs on N nodes, that belong to the family of so-called circulant graphs denoted by C N,k . We show that the expression for the clustering coefficient of C N,k reported in literature, only holds for sufficiently large N . Next, we consider three generalizations of the circulant graphs, either by adding some pendant links to C N,k , or by connecting, in two different ways, an additional node to some nodes of C N,k . For all three generalizations, we derive explicit expressions for the clustering coefficient. Finally, we construct a family of pairs of generalized circulant graphs, with the same number of nodes and links, but with different clustering coefficients.

Complex Networks

Introduction

To characterize key features of networks across a wide range of applications, network science has developed a wide range of metrics and mathematical indices [START_REF] Barrat | Dynamical processes on complex networks[END_REF]. Among the most commonly used metrics are the average shortest path length, degree distribution and clustering coefficient. The latter index was introduced by Watts & Strogatz in their seminal paper introducing the concept of "small-world" networks [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]. Specifically, nodes in a network are characterized by the structure of their local neighborhood or connectivity. The concept of clustering is associated with the tendency of some nodes to form cliques as observed in many real-world networks. This index accounts for the ratio between the actual number of relations among the neighbors of a node and the maximum possible number of such relations. The clustering coefficient is defined as the average of this ratio, when averaged over all nodes of the network.

Although the clustering coefficient is generally high in many small-world social and technological networks, it would be wrong to directly associate the small-world feature with high levels of clustering. For instance, random networks yield short path lengths-i.e. they exhibit the small-world feature-but fail to achieve high levels of clustering due to the random interconnectivity of nodes, which hinders the process of clique formation [START_REF] Barrat | Dynamical processes on complex networks[END_REF]. The empirical observation of very high levels of clustering in many real-world networks has therefore been a source of inquiry and to deal with this conceptual challenge, many network models with tunable clustering coefficient have been proposed [START_REF] Holme | Growing scale-free networks with tunable clustering[END_REF][START_REF] Sekunda | Interplay between signaling network design and swarm dynamics[END_REF]. For instance, varying clustering coefficients are encountered in gene expression networks [START_REF] Kalna | A clustering coefficient for weighted networks, with application to gene expression data[END_REF], and also in node spreading influence [START_REF] Wang | Identifying node spreading influence for tunable clustering coefficient networks[END_REF]. The spreading of complex contagious processes-e.g. social cooperation, protest events, the spread of consumer goods and technological innovations, the diffusion of health care preventive measures, the growth of violent-crime epidemics, the spread of institutional norms, the adoption of cultural practices, etc.-has been clearly tied to particular levels of clustering in social and co-presence networks [START_REF] Centola | Cascade dynamics of complex propagation[END_REF].

To study the impact of the clustering coefficient on dynamical processes taking place on any network, many authors have focused on the construction of graphs with tunable clustering coefficient, see for instance [START_REF] Holme | Growing scale-free networks with tunable clustering[END_REF][START_REF] Sekunda | Interplay between signaling network design and swarm dynamics[END_REF]. Typically, the process to reach a desired value for the clustering coefficient is realized through a stochastic growth process, e.g. following the preferential attachment procedure [START_REF] Newman | Clustering and preferential attachment in growing networks[END_REF] or alternatively through the rewiring of regular network lattices [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]. Due to the stochastic nature of these network models most of the results concerning the network properties are only valid for large networks.

In what follows, we give a summary of the state of the art associated with the construction of graphs with tunable clustering coefficient. In [START_REF] Mu | A novel edge rewiring strategy for tuning structural properties in networks[END_REF], the authors explore a "local" rewiring strategy, which allows for an efficient way to adjust the clustering coefficient, and steer it towards a desired value. However, this method depends on randomness in the choice of link pairs that are rewired and thus it is not known a priori if a certain target value for the clustering coefficient can effectively be attained. Krüger et al. [START_REF] Krüger | Unimodular lattice triangulations as small-world and scale-free random graphs[END_REF] use an approach of random "triangulations" of lattices and determine, amongst others, the clustering coefficient. However, the stochastic nature of the process that determines these so-called triangulations of the lattice imposes that the obtained results for the clustering coefficient are essentially (ensemble) average properties, and thus are only valid for large networks. A simple family of growing small-world networks is proposed by Shang [START_REF] Shang | Distinct clusterings and characteristic path lengths in dynamic small-world networks with identical limit degree distribution[END_REF]. The constructed graphs depends on growing the network by closing triangles, followed by a random process of link removal. By tuning the link removal, networks with different clustering coefficients but similar degree distribution are obtained. However, in this family of networks the clustering coefficient can only be determined explicitly for the extreme cases of the model, namely the path graph and the so-called Faray graphs. In [START_REF] Yang | Local-world and clustergrowing weighted networks with controllable clustering[END_REF], Yang et al. introduced a model that generates power-law distributions of degree and a tunable clustering coefficient. However, the construction process is based upon preferential attachment and hence stochastic in nature. In addition, the constructed networks are necessarily weighted. A variation of the Configuration Model (CM), which also takes the formation of triangles into account is proposed in [START_REF] Serrano | Tuning clustering in random networks with arbitrary degree distributions[END_REF]. Thanks to this approach, networks can be generated with a given degree distribution and a preassigned degree-dependent clustering coefficient. Unfortunately, for small-size networks multi-links and self-loops can occur, which often have to be discarded for some practical applications. Furthermore, the CM uses randomly selected stubs to generate links, so there is no control over the final network topology. The Block Two-Level Erdős-Rényi model (BTER) is introduced in [START_REF] Seshadhri | Community structure and scale-free collections of Erdős-Rényi graphs[END_REF]. The BTER model is derived to mimic real-world networks exhibiting community structure. This model also is based upon stochasticity, and again, the results are fundamentally asymptotic for large networks. In addition Ref. [START_REF] Seshadhri | Community structure and scale-free collections of Erdős-Rényi graphs[END_REF] focuses on the transitivity index and not the clustering coefficient.

In summary, the literature offers numerous models and family of networks with some forms of tuning or adjustment of the level of clustering. However, the vast majority of the available network-generating models are intrinsically stochastic in nature, and often assume large networks, with the control over the clustering coefficient and other network properties to be valid only over ensemble averaging.

In contrast, in this paper we consider a family of deterministic graphs whose clustering coefficient can be determined explicitly and exactly. This study is partially motivated by a number of practical applications in the field of networked robotics. Recently, Mateo et al. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] suggested to study the impact of the clustering coefficient on the responsiveness of multi-agent systems performing distributed consensus. Their results were backed up by experiments using swarm robotic networks, albeit these systems are small in size [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF][START_REF] Zoss | Distributed system of autonomous buoys for scalable deployment and monitoring of large waterbodies[END_REF][START_REF] Kit | Decentralized multi-floor exploration by a swarm of miniature robots teaming with wall-climbing units[END_REF].

Thus in order to conduct these experiments, it is important to have explicit expressions for the clustering coefficient, such that its impact on the dynamic processes considered can be easily determined. Since the network size is rather small-with 10 to 50 robots-the asymptotic properties of stochastic models cannot reliably be used.

The basis for our model is formed by k-regular graphs on N nodes, that belong to the family of so-called circulant graphs. In this family of graphs, which we will denote by C N,k , the N nodes form a cycle graph C N . In addition, every node also connects to all nodes at distance at most k/2 on the cycle, where it is assumed that k is even. This graph forms the basis for the construction of the Watts & Strogatz small-world graphs [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] and the clustering coefficient for C N,k is given by Newman in [START_REF] Newman | The structure and function of complex networks[END_REF]. However, we will show that the formula given by [START_REF] Newman | The structure and function of complex networks[END_REF] only holds if N is sufficiently large.

In addition, if one assumes that both ends of each edge are rewired with probability p, and by allowing both double and self-loops, then the clustering of the resulting small-world graph can also be determined [START_REF] Barrat | On the properties of small-world network models[END_REF]. The corresponding formula reported in Ref. [START_REF] Barrat | On the properties of small-world network models[END_REF] is the product of the clustering coefficient for C N,k and the factor (1p) 3 . It is worth noting that this formula is also incomplete in the sense that it presumes N to be sufficiently large. Although both Refs. [START_REF] Holme | Growing scale-free networks with tunable clustering[END_REF] and [START_REF] Barrat | On the properties of small-world network models[END_REF] provide a way to construct graphs with a given clustering coefficient, there is no control over the resulting graph given the fact that their methods are based upon a stochastic process. Moreover, the construction suggested by Ref. [START_REF] Barrat | On the properties of small-world network models[END_REF] has a non-negligible probability to lead to graphs with multiple links and self-loops.

In this paper, we determine the complete analytical expressions for the clustering coefficient of C N,k for any value of N . We also consider three generalizations of the circulant graphs C N,k . For the first generalization, we add a number of pendant links to C N,k . For the second generalization, we add a single node to C N,k , such that it connects to m nodes of C N,k , which are adjacent on C N .

Finally, the third generalization comprises of C N,k with one single node added which connects to m nodes on C N,k , which are exactly 1 + k 2 hops away on the cycle C N . For all three generalizations we derive explicit expressions for the clustering coefficient. The present article is partially motivated by the fact that a generalized circulant graph of Type I has the highest collective response at moderate frequency values, in a leader-follower system consisting of 11 agents, see Fig. 2 in [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF]. Note that in [START_REF] Tang | Generalized recursive circulant graphs, Parallel and Distributed Systems[END_REF], the authors propose a generalization of a class of recursive circulant graphs. However, it is worth pointing out that these generalized recursive circulant graphs belong to the set of circulant graphs, and hence are all vertex-symmetric. On the other hand, our generalized circulant graphs, do not belong the the class of circulant graphs, as they are not vertexsymmetric.

Lastly, we demonstrate how this family of graphs can be used to create a set of particular network topologies of the exact same size and number of edges, yet exhibiting vastly different levels of clustering. We demonstrate the impact of changing the network topology for a particular network dynamic, namely the leader-follower consensus, which shows how networked systems with a small number of nodes N are also strongly influenced by the chosen topology, and thus having a family of networks with explicitly known graph-metrics is of great use.

Clustering coefficient for the circulant graphs C N,k

For the computation of the clustering coefficient, two cases have to be considered. This becomes clear upon introducing some notations. Let the nodes of C N,k be labeled clockwise from 1 to N . Without loss of generality, let us focus on the clustering coefficient of node 1. It is obvious that node 1 is connected to the set of nodes S 1 = {2, 3, • • • , k/2 + 1} and to the node set

S 2 = {N -k/2 + 1, • • • , N -1, N }. Now, if
the distance between the last node in S 1 , i.e. node k/2 + 1 and the first node in S 2 , i.e. Nk/2 + 1, is more than k/2 on the cycle C N , then node k/2 + 1 does not connect to any node in S 2 . This condition translates to

N > 3k 2 . ( 1
)
We will now determine the clustering coefficient for node 1, under condition (1). Obviously, node 1 has a degree k by construction. Therefore, the maximum number of connections between its neighbors is k 2 . We now determine the number of connections between the neighbors of node 1, that are missing. Starting with node k/2 + 1, we have seen that k/2 links are missing.

Next, node k/2, is missing k/2 -1 links. Repeating this argument until node 2, we arrive at a total of k/2

+ 1 + k/2 + • • • + 1 = k 4 ( k 2 + 1) missing links. In conclusion, the clustering C 1 of node 1 satisfies C 1 = ( k 2 )-k 4 ( k 2 +1) ( k 2 )
. Given the symmetry of C N,k , all nodes of this graph have an identical clustering coefficient.

Simplifying the expression for C 1 we arrive at the following result.

Lemma 1. Consider the circulant graph C N,k , satisfying the condition N > 3k 2 . Then the clustering coefficient C of C N,k satisfies C = 3(k -2) 4(k -1) . ( 2
)
Note that Eq. ( 2) is given in [START_REF] Newman | The structure and function of complex networks[END_REF], albeit without mention of condition (1).

If k becomes large, then C approaches 3/4. This is why small-world networks constructed be rewiring a small percentage of the links in C N,k , have a clustering coefficient close to 3/4 for large k (see also Ref. [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]). Figure 1a depicts an example of a circulant graph satisfying the condition of Lemma 1. We now turn our attention to the remaining case:

N ≤ 3k 2 . (3)
Again, without loss of generality, we analyze the clustering coefficient of node 1.

Starting from the last node in the set S 1 , i.e. node 1+k/2, and moving clockwise k/2 hops on the cycle C N , we arrive at node 1 + k. Under condition (3), this node belongs to the set S 2 (see Fig. 1b).

For convenience sake, we introduce l = Nk -1 with l ≥ 0 since C N,k contains at least k + 1 nodes. Obviously, node 1 + k then corresponds to node N -l. As a result, moving clockwise on C N from 1+k/2, we cannot reach nodes

N -l + 1, • • • , N in set S 2 .
k 2 -l + 1 + l -1 + l -2 + • • • + 1 = kl 2 -l(l-1)
2 . Again, using symmetry arguments for C N,k , we arrive at the following result.

Lemma 2. Consider the circulant graph C N,k , satisfying the condition N ≤ 3k 2 . Then the clustering coefficient C satisfies C = k 2 -kl 2 + l(l-1) 2 k 2 , ( 4
)
where l = Nk -1.

Note that for the particular case l = 0, i.e. N = k + 1, the circulant C N,k is a complete graph, which therefor has the clustering coefficient C = 1. Figure 2 depicts an example of a circulant graph satisfying the condition of Lemma 2. Combining Lemmas 1 and 2, we obtain the following general result.

Theorem 3. Consider the circulant graph C N,k , and define r = min{Nk -

1, k 2 }. Then the clustering coefficient C satisfies C = k 2 -kr 2 + r(r-1) 2 k 2 . ( 5
)
Proof. First, assume that condition (1) holds. Then, it follows

N -k -1 > k/2, hence r = k/2.
An elementary calculation reveals that substitution of r = k/2 into Eq. ( 5) gives Eq. ( 2). Finally, if condition (3) holds, it follows that 5) corresponds to Eq. ( 4).

N -k -1 ≤ k/2, hence r = N -k -1 = l. Now Eq. (
This finishes the proof.

The above general expression [START_REF] Kalna | A clustering coefficient for weighted networks, with application to gene expression data[END_REF] for the the clustering coefficient of circulant graphs allows us to obtain the inequalities 3(k-2) 4(k-1) ≤ C ≤ 1 for any given k and C N,k . Furthermore, C attains k/2-1 values between the given bounds. Figure 3 depicts this for values of k up to 30. Mateo et al. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] studied topologies that optimize the collective response of multi-agent systems subjected to a dynamic linear leader-follower consensus protocol. Using simulated annealing optimization for the frequency response of a system of N + 1 = 11 networked agents-one leader and 10 followers, they found that for moderate values of the frequency ω of the variations of the leading agent, the generalized circulant graph C 7,6,4 (see Fig. 4b and Fig. 2 in [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF]) has the highest collective response at a frequency of ω = 0.4 [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF].

Theorem 4. Consider the graph C N,k,m and define r = min{N -k -1, k 2 } and A = k 2 -kr 2 + r(r-1)

2

. Then the clustering coefficient C satisfies

C = 2A(N k + N -2m) (N + m)k(k 2 -1) . ( 6
)
Proof. The m nodes attached to C N,k obviously have a clustering of 0 since they are of degree one. The N -m nodes that are not connected to the m nodes, have the same clustering as in the case of

C N,k . Using r = min{N -k -1, k 2 } and A = k 2 -kr 2 + r(r-1)

2

, according to Theorem 3, their clustering can be written as

A ( k 2 )
. Finally, we consider the m nodes on C N,k that are connected to one of the m pendant nodes (yellow nodes in Fig. 4). These nodes have k + 1 neighbors but the number of connections between these neighbors is the same as for the C N,k case. Therefore, the clustering for these m nodes satisfies A (

.

Summing over all nodes we obtain

C = (N -m) A ( k 2 ) + m A ( k+1 2 ) N + m .
Factorizing the latter expression leads to Eq. ( 6) and finishes the proof.

Note that for m = 0, we retrieve the result stated in Theorem 3. For any given N and k, upon increasing m, the clustering coefficient does become smaller. The smallest value is obtained for m = N . It is easy to verify that for

N > 3k 2 the clustering coefficient of C N,k,N satisfies C = 3(k -2) 8(k + 1) . (7)
Therefore, under the condition N > 3k 2 , the values 3(k-2) 8(k+1) and 3(k-2) 4(k-1) form an envelope for the clustering coefficient for C N,k,m (see Fig. 5). For each k, by varying m from 1 to N -1, we can obtain N -1 values for C within the envelope depicted in Fig. 5.

It is clear from the above result, that a lower bound for the clustering coefficient for C N,k,m is given by Eq. [START_REF] Centola | Cascade dynamics of complex propagation[END_REF]. If one seeks to generate clustering

r = min{N -k -1, k 2 } and A = k 2 -kr 2 + r(r-1)

2

. Then the clustering coefficient C satisfies

C = 2A N + ms N -m k(k -1) + m (k + s)(k + s -1) . ( 8
)
The proof follows from arguments very similar to those presented in the proof of Theorem 4 and is therefore omitted.

For any given N, k and s, the smallest value for the clustering coefficient is obtained for m = N . It is straightforward to show that for N > 3k 2 the clustering coefficient of C N,k,N,s satisfies

Degree k = 4 1 k + s = 6
C = 3k(k -2) 4(s + 1)(k + s)(k + s -1) . (9)
1 < m ≤ k 2 + 1. Define r = min{N -k -1, k 2 } and A = k 2 -kr 2 + r(r-1)

2

. Then the clustering coefficient C satisfies

C = (N -m) A ( k 2 ) + m (A+m-1) ( k+1 2 ) + 1 N + 1 . ( 10
C i = A+m-1 ( k+1 2 )
. As mentioned before, the central node has clustering 1, for m ≤ k 2 + 1. Summing over all nodes, we obtain Eq. ( 10). This completes the proof.

m > k 2 + 1. Define r = min{N -k -1, k 2 } and A = k 2 -kr 2 + r(r-1)

2

. Then the

clustering coefficient C satisfies C = (N -m) A ( k 2 ) + mA+ k 4 (3k-2)+(m-k)k ( k+1 2 ) + ( m 2 )- (m-k 2 )(m-k 2 -1) 2 ( m 2 ) N + 1 . ( 11
)
Proof. Following the same train of thought as in the proof of Lemma 6, the Nm nodes on C N,k that are not connected to the central node, have

a clustering A ( k 2 )
. When considering the set of nodes I on D N,k,m which are connected to the central node, it is convenient to consider two cases. We start with the case m ≥ k + 1 (see Fig. 9). Denote the nodes with index up to k 2 as I 1 and those with index k 2 + 1 up to index m -k 2 as I 2 . The remainder of the nodes in I are denoted as I 3 . We will now count the number of connections between the neighbors of all nodes in I, in addition to the A connections which are part of C N,k . Node 1 in I 1 induces k 2 additional connections. Node 2 induces k 2 + 1 additional connections. It can be deduced that the total number of additional connections due to I 1 is given by k

N + 1 1 m 2 m -1 k 2 m -k 2 + 1 k 2 + 1 m -k 2
2 + k 2 + 1 + • • • + k 2 + k 2 -1 = 3k 8 (3k -2). Every node in I 2 induces k additional connections, hence I 2 contributes (m -k)k additional connections.
Finally, the contribution of I 3 is similar to the contribution of I 1 , by symmetry.

Therefore, for the case m ≥ k + 1, the contribution to the clustering of all nodes in I amounts to

C I = mA + k 4 (3k -2) + (m -k)k k+1 2 . ( 12
)
For the case m < k + 1, the set of nodes I, can be split into two sets: the aforementioned set I 1 and the remaining nodes in I. With an argumentation similar to the one used in the case m ≥ k + 1, it can be shown that the contribution of the nodes in I to the clustering, is also given by Eq. [START_REF] Yang | Local-world and clustergrowing weighted networks with controllable clustering[END_REF]. We leave the details as an exercise to the reader.

Finally, we look at the clustering of the central node. We count the number of connections between its m neighbors, which are missing. Starting with node 1, we miss m-k 2 -1 connections. For node 2, we miss m-k 2 -2 connections. and so on. Therefore the total number of missing connections satisfies (m-k

2 -1)+(m- k 2 -2) + • • • + 1 = (m-k 2 )(m-k 2 -1) 2
. Therefore, the clustering of the central node is given by ( m 2 )-

(m-k 2 )(m-k 2 -1) 2 ( m 2 )
. Summing over all nodes we obtain Eq. ( 11).

Theorem 8. Consider the generalized circulant graph D N,k,m of Type II, with

m > 1. Define r = min{N -k -1, k 2 }, A = k 2 -kr 2 + r(r-1) 2 and let σ = min{m -1, k 2 }. Then the clustering coefficient C satisfies C = (N -m) A ( k 2 ) + mA+σ(3σ-1)+2(m-2σ)σ ( k+1 2 ) + ( m 2 )- (m-k 2 )(m-1-σ) 2 ( m 2 ) N + 1 (13)
Proof. First, assume that the condition m ≤ k 2 + 1 holds. Then, σ = m -1. Substitution of this value in Eq. ( 13) gives Eq. [START_REF] Krüger | Unimodular lattice triangulations as small-world and scale-free random graphs[END_REF].

Finally, if m > k 2 + 1, then, σ = k 2 .
Plugging this value into Eq. ( 13) leads to Eq. ( 11). This finishes the proof.

C = (N -m) A ( k 2 ) + mA ( k+1 2 ) N + 1 . ( 14
)
The proof of Theorem 9 is based on similar arguments as the proof of the previous theorems and is therefore omitted.

It is easy to see from Eq. ( 14) that C is a linearly decreasing function of m.

Similarly, subtracting Eq. ( 14) from Eq. ( 5) , we can show that the clustering coefficient of E N,k,m is always smaller than that of C N,k .

Discussion

In Ref. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF], Mateo et al. investigated the influence of the network topology on collective response. Specifically, they considered an archetypal model of distributed decision-making-the leader-follower linear consensus-and studied the collective capacity of the system to follow a dynamic driving signal (the "leader", an external perturbation locally disturbing the collective dynamics) for a range of network topologies. Mateo et al. reported a nontrivial relationship between the frequency of the driving signal and the optimal network topology [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF]. The emergent collective response of the networked dynamical system to a slow-changing perturbation increases with the degree of the interaction network, while the opposite is true for the response to a fast-changing one.

Although it is known that at high enough frequency the response is determined by the degree distribution [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF], a key question remains open: what is the importance of the structural properties-beyond just the degree distribution-of the network as predictors of a system's response at different frequencies.

Mateo et al. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] suggested to study the impact of other network characteristics, such as the clustering coefficient, but it is stated that it is not possible to change the degree distribution without also changing other characteristics, such as the clustering coefficient. For instance, Arenas et al. [START_REF] Arenas | Synchronization in complex networks[END_REF] stressed the existence of significant discrepancies in results in the literature for different network models when considering synchronization problems in complex networks. These reported discrepancies are attributed to the fact that multiple non-independent parameters (e.g. average degree, clustering coefficient, average shortest path) characterizing the network were concomitantly changed. This important issue stresses the challenges associated with any parametric studies of networked systems.

To avoid the pitfalls highlighted by Arenas et al. in Ref. [START_REF] Arenas | Synchronization in complex networks[END_REF]

N 1 + m 1 = N 2 + 1. ( 15
)
The condition for the two graphs to have the same number of links reads:

N 1 k 1 2 + m 1 = N 2 k 2 2 + m 2 . (16)
Solving Eq. ( 15) for N 2 , and substituting the solution into Eq. ( 16) leads to

N 1 (k 1 -k 2 ) = (m 1 -1)k 2 + 2m 2 -2m 1 . (17)
With the particular choice k 1k 2 = 2, the solution of the Diophantine equation ( 17) is given by

N 1 = (m 1 -1) k 2 2 + m 2 -m 1 . (18)
Note that N 1 is an integer because k 2 is even. Equation ( 18) gives rise to We can then use the two graphs generated by this approach, to analyze the same distributed linear leader-follower consensus as in Ref. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF]. By using these graphs, we can study the impact of the clustering coefficient on the collective frequency response H 2 (ω)-metric that can be interpreted as the number of agents that are able to respond or follow a leader evolving at frequency ωbut with equal average degree. Specifically, we follow the analytical approach outlined in Ref. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] to compute the collective response H 2 (see Materials and Methods section in [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF]). Since the graphs have the same number of nodes and edges, the collective frequency responses for the two graphs can then readily be compared so as to quantify the impact of the clustering coefficient. The results for this pair of networks are shown in Fig. 13 Given that both graphs have 14 nodes, there are 14 different possible nodes that can be given the "leader" role, hence we have a distribution of 14 collective frequency responses. In Fig. 13, we report the average collective response, H 2 , along with the minimum and maximum values. The average collective response for the classical circulant graph C 14,6 is provided for reference. These results show that although C 10,6,4,1 , D 13,4,8 and C 14,6 have the same average degree, they yield different average collective response H 2 . The differences are particularly marked in the midrange of frequencies (i.e. for 5 × 10 -2 < ω < 10 -1 ). It is quite clear that the low level of clustering of C 10,6,4,1 has a detrimental effect on the collective response of this networked system. Surprisingly though, the "best" leader node-i.e. the node such that when driven yields the maximum collective frequency responsein both proposed graphs show similar levels of collective frequency response, for the best leader in both networks (dotted lines), but at different cost to the average agents response (solid lines).

These conclusions for the collective response of these particular networked systems highlight the usefulness of the analytical results obtained for the clustering coefficients of the generalized circulant graphs of Type I, II and III. Even for small system sizes N = O [START_REF] Krüger | Unimodular lattice triangulations as small-world and scale-free random graphs[END_REF] where the statistical properties of grown networks are unreliable, the effect of the network topology has nonetheless a noticeable impact as appears clearly in Fig 13. This increases the usefulness of such deterministic network models, where network parameters can be varied greatly and known exactly, even for small networks. It is worth adding that many experiments with artificial networked systems are often on a limited scale, i.e. with a number of agents of the order of several tens [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF][START_REF] Kit | Decentralized multi-floor exploration by a swarm of miniature robots teaming with wall-climbing units[END_REF][START_REF] Zoss | Distributed system of autonomous buoys for scalable deployment and monitoring of large waterbodies[END_REF]. It is also possible to imagine that the new families of generalized circulant graphs introduced here could be used to study dynamical processes on more complex networks, such as multi-layered social networks [START_REF] Manivannan | Are the different layers of a social network conveying the same information?[END_REF] for instance. under the Grant #IDG31900101.

Figure 1 :

 1 Figure 1: (a) Circulant graph C 14,4 satisfying the condition N > 3k 2 .; (b) Relative positions of nodes 1 + k = Nl and N + 1k/2.

Figure 2 :

 2 Figure 2: Circulant graph C 12,8 satisfying the condition N ≤ 3k 2 .

= k + 1 k + 1 < N ≤ 3k 2 N > 3k 2 Figure 3 : 3 .

 112233 Figure 3: All possible values of clustering coefficient C for C N,k with k ≤ 30.

Figure 4 :

 4 Figure 4: (a) The graph C 14,4,3 from the Type I subfamily of circulant graphs.; (b) The generalized circulant graph C 7,6,4 .

Figure 5 : 2 . 5 .

 525 Figure 5: Envelope for the values of the clustering coefficient of the generalized circulant graphs C N,k,m with N > 3k 2 .

Figure 6 :

 6 Figure 6: Generalized circulant graph of Type I C 14,4,3,2 .

Figure 7

 7 Figure 7 shows values of the clustering coefficient for C N,k,N,s as a function of k, with N > 3k 2 and for several values of s.

Figure 7 :

 7 Figure 7: Clustering coefficient for generalized circulant graph of Type I, C N,k,m,s , with N > 3k 2 .

Figure 8 :Lemma 6 .

 86 Figure 8: Generalized circulant graphs of Type II: D N,k,m . Particular cases of D 14,4,2 (a) and D 14,4,5 (b).

) 2 ).

 2 Proof. The Nm nodes on C N,k that are not connected to the central node, have the same clustering as in C N,k , namely A ( k Denote the set of m nodes on D N,k,m which are connected to the central node by I. A node i in I has k + 1 neighbors. Because m ≤ k 2 + 1, among those neighbors there are A connections which are part of C N,k . There are m -1 additional connections between the neighbors of node i, namely from the central node to the nodes in I. Hence each node i in I has clustering

Lemma 7 .

 7 Consider the generalized circulant graph D N,k,m of Type II, with

Figure 9 :

 9 Figure 9: Relative positions of nodes for the case m ≥ k + 1.

Figure 10 Figure 10 : 2 . 4 . 9 . 2 .

 10102492 Figure 10 depicts the clustering coefficient of D 30,k,m , with m ∈ {0, 1, . . . , 30}. We observe the clustering coefficient drops for m = 1, example: C(D 30,4,1 ) < C(C 30,4 ) = 1 2 , while for m > 1 it holds that C(D 30,4,m ) > C(C 30,4 ).

Figure 11 :

 11 Figure 11: Generalized circulant graphs of Type III E N,k,m . The particular cases are E 14,4,3 (a) and E 14,4,4 (b).

Figure 12 :

 12 Figure 12: Two generalized circulant graphs of the same order and size.Particular cases of C 10,6,4 (a) and D 13,4,8 (b).

2 H 2 (

 22 C 10,6,4,1 ) H 2 (D 13,4,8 ) H 2 (C 14,6 ) min(C 10,6,4,1 ) max(C 10,6,4,1 ) min(D 13,4,8 ) max(D 13,4,8 )

Figure 13 :

 13 Figure 13: Collective frequency response for graphs with similar number of edges and nodes.The graph C 14,6 with 42 edges is included for the sake of comparison.

  Therefore, the number of nodes in S 2 that cannot be reached by node 1 + k/2 is l. The same holds for the nodes 1 + l, • • • , k/2. Next, node l in set S 1 does not reach l -1 nodes in S 2 . Continuing this argument, we arrive at node 2 in set S 1 , that only does not reach one node in S 2 , namely node

Nk/2 + 1. In summary, the total number of missing connections between sets S 1 and S 2 is given by l

  To this end, we now show how the classes of generalized circulant graphs introduced in the previous sections can be used to construct pairs of graphs of the same order (i.e. with the same number of nodes) and size (i.e. with the same number of edges). Such pairs of graphs thus have the same average degree, but as we will see different clustering coefficients. Let G 1 denote a generalized circulant graph C N1,k1,m1,1 of Type I and G 2 a generalized circulant graph D N2,k2,m2 of Type II. The graphs G 1 and G 2 have the same number of nodes, if the following condition is satisfied:

and to carry out a meaningful parametric study of networked systems with different graph topologies, one could look at pairs of networks with the same average degree, but with different clustering coefficients. Families of fully deterministic graphs, such as the generalized circulant graphs presented here, provide a useful tool in determining the impact of network structure on particular complex dynamical systems. Given their deterministic character, it can be known exactly how much a specific graph topological metric, such as the clustering coefficient, varies from one graph to the other, thereby making it easier to test and analyze new hypotheses.

Acknowledgments

This work is partially supported by the National Research Foundation (NRF), Prime Minister's Office, Singapore, under its National Cybersecurity R&D Programme (Award No. NRF2014NCR-NCR001-040), administered by the National Cybersecurity R&D Directorate. The authors also acknowledge the support from the SUTD-MIT International Design Center (http://idc.sutd.edu.sg)

Data and materials availability

All data needed to evaluate the conclusions in the paper are present in the paper. Additional data related to this paper may be requested from the authors.

Authors Contribution

R.K. and R.B. designed the study. R.K. and N.H. developed the analytical and numerical tools. All authors wrote the manuscript.

Competing Interests

The authors declare that they have no competing interests.