Blending research methods: Qualitative and quantitative approaches to researching computer corpora for language learning.

Blending research methods: Qualitative and quantitative approaches to researching computer corpora for language learning Alex Boulton (CRAPEL-ATILF/CNRS, Nancy University)

I. INTRODUCTION

Language features high on the list of daily human activities, so popular ideas about it inevitably develop, many of which are little short of "language myths" (Bauer & Trudgill 1998). As Crystal (2011: 3, 6) points out, "an important part of the linguist's job is to eliminate popular misconceptions" or "demythologize" linguistic concepts. The same applies to education, where every politician, parent and child has their own ideas about the 'best method' -especially perhaps in language teaching. Here even teachers, despite (or perhaps because of)

their training and heard-earned practical experience, may be inclined to favour their intuitions about good practice rather than keep up with research. Teachers and researchers are notoriously bad at talking to each other (M. McCarthy 2008) -and who is to say that the expert researcher is better placed for this than the expert teacher (or vice versa)? Intuition may have been creatively productive in the 1970s, where "the so-called designer methods movement -Suggestopedia, , the Silent Way, Community Language Learning, and so on… were 'data free', drawing sustenance from rhetoric rather than empirical support" (Nunan 2007: 9). But even today, many research papers in applied linguistics are essentially based on the (teacher-)researcher's personal experience, subjective intuitions, selective observation and anecdote. Though they may describe innovative tools, techniques and practices and feature the traditional 'literature review', the body of many such papers is mainly reflective or descriptive with little if any rigorous data collection. Many current tenets of language teaching theory have come in for criticism for neglecting evidence or even ignoring it -communicative language teaching [START_REF] Swan | A Critical Look at the Communicative Approach[END_REF][START_REF] Decoo | On the Mortality of Language Learning Methods[END_REF], constructivism [START_REF] Kirschner | Why Minimal Guidance during Instruction does not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching[END_REF][START_REF] Matthews | Constructivism in the Classroom: Epistemology, History, and Empirical Evidence[END_REF][START_REF] Mayer | Should there be a Three-Strike Rule against Pure Discovery Learning? The Case for Guided Methods of Instruction[END_REF], taskbased learning [START_REF] Swan | Legislation by Hypothesis: The Case of Task-Based Instruction[END_REF], noticing [START_REF] Truscott | Noticing in Second Language Acquisition: A Critical Review[END_REF][START_REF] Robinson | Individual Differences and the Fundamental Similarity of Implicit and Explicit Adult Second Language Learning[END_REF], induction [START_REF] Decoo | The Induction-Deduction Opposition: Ambiguities and Complexities of the Didactic Reality[END_REF], the lexical approach [START_REF] Lindstromberg | My Good-Bye to the Lexical Approach[END_REF], and the use of authentic documents [START_REF] Widdowson | On the Limitations of Linguistics Applied[END_REF][START_REF] Cook | The Philosopher Pulled the Lower Jaw of the Hen.' Ludicrous Invented Sentences in Language Teaching[END_REF][START_REF] Gilmore | Authentic Materials and Authenticity in Foreign Language Learning[END_REF], to name but a few.

II. CORPORA IN LANGUAGE LEARNING / TEACHING

This paper looks at the nature of evidence as applied to uses of corpora in language teaching and learning, often referred to as "data-driven learning" or DDL, a term first coined by Tim [START_REF] Johns | From Printout to Handout: Grammar and Vocabulary Teaching in the Context of Data-Driven Learning[END_REF]. In this approach, corpora are used not merely to inform the syllabus, materials or language input (M. McCarthy 2004), but themselves represent a language resource for the learners to explore via dedicated software or materials overtly derived from them (Boulton 2010a). For many, this is strongly rooted in the concordance, but computers can also help to present the data in other ways that may be beneficial to language learners -frequency lists of word forms or lemmas, clusters and collocates, distribution plots over the corpus, keywords in different subcorpora or for a specialist corpus compared to a reference corpus, and so on. The ability to switch between this kind of quantitative view and the wider context is also increasingly being used to promote a more discourse-oriented approach to corpus use for language learning (e.g. It is of course one thing to see the potential of corpora and associated software and techniques (or indeed any other approach or technology), quite another to demonstrate their effectiveness, let alone their efficiency.

Evidence is essential for any innovation, but for anything related to corpus linguistics -that most empirical of fields in linguistics [START_REF] Carter | Obituary: Professor John Sinclair (1933-2007[END_REF]) -its lack would be ironical in the extreme (cf. Boulton 2010b). Curiously, then, it has become a commonplace to lament the lack of empirical DDL studies (e.g. Ma 1993: 24;Aston 1998: 14;Gaskell & Cobb 2004: 302;Conrad 2005: 401-402;O'Keeffe et al. 2007: 24;Johansson 2009: 41;Chang 2010: 61). Certainly there seems to be no end to papers extolling the potential advantages (and occasional outright criticism, e.g. [START_REF] Dellar | What have Corpora ever done for us?[END_REF]; but what of scientific investigation? Without this, as P. [START_REF] Mccarthy | Review of L. Gavioli[END_REF] n.p.) puts it, "texts on corpus analysis that ignore quantitative evidence are doomed to endless instances of phrases such as 'it seems to me'."

With broad definitions of DDL (i.e. "language users exploiting language corpora"; Boulton in press: n.p.) and of empirical studies (i.e. subjecting "some aspect of DDL to observation or experimentation with some kind of externally validated evaluation other than the researchers' own intuition"; While it is certainly possible to find evidence to back up any such pre-existing position, it is more fruitful perhaps to focus on the bigger picture -not any one study in isolation, but the weight of evidence as a whole.

One approach to this would be to conduct a meta-analysis, combining the results of different studies as is frequently the case in medicine, for example, but relatively rare in fields related to ours (e.g. [START_REF] Masgoret | Attitudes, Motivation and Second Language Learning: A Meta-Analysis of Studies Conducted by Gardner and Associates[END_REF] for motivation in language learning; Means et al. 2010 for on-line learning). The reason this is not more widespread is not hard to see, and the empirical studies in DDL are a case in point: they are so diverse in terms of setting, participants, tools, materials, design, data gathering, aims, and so on that they are insufficiently comparable to be combined into a single meta-analysis. However, if such a meta-analysis were possible, it seems very probable that they would, together, produce highly significant results.

III. TOWARDS A COMPLEX SYSTEMS PARADIGM FOR DDL

A third reaction is simply to wonder what is going wrong. The researchers and authors in these studies are overwhelmingly corpus users and enthusiasts: if they can't get significant results, who can? In fact, the question seems to provide part of the answer: the enthusiasts are those who cannot afford to be overly enthusiastic, and the various studies abound in hedging such as:

Overall, given that the students were advanced and the items already partially known it is possible to conclude, albeit tentatively, that, given language items at the right level, DDL has an observable (though slight) positive effect on actual use. (Cresswell 2007: 280) But actually there may be nothing special about DDL in this regard. The vast majority of empirical research in applied linguistics designed to investigate different methods and approaches, tools and techniques, are similarly hedged. And this is where it gets interesting. Larsen-Freeman and Cameron (2008: 243), for example, ask:

Who can say, for example, on the basis of a pre-test / post-test design that a particular experimental treatment works or does not work? If the results are non-significant, the effects of the treatment may not yet be manifest; if the results are significant, they may have followed from an experience prior to the pre-test.

De Bot (2008: 173) puts this into perspective with the example of a study designed to evaluate the impact of L1 use in class:

In a carefully designed quasi-experimental study, no differences among the conditions were found, which was of course disappointing for the researchers, but not really surprising: in an FL classroom many variables play a role in the acquisition of vocabulary, and the use of the L1 is likely to be secondary to many other factors that play a role in the process. In a way, it would have been surprising if such a single factor had explained differences in learning success. This is not a critique of the… study, which is well designed and carried out carefully. The null effect found merely supports the fallacy of focusing on a single explaining factor in a setting in which there are clearly many potentially relevant factors. In this paradigm, language (and, a fortiori, language learning) is "dynamic, complex, nonlinear, chaotic (at times), unpredictable, sensitive to initial conditions, open, self-organising, feedback sensitive, adaptive, characterised by strange attractors, which are fractal in shape" (Ellis & Larsen-Freeman 2006: 576). DST recognises the "unknowableness and interconnectedness of systems" (Larsen-Freeman & Cameron 2008: 232), including language whose emergent properties cannot be reduced to a simple series of rules (Beckner et al. 2009: 2). The same is true of language learning, hence the inherent difficulty of isolating variables or establishing direct cause-effect relationships. Language acquisition is essentially "exemplar based" (Ellis & Larsen-Freeman 2006: 565), a bottom-up process deriving from formal or informal exposure to the language where "every instance of language use changes an idiolect's internal organization" (Beckner et al. 2009: 16).

While this highlights the difficulties facing the researcher, it does tie in well with DDL which entails massive exposure to the language via the corpus, including through organized extracts in the form of concordances (cf. Gaskell & Cobb 2004: 304).

The new research framework goes a long way towards explaining the problems inherent in traditional research,

but exactly what contribution it can make to overcoming those problems is admittedly "challenging" (Larsen Freeman 2006: 594). The large-scale mathematical models proposed by Beckner et al. (2009: 12) are likely to be nigh on impossible to implement in many cases, not least because researchers in applied linguistics simply do not have the necessary skills and resources to implement them. Of more immediate relevance is the insight that qualitative and quantitative approaches need to be combined to obtain a more complete picture, though precisely

how has yet to be determined. Many of the early empirical DDL studies tended to be small-scale and qualitative in nature [START_REF] Chambers | Popularising Corpus Consultation by Language Learners and Teachers[END_REF]): on the one hand, they provide useful insights to the processes / performance and not just the product / outcomes (Moreno Jaén 2010: 243) and allow a certain depth of understanding of various factors involved; on the other, they tend towards "excessive complexity that might discourage use through its impenetrability and unwieldiness" (Leakey 2011: 251) and which often makes it difficult to extrapolate to other cases. Quantitative research, by contrast, may be more generalizable as it irons out some individual differences;

but that is also its disadvantage as it can result in "over-simplicity, [making] it a blunt and meaningless instrument" (Leakey 2011: 251). Variation between individuals is crucial and not just "noise" (Ellis & Larsen-Freeman 2006: 564), and over time, "it is difficult to imagine a dynamic systems study that does not have a prominent longitudinal aspect" (Dörnyei 2009: 242).

Partisans of a DST approach applied to language learning recognise these difficulties, but remain optimistic. For Dörnyei (2009: 245), "the absence of ready-made research models and templates is not an indication of the inadequacy of a dynamic approach but only of the transitional problems that are bound to accompany a major paradigm shift." We should perhaps expect fewer studies that are exclusively quantitative in nature in their attempt to isolate variables and establish direct causal relationships, but that does not mean that a quantitative approach should be abandoned altogether. According to de Bot (2008: 169), "some of the proponents of DST are overstating their claims, and there is not yet enough substantial evidence to abandon traditional cognitive science in favour of a DST based approach." Though he admits that it is an impossible endeavour to control all potential variables, it should be able to reduce them to something more manageable and relevant (p. 175). In the same vein:

Arguments for complexity are not arguments against simplicity. Some things can be understood by the analytic and reductionist program and where that program works it has done great service in elucidating causality. The problem is that it works where it works and it does not work everywhere. [START_REF] Byrne | Interpreting Quantitative Data[END_REF]: 101-102, cited in Dörnyei 2009: 241) Research to date is thus not without interest and the results should not be ignored: they provide precious insights to many aspects of language learning. Quantitative research is currently seen as more prestigious, at least insofar as it dominates empirical research in many prestigious journals as highlighted in two recent analyses [START_REF] Benson | Qualitative Research in Language Teaching and Learning Journals, 1997-2006[END_REF][START_REF] Richards | Trends in Qualitative Research in Language Teaching since 2000[END_REF]) -just as it does in the study of learning outcomes from DDL presented above. But as Byrne points out in the preceding quotation, the methodology is limited and constraining. Quantitative research by definition focuses on the quantifiable, with a corresponding emphasis on short-term outcomes (cf. Leakey 2011); in the case of DDL, it is notable that much of the research so far focuses on targets that are easy to measure in a highly controlled experimental environment -short-term learning outcomes in vocabulary and lexico-grammar, as well as error-correction and Likert-scale questionnaires of learner attitudes, etc. At the same time, there is a notable dearth of studies looking at the major advantages that are generally attributed to DDL as seen earlier -the long-term effects on learner autonomy, responsibility, life-long learning, constructivism, cognitive and metacognitive development, language awareness, skills and communicative ability, the capacity to work with authentic data, and so on.

IV. CONCLUSION AND IMPLICATIONS

A complex dynamic systems perspective does not in itself imply that empirical research should be abandoned, though it may need tackling in new ways [START_REF] Dörnyei | Researching Complex Dynamic Systems: 'Retrodictive Qualitative Modelling' in the Language Classroom[END_REF]. As the DST paradigm evolves, we can expect new types of research, combining qualitative and quantitative into an integrated whole rather than being treated separately, which may in turn lead to new research questions and open up new areas to explore. It remains to be seen exactly how: to date, only [START_REF] Thorne | Mediation as Objectification in the Development of Professional Academic Discourse: A Corpus-Informed Curricular Innovation[END_REF] have worked in this direction in DDL. In the meantime, the body of empirical DDL research suggests there is good reason to believe that, thoughtfully applied with sensitivity to the local context, corpora can be useful and relevant to many learner populations and individuals with different needs and for different purposes. Current research provides a solid background to inform ideas of what is likely to work, but each individual study is inevitably specific in that it derives from a particular context, while the overview is inevitably generic and may not apply in every case. Contrary to popular belief, there can be no 'best method' [START_REF] Prabhu | There is no Best Method -Why[END_REF], and one size never fits all: no innovation will suit all learners perfectly for all language points on all occasions in every context, so it is a local decision as to whether a new technique is 'sufficient' to warrant further exploration for given purposes with given learners in given conditions. DDL researchers certainly make no claim it is a panacea (cf. [START_REF] Boulton | Data-Driven Learning: Reasonable Fears and Rational Reassurance. CALL in Second Language Acquisition: New Approaches for Teaching and Testing[END_REF], or that it should be used to the exclusion of other techniques. Against this background, teachers / researchers who are interested in seeing what DDL (or indeed any other innovation) can bring to their own learners should not ignore the evidence available, but ultimately have but one option: to try it out.

  These authors are proponents of a new research paradigm in applied linguistics, commonly and interchangeably known as dynamic systems theory (DST) or complex adaptive systems. Though drawing on earlier foundations, the movement began in earnest in the 1990s with work by Larsen-Freeman (1997), attaining widespread interest with the book Complex Systems and Applied Linguistics (Larsen-Freeman & Cameron 2008) and a number of special issues of reputable journals: Applied Linguistics (Ellis & Larsen-Freeman, 2006); The Modern Language Journal (de Bot, 2008); Language Learning (Ellis & Larsen-Freeman, 2009).

  on the whole, learning seems to happen and compares favourably with control groups or other approaches -i.e. DDL is both effective and efficient (cf. Boulton 2010b: 139); mainly quantitative data obtained from tests, written samples or other productions; d) 24 studies of using corpora as a reference resource: on the whole, learners are able to refer to corpora to help with their writing, error-correction or translation for specific purposes (e.g. research papers), though as with dictionaries and other tools, this says nothing about whether learning does or does not happen; data mainly based on written samples or other productions.Though the overwhelming majority report largely favourable findings, it will be noted that all four categories above are prefaced by 'on the whole'. This is partly because a very small number of researchers are clearly disappointed with the results overall (e.g.[START_REF] Vannestål | Learning English Grammar with a Corpus: Experimenting with Concordancing in a University Grammar Course[END_REF]. More importantly, and with

Boulton 2010b: 130

) it has been possible to collect over 90 empirical studies of DDL (see

Boulton 2010b [supplement]

for a complete list of references, overview and summaries). It is of course not possible to go into much detail here, but one angle is to group them according to their major objectives (most studies have more than one focus): a) 54 studies of learners' attitudes: on the whole, learners are generally receptive to the approach and perceive corpora favourably; data usually collected via questionnaires or other feedback, qualitative and quantitative; b) 44 studies of learners' behaviour: on the whole, learners work successfully with corpora, thinking and acting like corpus linguists according to the 'learner as researcher' metaphor; mostly a qualitative analysis based on observation and feedback; c) 32 studies of learning outcomes: very few exceptions (e.g.

[START_REF] Koosha | Data-Driven Learning and Teaching Collocation of Prepositions: The Case of Iranian EFL Adult Learners[END_REF]

, most individual papers are mitigated in their conclusions -DDL does generally do what the researchers thought or hoped it would, but not always as well as they might have expected, or on all the points covered. There are a number of reactions one might have to this, the first being rather subjective: if you begin with a positive attitude to corpora, you are probably more likely to note the positive points; if you are sceptical, you are more likely to pick up on the negative aspects. But this doesn't really take us much beyond the initial problems inherent in non-empirical evidence.


C Leroy

G Hakhumyan

Y Pashayan-Leroy

D Sarkisyan

Alkali metal atomic transition probability in strong external magnetic field and its application

Keywords: Zeeman Hamiltonian, atomic transition intensity, frequency shift, submicron thin vapor

Interaction of alkali atoms with external magnetic field induced a splitting and a shift of their energy levels. We have study this interaction for external field from 0 to 5000 Gauss when the alkali vapor is confined in submicron thin vapor cell with thickness L = λ/2. Rubidium and Sodium vapors have been studied. The Hamiltonian can be expressed as the sum of the unperturbated atomic Hamiltonian and the so-called Zeeman Hamiltonian. The probability of a transition, induced by the laser electric field is proportional to the square of the transfer coefficients modified by the presence of the magnetic field. We will show that the strong nonlinearity of the transition intensities versus the external magnetic field intensity is obvious for B > 100 G. Some possible application for Laser Spectroscopy of the Rb and the Na atoms are presented

INTRODUCTION

Optical nano-cells (NCs) containing atomic vapor of alkali metal have already proved their efficiency to study atomic phenomena with a precision that normal size cells cannot achieve. One could even define as elegant tool such a device when used for the interpretation of interaction between atom and light, atom and surface or atom and atom [START_REF] Sarkisyan | Sub-Doppler spectroscopy by sub-micron thin Cs vapour layer[END_REF][START_REF] Dutier | Collapse and revival of a Dicke-type coherent narrowing in a sub-micron thick vapor cell transmission spectroscopy[END_REF][START_REF] Sarkisyan | Sub-Doppler Fluorescence on atomic D 2 line of Sub-Micron Rubidium Vapor Layer[END_REF][START_REF] Sarkisyan | Spectroscopy in an extremely thin vapor cell: Comparing the cell-length dependence in fluorescence and in absorption[END_REF][START_REF] Papoyan | Magnetic field-induced mixing of hyperfine states of the Cs 6P level observed with a submicron vapor cell[END_REF][START_REF] Sarkisyan | Hyperfine structure Zeeman effect on of an atomic D1 line of a sub-micron 87 Rb vapor layer[END_REF][START_REF] Sarkisyan | Fluorescence of rubidium in a submicrometer vapor cell: spectral resolution of atomic transitions between Zeeman sublevels in a moderate magnetic field[END_REF][START_REF] Weissbluth | Atoms and Molecules[END_REF] .

A NC has one of his dimensions L of nanometric thickness, that is, L can vary continuously from 0.1 to 10 λ where λ is the laser wavelength resonant with the considered atomic transition while the two others dimensions are of centimeter order. Furthermore, one of the main advantages of this elegant tool appears when one considers the atomic transitions of alkali atoms in magnetic fields. Indeed, for centimeter size thickness cell, the splitting of these transitions into their Zeeman components is relatively bad resolved due mainly to Doppler broadening and the resolution becomes even worst when the external magnetic field increases: well-known saturation absorption method allows only studying the behaviour of a total set of hyperfine transitions and becomes useless for magnetic field such that B > 100 G if one wants to describe individual hyperfine transition.

The high resolution atomic spectra obtained with NCs come from the possibility to highly reduce the Doppler broadening. This may be seen if one examines the efficiency of the optical pumping which is given by the following coefficient

2 2 2 ) . ( ~     v k t N     (1)
where t is the time of interaction of the radiation with the atom,  is the detuning of the resonance,  is the sum of homogeneous and non-homogeneous broadenings, N  is the spontaneous decay of the considered transition,  is the Rabi frequency of the laser radiation which direction of propagation is k  and v  is the atomic velocity.

An obvious way to increase the coefficient (1) is to be as close as possible to the resonance 0) ( and to perform a spectrum such that the experimental condition 0

.  v k  
is fulfilled. NC is an elegant and powerful device that realizes this condition: for the atoms flying perpendicularly to the laser beam the interaction time

is v D t  
, where D is the diameter of the laser beam, while for the atoms flying parallel to the laser beam, the

interaction time is v L t  //
. Thus, with a laser beam of diameter and expression (1) takes the maximum value at  = 0, the VSOP peaks are exactly at the atomic transition frequencies.

To demonstrate briefly the efficiency of the use of NCs to reduce the Doppler broadening, we show on Fig. 1 and2 (respectively) the spectra for ordinary cell and NC (respectively) for the transition of 87 In what follows, we will summarize the theoretical ingredients used to study the effect of an external magnetic field on the atomic transitions. Shifts of the Zeeman components and intensity of the corresponding transitions will be examined. Numerical applications to Rb and Na vapors confined in submicron thin vapor cell of thickness L = λ/2 are compared with experimental results.

EXPERIMENTAL PART

Sketch of the experimental setup is presented in Fig. 3a. The circularly polarized light of extended cavity diode laser (ECDL,  = 780 nm, P L ~ 30 mW, γ L < 1 MHz) resonant with 87 Rb D 2 transition frequency, after passing through Faraday isolator is directed onto the Rb NC 2 with the thickness of vapor column L = /2, at an angle close to normal.

The needed temperature regime of the NC (T SA ~ 110 -120 ºC, T W ~ 140 -150 ºC, corresponding to N ~ 6  10 12 -1.5  10 13 atoms/cm 3 ) is provided by a special oven with 3 openings: 2 inlets for laser beam passage and one orthogonal inlet for side fluorescence detection. This geometry allows simultaneous recording of fluorescence and transmission spectra. The fluorescence signal is detected by a photodiode 4 with an aperture of 1 cm 2 which is placed at 90 angle of the laser propagation direction. The photodiode signal is recorded by a digital four channel storage oscilloscope, Tektronix TDS 2014B.

A Glan prism is used to purify initial linear radiation polarization of the laser radiation; to produce a circular polarization a /4 plate 1 is utilized. The geometrical configuration of the experiment is shown in Fig. 3b. Magnetic field is directed along the laser radiation propagation direction k (B // k). With the help of a beam splitter, 50% of the pump beam is directed on the auxiliary 2' NC with Rb used as a frequency reference for B = 0. In some cases the reference spectrum is transmission spectrum [START_REF] Papoyan | Magnetic Field-Induced Mixing of the Cs 2 3 2 6 P Level Observed with a Submicron Vapor Cell[END_REF] obtained with an auxiliary NC of thickness L = . The longitudinal magnetic field is applied to the NC by a system of Helmholtz coils (not shown in Fig. 3a). The B-field strength is measured by a calibrated Hall gauge. Among the advantages of this setup is the possibility to apply strong magnetic field using widely available strong permanent magnets (PM). In spite of strong inhomogeneity of B-field, due to small thickness of the atomic vapor column inside NC (L = 390 nm) the variation of B inside the column is less than 0.1 G, i.e. by several orders less than the applied B value.

THEORETICAL BACKGROUND AND NUMERICAL CALCULATIONS

The so-called Dirac equation [START_REF] Weissbluth | Atoms and Molecules[END_REF] for an electron to order 2 2 / c v is the starting point to study the spectra of an atom within an electromagnetic field describes by the potentials   V A,  :

                    E eV p V S c m e V c m e c m p A S mc e A c e p m ) .( 2 8 8 . ) ( 2 1 2 2 2 2 2 3 4 2          (2)
with

    2  S
where ) , , (

z y x       , i  ) , , ( z y x i  are the matrices of Pauli. For a potential r e Z r V V   ) ( , the
term ) .( 2
2 2 p V S c m e      
describing the spin-orbit interaction is rewritten on a more compact

form S L r H SO   . ) (  
. It is worth to notice that this term leads naturally to calculate the Hamiltonian matrix in the basis

F F m F m F J L n , , , , ,  with J I F      and S L J     
. Thus for the Zeeman interaction, defined by can be chosen as a good basis to the first order if the condition

B B B H I S L Z       . . .        with L L        the magnetic orbital momentum, S S      2   the
SO Z E E  is fulfilled.
For the magnetic field that we apply to the nano-cells, this condition is respected as the magnetic field will vary from 0 to 10 4 G. One should note that all the previous discussion is valid for a magnetic field

B  supposed uniform ( ) ( 2 1 r B A     
), which is what we have supposed in the development of the Zeeman interaction. Although interaction of atoms with laser radiation field in such cells is strictly anisotropic, however due to nanometric longitudinal size of the NCs, one may consider this hypothesis of uniformity as valid. This assumption is conformed by numerical calculation for inhomogeneity of B-field 150 G/mm.

The Hamiltonian of interaction

B B B H I S L Z       . . .       
may be expanded using different forms [START_REF] Tremblay | Absorption profiles of alkali-metal D lines in the presence of a static magnetic field[END_REF][START_REF] Aleksandrov | Interference of Atomic States[END_REF][START_REF] Papoyan | Magnetic Field-Induced Mixing of the Cs 2 3 2 6 P Level Observed with a Submicron Vapor Cell[END_REF][START_REF] Shore | The Theory of Coherent Atomic Excitation, Multilevel Atoms and Incoherence[END_REF][START_REF] Sarkisyan | Sub-Doppler spectroscopy of Rb atoms in a sub-micron vapour cell in the presence of a magnetic field[END_REF] , we chose a formulation 10 which leads to

) I g S g L ( . B H I S B Z                 (3)
where S g , I g are respectively the spin and the nuclear Landé factor.

We adopt a matrix representation in the coupled basis, that is, the basis of the unperturbated atomic state vectors

F F m F m F J L n , , , , ), 3 (  
to evaluate the matrix elements of the Hamiltonian (3). In this basis, the diagonal matrix elements are given by

Z F F B F F B m g F E m F H m F    ) ( , , 0 (4)
where ) ( 0 F E is the initial energy [START_REF] Daniel | Sodium D Line Data[END_REF] of the sublevel

F m F,
and F g is the effective Landé factor [START_REF] Daniel | Sodium D Line Data[END_REF] .

Non-zero off-diagonal matrix elements verify the selection rules

1 , 0          F m J L F
and are given by

    2 1 2 2 2 1 2 2 2 2 ) 1 2 )( 1 2 ( ) ( ) 1 ( ) ( 2 , 1 , , , 1
                               F F F m F F I J F F I J g g B m F H m F m F H m F F I J Z B F F F F  (5)
The Hamiltonian matrix is then block diagonal where each block corresponds to a given value of F m . The diagonalization of the Hamiltonian matrix allows one to find the eigenvectors and the eigenvalues, that is to determine the eigenvalues corresponding to the energies of Zeeman sublevels and the new states vectors which can be expressed in terms of the initial unperturbed atomic state vectors, In figure 4a we present the energy levels obtained by the Hamiltonian matrix diagonalization for 1 D line of 87 Rb atom. Figure 4b simply gives the definition of the notation used to label the transitions in Fig. 4a.

              g g g g e e e e e g g g g g g g g g J F J F F e F z e F F F e J F J F F g F z g F F F g m F m B m F m F m B m F 2 3 2 3 ' 2 3 2 3 ' , ) , ( ) , ( and , ) , ( ) , ( ' ' 
The transition frequency shift values are obtained on a quasi continuous variation of the B field intensity as we diagonalize 50 000 matrices. From low B field intensity to high B field variation, this diagonalization procedure allows us to follow exactly the eigenvalues by analyzing precisely the jump of the eigenvectors using a standard derivative analysis method. One should note that Fig. 4a shows the observed frequency shift, measured experimentally (which are obtained from Fig. 2) while simultaneously on the same curves have been superposed the calculated ones. As previously written, the hypothesis of locally homogeneous and uniform magnetic field is verified.

The probability of a transition, induced by the interaction of the atomic electric dipole and the oscillating laser electric field is proportional to the spontaneous emission rate of the associated transition g e A , that is, to the square of the transfer coefficients modified by the presence of the magnetic field

' 2 2 3 0 2         3 8 '
where

q D ( 1 , 0 , 1   q
) denote the standard components of the electric dipole moment

  q q q e D e D   .
.

In eq. ( 7), the transfer coefficients are expressed as

    ) , ( ; , ; , ) , ( ); , ( ); , ( ' ' , ' '
F F g F e F F F z e F F F g F e m B q m F m F a m B q m F m F a       , (8)
where the unperturbated transfer coefficients have the following definition [START_REF] De Clercq | Laser diode optically pumped caesium beam[END_REF]  

                         e J I g J g F e F g F m q e F

CONCLUSION

The /2-Zeeman technique is shown to be convenient and robust method for the study of individual transitions between the Zeeman sublevels of hyperfine levels in an external magnetic field. NC in external magnetic field allows exhibiting obviously the several Zeeman components due mainly to the suppression of the Doppler broadening. One of the goals of the construction of such an elegant tool was the possibility to remove this effect and it succeeds perfectly. As a result, the use of a NC as a nanomagnetometer seems a successful application of this device as it allows the measurement of extremely large gradient of magnetic field. Calculated and experimental results are in very good agreement.

Fig. 1 Fig. 2

 12 Fig. 1 Spectra of transmission in an ordinary cell for 87 Rb in the case B = 0 and B = 200 G for the radiations with polarisation σ + and σ -.Fig. 2 Spectra of fluorescence in a NC for 85 Rb and 87 Rb for four values of B from 0 G to 1875 G.

Fig. 3 a

 3 Fig.3 a -Sketch of the experimental setup. FI -Faraday isolator, 1 -λ/4 plate, 2 -STC and the oven, 2'-the auxiliary NC and the oven, 3 -RMs, 4 -photodetectors, 5 -digital storage oscilloscope; b -mutual orientations of laser excitation and detection of fluorescence emitted by a NC of L = λ/2.



  the unperturbated state vectors, respectively, for the excited and the ground states. The coefficients ) are mixing coefficients, respectively, for the excited and the ground states; they depend on the field strength and magnetic quantum numbers e m or g m .

b

  Fig. 4a Transition frequency shift versus magnetic field for 87 Rb, D 1 line Fig. 4b Diagram of the labels 1, 2, 3 and 1, 2

b

  Fig.5a D 1 Line intensities for Na, transitions F g = 1 → F e = 1, 2 Fig.5b Diagram of the labels 1, 2, 3 and 1, 2

Acknowledgement

The authors are grateful to A. Sarkisyan for his valuable participation in fabrication of the STC as well as to A. Papoyan for useful discussions. Research conducted in the scope of the International Associated Laboratory IRMAS. Armenian team thanks for a research grant PS 1868 from the Armenian National Science and Education Fund (ANSEF) based in New York, USA.


G Hakhumyan

A Sargsyan

C Leroy

Y Pashayan-Leroy

D Sarkisyan

Study of Rb atomic transitions D 1, 2 lines in strong magnetic field based on fluorescence spectra of sub -micron thin cell

The so-called "/2-Zeeman technique" (HLZT) for studies of individual optical transition between Zeeman sublevels of atomic hyperfine structure in an external magnetic field of B = 10 -2500 G is presented. Particularly, implementation of HLZT allows one to realize a direct determination of frequency shift and a strong modification of an optical transition probability between Zeeman components in a B-field. The main advantages of the method compared to "-Zeeman technique" (LZT) is that it allows one to study weak transitions. Particularly, with the help of fluorescence on 87 Rb, D 2 line, F g = 1 → F e = 3 transitions, three "forbidden" transitions in magnetic field B are detected and studied. Also, on 87 Rb D 1 line, F g = 1, m F = 0 → F e = 1, m F = 0 "forbidden" transition is detected when B ~ 400 G. A strong modification of the probability for these "forbidden" transitions is revealed. The theoretical model well describes the observed results.

INTRODUCTION

The development of atomic vapor cells of sub-micrometric thickness (STC) opens a new possibility to realize sub-Doppler spectroscopy. Interaction of atoms with laser radiation field in such cells is strictly anisotropic. Atoms with nonzero longitudinal velocity projection experience frequent quenching collisions with the cell walls (windows). As a result, the interaction is favored for the atoms with zero longitudinal velocity, i.e. those yielding sub-Doppler fluorescence response [START_REF] Sarkisyan | Sub-Doppler spectroscopy by sub-micron thin Cs vapour layer[END_REF][START_REF] Dutier | Collapse and revival of a Dicke-type coherent narrowing in a sub-micron thick vapor cell transmission spectroscopy[END_REF][START_REF] Sarkisyan | Sub-Doppler Fluorescence on atomic D 2 line of Sub-Micron Rubidium Vapor Layer[END_REF][START_REF] Sarkisyan | Spectroscopy in an extremely thin vapor cell: Comparing the cell-length dependence in fluorescence and in absorption techniques[END_REF] . With the proper choice of parameters, it is possible to achieve a sevenfold narrowing of the spectrum.

It is well-known that atomic transitions of alkali metals split in magnetic fields into Zeeman components, whose frequency shifts deviate from the linear behavior in already moderate magnetic fields. As this takes place, the probabilities of atomic transitions usually change significantly as well [START_REF] Tremblay | Absorption profiles of alkali-metal D lines in the presence of a static magnetic field[END_REF][START_REF] Papageorgiou | High-resolution selective reflection spectroscopy in intermediate magnetic fields[END_REF][START_REF] Papoyan | Magnetic field-induced mixing of hyperfine states of the Cs 6P level observed with a submicron vapor cell[END_REF] . Previous papers [START_REF] Sarkisyan | Hyperfine structure Zeeman effect on of an atomic D1 line of a sub-micron 87 Rb vapor layer[END_REF][START_REF] Sarkisyan | Fluorescence of rubidium in a submicrometer vapor cell: spectral resolution of atomic transitions between Zeeman sublevels in a moderate magnetic field[END_REF][START_REF] Sarkisyan | Sub-Doppler spectroscopy of Rb atoms in a sub-micron vapor cell in the presence of a magnetic field[END_REF][START_REF] Varzhapetyan | Study of atomic spectral lines in a magnetic field with use of a nanocell with the thickness L = [END_REF] demonstrated that the use of resonance fluorescence spectra of a STC filled with atomic vapor and of thickness L = 0.5λ (where λ = 794 nm (or 780 nm) is the wavelength of the laser radiation whose frequency is resonant with the atomic transition of the D 1,2 lines of Rb) allows one to separate and study atomic transitions between the levels of the hyperfine structure of the D 1,2 lines of 87 Rb atoms in magnetic fields with B varying in the range of 10 -200 G. The achieved high sub-Doppler spectral resolution is caused by the effect of a strong narrowing of the fluorescence spectrum of a nanocell with the atomic vapor column thickness L = 0.5λ (the method was called HLZM) compared to normal 1cm-long cells (for which the Doppler width is ~ 500 MHz). It is known that, with the saturated absorption (SA) technique, the sub-Doppler spectral resolution can also be achieved using cm-long cells (when the parameters are properly chosen, it is possible to obtain peaks of reduced absorption with a width close to the natural one ~ 6 MHz). In the publications [START_REF] Momeen | Variations of intensity in Rb D2 line at weak/intermediate fields[END_REF][START_REF] Školnik | Optical pumping of Zeeman components in rubidium vapor[END_REF] , the SA technique is used to study spectra of D 1 and D 2 lines of Rb atoms. One of the significant disadvantages of the application of the SA technique is the presence of so-called cross-over resonances in the spectra. In a magnetic field, these resonances split into numerous components, making the spectrum very difficult to process. This restricts the magnitude of acceptable magnetic fields and, as a rule, B should be below 100 G. Another significant disadvantage is the fact that SA is strongly nonlinear and, therefore, the peak amplitudes of the decreased absorption do not correspond to the probability of atomic transitions at which frequencies these peaks are formed. This additionally complicates the processing of spectra. At the same time, using the HLZM technique, the peak value of fluorescence (up to laser intensities of ~ 20 mW/cm 2 ) corresponds to the probability of atomic transitions. This allows one to directly study the dependence of the probability on B. Another important advantage of the HLZM method is the possibility of using strong permanent magnets (PMs) that can generate fields of several thousand gausses at distances of several centimeters. The fields of these PMs are strongly non-homogeneous and the gradient can achieve ~ 100 -200 G/mm, which makes impossible to use cm-long cells. At the same time, due to the small thickness of the STC (~ 400 nm), the gradient of B inside the STC is four to five orders of magnitude smaller than the measured value of B. This remarkable feature of a STC is used for the study of fluorescence spectra [START_REF] Hakhumyan | Investigation of Rb D1 Atomic Lines in Strong Magnetic Fields by Fluorescence from a Half-Wave-Thick Cell[END_REF] as well as for absorption spectra of atomic vapors within B fields from 10 -2500 G at the vapor column thickness L = λ (the method was called LZM) [START_REF] Sargsyan | A novel approach to quantitative spectroscopy of atoms in a magnetic field and applications based on an atomic vapor cell with L = [END_REF] .

The aim of this work is to experimentally demonstrate that with the help of HLZM it is possible to study the "forbidden" transitions on D 1, 2 lines of the 87 Rb. Also, a theoretical consideration is provided.

EXPERIMENTAL PART

Sketch of the experimental setup is presented in Fig. 1a. The circularly polarized light of extended cavity diode laser (ECDL,  = 780 nm, P L ~ 30 mW, γ L < 1 MHz) resonant with 87 Rb D 2 transition frequency, after passing through Faraday isolator is directed onto the Rb STC 2 with the thickness of vapor column L = /2, at an angle close to normal. The needed temperature regime of the STC (T SA ~ 110 -120 ºC, T W ~ 140 -150 ºC, corresponding to N ~ 6  10 12 -1.5  10 13 atoms/cm 3 ) is provided by a special oven with 3 openings: 2 inlets for laser beam passage and one orthogonal inlet for side fluorescence detection. This geometry allows simultaneous recording of fluorescence and transmission spectra.

The fluorescence signal is detected by a photodiode 4 with an aperture of 1 cm 2 which is placed at 90 angle of the laser propagation direction. The photodiode signal is recorded by a digital four channel storage oscilloscope, Tektronix TDS 2014B 5. A Glan prism is used to purify initial linear radiation polarization of the laser radiation; to produce a circular polarization a /4 plate 1 is utilized. The geometrical configuration of the experiment is shown in Fig. 1b. Magnetic field is directed along the laser radiation propagation direction k (B // k).

With the help of a beam splitter, 50% of the pump beam is directed on the auxiliary 2' STC with Rb used as a frequency reference for B = 0 . In some cases the reference spectrum is transmission spectrum [START_REF] Sargsyan | A novel approach to quantitative spectroscopy of atoms in a magnetic field and applications based on an atomic vapor cell with L = [END_REF] obtained with an auxiliary STC of thickness L = . The small longitudinal magnetic field (B < 250 G) is applied to the STC by a system of Helmholtz coils (not shown in Fig. 1a). The B-field strength is measured by a calibrated Hall gauge. Among the advantages of HLZM is the possibility to apply much stronger magnetic field using widely available strong PMs. In spite of strong inhomogeneity of B-field (in our case it can reach 150 G/mm), due to small thickness of the atomic vapor column inside STC (L = 390 nm) the variation of B inside the column is less than 0.1 G, i.e. by several orders less than the applied B value.

Implementation of HLZT method for the 87 Rb D 2 line F g = 1  F e = 3 forbidden transitions study

The optical transitions between magnetic sublevels of hf states for 87 Rb D 2 line in the case of σ + (left circular) polarized excitation and selection rules Δm F = +1 are depicted in Fig. 2.

Examination of the fluorescence peak formed in a STC allow one to obtain, identify and investigate the atomic transitions between the Zeeman sublevels in the transmission spectrum of the 87 Rb D 2 line in the range of magnetic fields 10  2500 G, i.e., the transitions fluorescence peak numbered 6. It is important to note, that both the frequency position as well as the fluorescence peaks 6, 7, 8 and 9 are well described by the theoretical curves presented in Figures 5 and6 in a previous work [START_REF] Sargsyan | A novel approach to quantitative spectroscopy of atoms in a magnetic field and applications based on an atomic vapor cell with L = [END_REF] . Thus, HLZT is a convenient and robust tool to study forbidden atomic transitions in an external magnetic field. In order to study the 87 Rb D 1 line F g = 1, m F = 0  F e = 1 m F = 0 forbidden transition we use a -polarized excitation (selection rules Δm F = 0 are depicted in Fig. 4). For this case the magnetic field B is directed along the laser electric field E (B  E ) and the fluorescence detection is in the direction perpendicular to E (see Fig. 1b). As it is seen all the five allowed transitions are well seen, while forbidden transition 5 is absent. For B = 0 the ratio of 1, 2, 3 amplitudes is 3:4:3 and for 4, 6 amplitudes is 1:1 (see Fig. 6). As it is seen for the magnetic field B  200 G there already occur modifications of the amplitudes.

F g = 1  F e = 0, F g = 1  F e = 1, F g = 1  F e =

Use of HLZT method for the

Fig. 6 (the upper curve) presents the fluorescence spectrum of STC for the atomic 87 Rb, F g = 1  F e = 1, 2 transitions when B  410 G. The lower grey curve is the reference fluorescence spectrum obtained with the auxiliary STC of thickness L = /2 for B = 0. As it is seen all the six transitions are well seen, including forbidden (when B = 0) transition

F g = 1, m F = 0  F e = 1 m F = 0 numbered 5.
Thus, a strong modification of atomic transitions in magnetic field is seen.

It is important to note, that both the frequency position as well as the fluorescence peaks 1, 2, 3, 4, 5 and 6 are well described by the theoretical curves presented in Figures 7 and8. Particularly, the experimentally detected frequency shift, from the initial position when B = 0, for the components 1, 2, 3, 4 and 6 is well described by the theoretical curves shown in Fig. 7. Note, that the component 5 has practically zero shift from the initial frequency position

F g = 1  F e = 1.
As to the modification of the probability of the component 5 as a function of B value, as it is predicted theoretically by the curves presented in Fig. 8, for B > 400 G it's probability becomes larger than that of the component 6 . This is well seen from the experimental curves shown in Fig. 6.

F e = 2 F e = 1 F g = 1

Theoretical model

When an external magnetic field is applied, the Hamiltonian for the atom in the presence of a magnetic field can be expressed as the sum of the unperturbated atomic Hamiltonian 0 H and the so-called Zeeman Hamiltonian Z H :

Z H H H   0 . (1)
The Hamiltonian of interaction Z H may be expanded using different forms [START_REF] Tremblay | Absorption profiles of alkali-metal D lines in the presence of a static magnetic field[END_REF][START_REF] Papoyan | Magnetic field-induced mixing of hyperfine states of the Cs 6P level observed with a submicron vapor cell[END_REF][START_REF] Sarkisyan | Sub-Doppler spectroscopy of Rb atoms in a sub-micron vapor cell in the presence of a magnetic field[END_REF] , in this way [START_REF] Tremblay | Absorption profiles of alkali-metal D lines in the presence of a static magnetic field[END_REF] Z H may be expressed as

) ( I S L I S B Z g g           . B H   , ( 2
)
where L is the orbital angular momentum, S is the spin momentum and I the nuclear spin momentum, B  is the Bohr magneton and S g , I g are respectively the spin and nuclear Landé factor.

We adopt a matrix representation in the coupled basis, that is, the basis of the unperturbated atomic state vectors

F m F J L n , , , ), 3 (  to
evaluate the matrix elements of the Hamiltonian (1). In this basis, the diagonal matrix elements are given by

Z F F B F F B m g F E m F m F    ) ( , , 0 H , (3)
where

) ( 0 F E is the initial energy of the sublevel F F m F m F J L n , , , , ), 3 ( 
    2 1 2 2 2 1 2 2 2 2 ) 1 2 )( 1 2 ( ) ( ) 1 ( ) ( 2 , 1 , , , 1                                F F F m F F I J F F I J g g B m F m F m F m F F I J Z B F F F F  H H . (4)
The Hamiltonian matrix is then block diagonal where each block corresponds to a given value of F m .

The diagonalization of the Hamiltonian matrix allows one to find the eigenvectors and the eigenvalues, that is to determine the eigenvalues corresponding to the energies of Zeeman sublevels and the new states vectors which can be expressed in terms of the initial unperturbed atomic state vectors,

and

       g g g g g g g g g J I F J I F F g F z g F F F g m F m B m F , ) , ( ) , ( ' '  . ( 6
)
The state vectors

' ' 2 2 3 0 2         , 3 8
where q D denote the standard components of the electric dipole moment:

   q q q e D e D , (8) with 1
, 0 , 1   q .
In eq. ( 7), the transfer coefficients are expressed as

F F g F e F F F z e F F F g F e m B q m F m F a m B q m F m F a       , (9)
where the unperturbated transfer coefficients have the following definition

                           
the parenthesis and curly brackets denote, respectively, the j  3 and j  6 symbols, g and e point respectively ground and excited states.

Conclusion

The "/2-Zeeman technique" (HLZT) is shown to be a convenient and robust tool for the study of individual transitions between the Zeeman sublevels of hyperfine levels in an external magnetic field of 10 -2500 G. HLZT is based on resonant fluorescence spectrum emitted by a STC with L = /2, where  is the resonant wavelength (780 nm for D 2 and 794 nm for D 1 line of Rb). Narrow fluorescence peak of spectral width ~ 70 MHz is split into several components in a magnetic field; their frequency positions and probabilities depend on the B-field. The implementation of HLZT allows us to reveal strong modification of the probability of the atomic transition. Also, the HLZT allows us to detect strong modification of three "forbidden" transitions with the help of fluorescence on 87 Rb, D 2 line, F g = 1 → F e = 3 transitions. On 87 Rb, D 1 line, F g = 1, m F = 0 → F e = 1, m F = 0 "forbidden" transition probability also demonstrates a strong modification for the magnetic field B > 300 G. The experimental results are in a good agreement with the theoretical values.

Fig. 1 a

 1 Fig.1 a -Sketch of the experimental setup. FI -Faraday isolator, 1 -λ/4 plate, 2 -STC and the oven, 2'-the auxiliary STC and the oven, 3 -RMs, 4 -photodetectors, 5 -digital storage oscilloscope; b -mutual orientations of laser excitation and detection of fluorescence emitted by a STC of L = λ/2.

  2 as well as F g = 1  F e = 3 for  + polarizations of the exciting laser radiation. It is revealed experimentally that in relatively weak magnetic fields (about 100 G) with the  + -polarized laser also those atomic transitions are recorded, for which new selection rules with respect to the quantum number F take place: F = 2, m F = +1. The probability of these transitions increases with the increase of the magnetic field strength. Note that for B = 0 the F = 2 transitions are strongly forbidden. In the field of B  200 G the probability of the transition F g = 1, m F = 1  F e = 3, m F = 0 (Fluorescence peak 7) becomes equal to the probability of the transition F g = 1, F = +1  F e = 2, m F = +2 (Fluorescence peak 6). Also with a field B > 400 G the transitions F g = 1, m F = 0  F e = 3, m F = +1 (Fluorescence peak 8) and F g = 1, m F = +1  F e = 3, m F = +2 (Fluorescence peak 9) are easily recorded.

Figure 3 Fluorescence 2 FF e = 3 F e = 2 F 1 F g = 1 F e = 0 Fig. 2 .Fig. 3 .

 323211023 Figure3shows the fluorescence spectrum of STC for the atomic F g = 1  F e = 2, 3 transitions. The fluorescence peaks under number 7, 8, 9 represent forbidden (when B = 0) atomic transitions F g = 1  F e = 3. Note, that the amplitudes of the Fluorescence peaks 7, 8, 9 are comparable to the peak numbered 6 (F g = 1, m F = +1  F e = 2, m F = +2) which is the strongest atomic transition when B = 0. Moreover, the amplitude of the fluorescence peak 9 is even larger than that of the

  87 Rb D 1 line F g = 1, m F = 0  F e = 1 m F = 0 forbidden transition study

Fig. 5 (

 5 Fig.5 (the upper curve) shows the fluorescence spectrum of STC for the atomic 87 Rb, F g = 1  F e = 1, 2 transitions when B  200 G. The lower grey curve-reference fluorescence spectrum obtained with the auxiliary STC of thickness L = /2.As it is seen all the five allowed transitions are well seen, while forbidden transition 5 is absent. For B = 0 the ratio of 1, 2, 3 amplitudes is 3:4:3 and for 4, 6 amplitudes is 1:1 (see Fig.6). As it is seen for the magnetic field B  200 G there already occur modifications of the amplitudes.

Fig. 4 .Fig. 5 .

 45 Fig.4. Energy level diagram of D 1 line of 87 Rb in a magnetic field and possible atomic transitions for -polarized excitation. Transition number 5 is forbidden when B = 0.

Fig. 6 .

 6 Fig.6. Fluorescence spectrum of D 1 line of the 87 Rb, F g = 1  F e = 1, 2 transitions, B  410 G. Fluorescence peaks 1, 2, 3, 4, 5 and 6 represent atomic transitions shown in diagram Fig.4. As it is seen the fluorescence peak under number 5 already exists. The lower grey curve-reference fluorescence spectrum is obtained with an auxiliary STC of thickness L = /2 for B = 0.



  the unperturbated state vectors, respectively, for the excited and the ground states. The coefficients ) are mixing coefficients, respectively, for the excited and the ground states; they depend on the field strength and magnetic quantum numbers e m or g m .The probability of a transition, induced by the interaction of the atomic electric dipole and the oscillating laser electric field is proportional to the spontaneous emission rate of the associated transition g e A , that is, to the square of the transfer coefficients modified by the presence of the magnetic field

  

														1  1'		1  2'
												0.042			0.044	0.046	0.048
																	and F g is the effective Landé
	factor.
	The		off-diagonal		matrix		elements	are	non-zero	for	levels	verifying	the	selection	rules
		L		, 0		J		, 0		F		, 1 		F m		0

Acknowledgement

The authors are grateful to A. Sarkisyan for his valuable participation in fabrication of the STC as well as to A. Papoyan for useful discussions. Research conducted in the scope of the International Associated Laboratory IRMAS. Armenian team thanks for the ANSEF PS 1868 support.


Chantal Cherifi

Vincent Labatut
email: vlabatut@gsu.edu.tr

Jean-François Santucci
email: santucci@univ-corse.fr

WEB SERVICES DEPENDENCY NETWORKS ANALYSIS

Keywords: Dependency Networks, Semantic Web Services, Complex Networks

Along with a continuously growing number of publicly available Web services (WS), we are witnessing a rapid development in semantic-related web technologies, which lead to the apparition of semantically described WS. In this work, we perform a comparative analysis of the syntactic and semantic approaches used to describe WS, from a complex network perspective. First, we extract syntactic and semantic WS dependency networks from a collection of publicly available WS descriptions. Then, we take advantage of tools from the complex network field to analyze them and determine their topological properties. We show WS dependency networks exhibit some of the typical characteristics observed in real-world networks, such as small world and scale free properties, as well as community structure. By comparing syntactic and semantic networks through their topological properties, we show the introduction of semantics in WS description allows modeling more accurately the dependencies between parameters, which in turn could lead to improved composition mining methods.

INTRODUCTION

A Web Service (WS) is a self-describing, selfcontained, modular application accessible over the web. It can be published and discovered in a WS registry and invoked for remote use. Its interface is exposed throughout a so-called WS description, which lists the implemented operations. Currently, production WS use syntactic descriptions expressed with the most prevalent description language, WSDL, a W3C (World Wide Web Consortium) standard. Along with textual information and some low level access directives, descriptions basically contain the names of the operations and their parameters names and data types. WS were initially designed to interact with each other, in order to provide a composition of WS able to offer higher level functionalities [START_REF] Benatallah | Towards patterns of web services composition[END_REF]. The keyword-based techniques used in current production discovery are not suitable as they often lead to false positives and false negatives [START_REF] Pilioura | Using WSDL/UDDI and DAML-S in Web Service Discovery[END_REF]. A false positive takes the form of an irrelevant service which includes the searched keywords in its description. On the contrary, different descriptions can contain syntactically different but semantically equivalent words, leading to false negatives. Furthermore, keyword-based techniques do not allow to perform any form of inference nor flexible match [START_REF] Sycara | Automated Discovery, Interaction and Composition of Semantic Web Services[END_REF].

The key underlying problem is that keywords are a poor way to capture the semantics of a search or an advertisement. More advanced research (nonproduction) approaches rely on comparing structured data such as parameters types and names, or analyzing unstructured textual comments [START_REF] Stroulia | Semantic structure matching for assessing Web-service similarity[END_REF][START_REF] Wu | Similarity-based Web service matchmaking[END_REF][START_REF] Ma | Web Services Discovery Based on Latent Semantic Approach[END_REF]. This is generally not enough to distinguish WS in terms of functionality, and consequently makes it difficult, or even impossible, to use these methods to automate WS composition. Indeed, syntactically discovered WS must be manually validated to ensure they implement the desired behavior, leading to static, a priori compositions.

To solve this limitation, a different mechanism is needed, one that entails retrieving WS on the basis of the functionalities they provide. The research community followed the current semantic Web trend by introducing semantics in WS descriptions, in order to enrich them. Several initiatives for new semantic description languages exist among which we can distinguish purely semantic descriptions with OWL-S (also a W3C standard), from annotated WSDL descriptions with WSDL-S and SAWSDL. Although those languages allow associating ontological concepts with various elements of the description, the research community has been focusing only on those qualifying the operations inputs and outputs. But retrieving semantic information is far more costly than collecting syntactic descriptions, even when considering only parameters. The latter can be performed quickly and completely automatically, whereas the former is a long task, requiring human intervention to label each parameter with the proper concept. Annotation tools exist to help, but they are clearly not mature yet, and often defined for specific collections or languages [START_REF] Hess | ASSAM: A Tool for Semi-Automatically Annotating Semantic Web Services[END_REF][START_REF] Gomadam | Radiant: A tool for semantic annotation of Web Services[END_REF]. Maybe for these reasons, no semantic annotation language emerged as an industry standard, although they appeared more than five years ago now: all production WS still rely on WSDL. Even at a research level, no publicly available significant collection of semantically annotated WS exists, making it very difficult to test new algorithms.

This situation leads to one question: is describing WS semantically worth the cost? To our knowledge, no one did ever compare the information underlying syntactic and semantic WS descriptions. In this work, we try to tackle this problem from the perspective of parameters dependency, through the use of complex networks. We model parameters spaces by building so-called dependency networks, based on syntactic and semantic descriptions of a single WS collection. We make the assumption the information conveyed by the two different kinds of descriptions appears in the corresponding dependency networks. We then compare the syntactic and semantic descriptive approaches through the networks topological properties. Our main contributions are an extended investigation of the parameters networks topology and the comparison of syntactic and semantic networks. The rest of the paper is organized as follow. In section 2, we present complex networks and their main topological properties. Section 3 introduces dependency networks and explains how they can be extracted from WS descriptions. Section 4 is dedicated to the presentation and discussion of our experimental results, i.e. the extracted networks, their topological properties and how they compare. Finally, in section 5, we emphasize the original points of our work, discuss its limitations and their possible solutions, and explain how it can be carried on.

COMPLEX NETWORKS

Complex networks are a specific class of graphs, characterized by a huge number of nodes and non trivial topological properties. They are used in many different fields to model real-world systems [START_REF] Costa | Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications[END_REF], and have been intensively studied both theoretically and practically [START_REF] Newman | The structure and function of complex networks[END_REF]. Because of their complexity, specific tools are necessary to analyze and compare them. This is usually performed through the comparison of several well-known properties, supposed to summarize the essential of the network structure.

Distance-Based Measures

The distance between two nodes is defined as the number of links in the shortest directed path connecting them. At the level of the whole network, this allows to process the average distance and the diameter. The former corresponds to the mean distance over all pairs of nodes [START_REF] Newman | The structure and function of complex networks[END_REF]. This notion is related to the small-world property, observed when this distance is relatively small. The classic procedure to assess this property consists in comparing the average distance measured in some network of interest to the one estimated for an Erdős-Rényi (ER) network [START_REF] Erdos | On random graphs[END_REF] containing the same numbers of nodes and links, since this random generative model is known to produce networks exhibiting the small-world property [START_REF] Newman | The structure and function of complex networks[END_REF]. In terms of dynamic processes, the existence of shortcuts between nodes can be interpreted as propagation efficiency [START_REF] Watts | Collective dynamics of small-world networks[END_REF]. Most real-world networks have the small-world property. The diameter is the greatest distance over all pairs of nodes in the network. In real-world networks, a small diameter is synonymous to rapid information propagation [START_REF] Cherifi | Mesh-Like : An Interconnection Network Model[END_REF].

Transitivity

A network transitivity (also called clustering) corresponds to its density of triangles, where a triangle is a structure of three completely connected nodes. It is measured by a so-called transitivity coefficient, which is the ratio of existing triangles to possible triangles in the considered network [START_REF] Watts | Collective dynamics of small-world networks[END_REF]. The higher this coefficient, the more probable it is to observe a link between two nodes which are both connected to a third one. A real-world network is supposed to have a higher transitivity than the corresponding ER network by an order of magnitude corresponding to their number of nodes, meaning their nodes tend to form densely connected groups.

Degree-Based Measures

The degree of a node corresponds to the number of links attached to it. In a directed network, one can distinguish in and out degrees, i.e. the numbers of incoming and outgoing links, respectively. Nodes with a high in (resp. out) degree are called authorities (resp. hubs). The most basic degreebased measure is certainly the average degree over the whole network. When comparing networks containing the same number of nodes, it is related to their link density. The degree distribution of a network is particularly revealing of its structure. Most real-world networks have a power law degree distribution [START_REF] Albert | The diameter of the world wide web[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF][START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF], resulting in the so-called scale-free property. In other terms, real-world networks contain a very few nodes with extremely high degree, and a large number of nodes with very small degree.

The degree correlation of a network constitutes another interesting property. The question is to know how a node degree is related to its neighbors'. Real networks usually show a significantly different from zero degree correlation. If it is positive, the network is said to have assortatively mixed degrees whereas if it is negative, it is disassortatively mixed [START_REF] Newman | The structure and function of complex networks[END_REF]. According to Newman, social networks tend to be assortatively mixed, while other kinds of networks are generally disassortatively mixed.

Component Organization

A component is a maximal connected sub-graph, i.e. a set of interconnected nodes, all disconnected from the rest of the network. The component distribution and, more specifically, the size of the largest component are important network properties. Indeed, depending on the applicative context, the fact the network is split in several separated parts with various sizes can be considered as an indirect representation of the modeled system effectiveness. For example, in a communication network like the Internet, the size of the largest component represents the largest fraction of the network within which communication is possible and hence it reflects the effectiveness of the network at doing its job [START_REF] Newman | The structure and function of complex networks[END_REF]. Most real-world networks have a so-called giant component, whose size is overwhelming greater than the other components.

Community Structure

A community is defined as a subset of nodes densely interconnected relatively to the rest of the network. Unlike components, communities are not necessarily disconnected from each other (and generally, they are significantly connected). Many real-world networks have a community structure [START_REF] Newman | The structure and function of complex networks[END_REF]. Specific community detection algorithms must be used to identify them, leading to a partition of the overall nodes set. Almost all of them are dedicated to undirected networks, and only a very few recent ones can use the information conveyed by directed links. In this work, we chose to use a well tested program, and therefore focused on undirected links. We selected the Walktrap algorithm which exhibits good performances according to recent benchmarks [START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF].

To assess the quality of a network partition, the standard measure is Newman's modularity [START_REF] Newman | Finding and evaluating community structure in networks[END_REF], whose value also depends on the considered network structure. Consequently, its theoretical maximal value of (perfect community structure and partition) is rarely reached, and in practice values between and are considered high [START_REF] Newman | Modularity and community structure in networks[END_REF]. A value of represents a random partition or the absence of community structure.

DEPENDENCY NETWORKS

Network Definition

We define a dependency network as a directed graph whose nodes correspond to depending objects and links indicate the head nodes depends on the tail nodes. They can be considered as complex networks, and a few authors used similar approaches to model collections of WS based on syntactic [START_REF] Kil | On the Topological Landscape of Semantic Web services Matchmaking[END_REF][START_REF] Oh | Effective Web Services Composition in diverse and large-scale services networks[END_REF] or on semantic [START_REF] Hashemian | A Graph-Based Approach to Web Services Composition Symposium on Applications and the Internet (SAINT'05)[END_REF] descriptions. In the resulting parameters networks, each node corresponds to a parameter, and the links between them represent operation-related dependences. In this work, our goal is to compare the two types of WS descriptions; hence we used both syntactic and semantic descriptions.

As stated before, a WS interface is defined under the form of a set of operations. An operation represents a specific functionality, described independently from its implementation for interoperability purposes. Besides its functionality, it is characterized by two sets of input and output parameters, noted and , respectively. In a syntactic description, each parameter has a name and a type. This type is also defined independently from any implementation, again for interoperability reasons. Most of the time, the XML schema definition language (XSD) is used. In a semantic description, name and type are also generally specified, and an additional ontological concept is associated to the parameter, in order to give it a precise meaning. The most popular language used to describe these concepts is OWL, the Web Ontology Language designed by the W3C Web Ontology Working Group.

In the context of dependency networks, each operation is formally defined as a triplet [START_REF] Hashemian | A Graph-Based Approach to Web Services Composition Symposium on Applications and the Internet (SAINT'05)[END_REF], where denotes the set of dependencies defined by the operation. We consider each output parameter depends on each input parameter. The left side of Figure 1 represents three operations , and , with their respective inputs and outputs under the form of parameters named with letters ranging from to . As an example, consider operation : it is defined as where: , and (i.e. and are both dependent on and ). When considering not only a single operation, but a whole collection, one can say a parameter depends on another parameter iff an operation exists such as and . Parameters dependency network capture this three-part information: their nodes represent the parameters ( and ) and their links stand for the dependencies ( ). To build such a network, we first create one node for each parameter present in the whole collection. Then, links are created by considering each operation separately: a link is added between each one of its input parameters and each one of its output parameters. The right side of Figure 1 represents the dependency network corresponding to the three operations described on the left side. For example, for each input parameter belonging to , there exists a link directed to each one of its output parameters . In this network, the presence of a link from a node towards another node indicates at least one operation uses the parameter corresponding to as an input, and the parameter corresponding to as an output. This can also be interpreted in terms of production: we say such a link means one or several operations allow producing provided is already available. Connectivity in a dependency network is caused by the fact one parameter may be used by several operations, either as an input or an output. For example, parameters appear more than once in the collection, either as inputs or outputs for several operations, but only one node stands for each of them in the resulting dependency network. The parameters described in the collection of WS descriptions (left side in Figure 1) are called parameter instances. Those represented by nodes in the dependency network (right side) are called parameter archetypes. One parameter archetype represents a group of parameter instances supposed to convey the same information. Consequently, deciding if two instances correspond to the same archetype is a central task in extracting a dependency network. This depends on the nature of the considered parameters (syntactic vs. semantic description) and on how the notion of similarity is defined. These factors are implemented under the form of a so-called matching function.

Matching Function

A matching function takes two parameters and , and determines their level of similarity [START_REF] Shvaiko | A Survey of Schema-Based Matching Approaches[END_REF] . This function can be either symmetrical or asymmetrical, and its output can be either binary or real. When comparing two parameters, a real output allows representing various levels of similarity, which is a good thing because it conveys a more subtle information than a raw binary value. But it also results in a more complex processing during network generation. So we selected only binary matching functions in order to avoid this situation.

Because of the different nature of the concerned information, we used different matching functions to compare syntactically and semantically described parameters, resulting in so-called syntactic and semantic dependency networks, respectively. For syntactic descriptions, we compare parameters names: two parameters are said to be similar if their names are the exact same strings. The semantic matching is performed against the concepts associated to the parameters. It is based on the classic exact operator used in previous WS-related works to compare ontological concepts [START_REF] Paolucci | Semantic Matching of Web Services Capabilities[END_REF]. It considers two parameters to be identical iff their associated concepts match perfectly. Note both matching functions are symmetrical. Our goal is to compare syntactic and semantic descriptions, not matching functions, so we opted for standard and simple tools. In summary, we can extract two distinct networks: syntactic equal and semantic exact, noted and , respectively.

RESULTS AND DISCUSSION

Data

We extracted interaction networks from the SAWSDL-TC1 collection of WS descriptions [START_REF] Klusch | Semantic web service selection with SAWSDL-MX[END_REF][START_REF] Sawsdl-Tc | [END_REF]. This test collection provides 894 semantic descriptions written in SAWSDL, and distributed over 7 thematic domains (education, medical care, food, travel, communication, economy and weapon). It originates in the OWLS-TC2.2 collection, which contains realworld services descriptions retrieved from public IBM UDDI registries, and semi-automatically transformed from WSDL to OWL-S. This collection was subsequently resampled to increase its size, and converted to SAWSDL. An SAWSDL file describes a service both syntactically and semantically. This allowed us to extract our syntactic and semantic networks from the same collection, and to compare them consistently.

Networks Structure

The whole SAWSDL-TC1 test collection contains parameters instances, represented by nodes in and nodes in . The proportion of isolated nodes, and in and respectively, is quite small in both networks, and even a little smaller in the semantic one. Both networks exhibit a giant component. In it contains of the remaining nodes and of the remaining links. The others components are much smaller ranging from to nodes. is separated in distinct components, the giant one containing of the nodes and of the links (see Figure 2).

The fact that many parameters instances appear many times in the collection, like for example, _PRICE ( occurrences) or _AUTHOR ( occurrences), is the reason why there is a significant difference between the number of instances and the number of parameter archetypes (i.e. nodes in the networks). Moreover, we used different matching functions to build the syntactic and semantic networks, so the resulting archetypes are different (i.e. they do not correspond to the same sets of instances), which explains the difference in the number of nodes between and . The number of nodes is smaller for the later, which indicates semantic matching allows associating more instances.

This highlights the presence of false negatives (FN) in the syntactic network. FN are instances associated to different archetypes in the dependency network, whereas they are actually conveying the same information, and should therefore be represented by the same archetype. These FN usually are syntactically different (different names) but are associated to the same ontological concept (same meaning). For example parameter instances _AUTHOR, _AUTHOR1 and _AUTHOR2 are represented by three distinct nodes in the syntactic network, whereas they are associated to a unique node in the semantic network, as they all are associated to the same #author concept. The semantic matching also allows eliminating some false positives (FP). FP correspond to instances represented by the same archetype whereas they do not represent the same information. For example, many instances are simply called PARAMETER but are associated to very different concepts. The syntactic matching will improperly associate them to a common archetype, whereas the semantic matching will not.

Globally, the semantic matching results in less isolated nodes and small components, and a larger giant component, both in terms of nodes and links. The fact distinct components exist reflects the decomposition of the collection into several noninteracting groups of parameters. The presence of the giant component is a good property. It means the number of dependencies in which several operations are implied is high, allowing a large proportion of parameters to interact. In the rest of this section, we focus on the giant components properties, discarding isolated nodes and smaller components. Table 1 lists which properties we discuss hereafter. For average distance and diameter, values are given for both directed and non-directed networks.

Distance-Based Measures

Both syntactic and semantic networks exhibit small average distances: and , respectively. By comparison, this distance is approximately in ER random networks of comparable size, which means the dependency networks possess the small world property. In other terms, many shortcuts exist in the networks, indicating one can find dependency paths using a relatively small number of parameters. This can be interpreted in terms of WS composition, meaning one can produce some parameters of interest using a relatively small number of operations. According to the results, we can say semantic matching generates a more distinct smallworld property.

The component diameter is a good indicator of the largest dependency path, which is for and for . Observing this significant difference between the syntactic and the semantic networks allows us to say producing parameters is more efficient in terms of number of required operations. The fewer operations there are, the shorter the production time and the smaller the chance to meet unavailable operations on the path. Another point is the directed nature of the network, which leads to sensibly different results: from to and from to for the respective average distances of and , and from to and to for their respective diameters. 2, parameters are organized very hierarchically, in the form of trees rather than triangles, which explains these low values. This structure favors the apparition of hubs and authorities, the former corresponding to parameters used as an input by many operations and the latter to parameters being outputs for many operations. They play a central role in the parameters production process, and their absence can be critical. If a parameter is a hub, the production of many others depends on its presence. If it becomes unavailable, all these parameters cannot be produced anymore. If a parameter is an authority, its production depends on many others; and there are many operations able to produce it. For example, _COUNTRY and _PRICE are such remarkable parameters used or produced by several operations.

Degree-Based Measures

An empirical analysis of the network shows few parameters have a huge number of links while the majority has only a few, which is characteristic of a power law degree distribution. To confirm this observation, we used the method proposed in [START_REF] Clauset | Power-Law Distributions in Empirical Data[END_REF] to fit our data. We obtained high p-values for the global degree distribution (all) as showed in Table 1. Hence, the null hypothesis cannot be rejected, allowing us to suppose this distribution follows a power law (see Figure 1).

The p-values are lower for the in and out degrees distributions. They still do not allow rejecting the power law distribution, except for the syntactic out degree if we take a threshold of . Indeed, one can observe the existence of few hubs and authorities.

Figure 3: Degree distribution for , on a log-log scale.

These results are in accordance with average degree and degree correlation values. In Table 1, negative values of degree correlation indices indicate that nodes are significantly disassortatively mixed. Strongly connected nodes, as hubs and authorities are preferentially linked with lightly connected ones. While the value for the maximum degree is in both and , the average degrees values are relatively low, due to the presence of a high number of poorly connected nodes.

Community Structure

The Walktrap algorithm detected communities with a good modularity ( ) for equal and exact networks (values greater than 0.3 are considered high [START_REF] Newman | Modularity and community structure in networks[END_REF]. This community structure seems to reflect the collection domains, i.e. there is a partial correspondence between the groups of parameters retrieved from the network structure and those belonging thematically to specific domains. Indeed, it makes sense to observe denser relationships between parameters belonging to the same application field, because it is likely the related operations were designed to interact with each other. Parameters which are responsible to bridges between communities may be remarkable nodes such as hubs and authorities. For example _COUNTRY and _PRICE are transverse to the collection, i.e. they are produced or used by many operations across several domains. We did not notice any significant difference between the two considered networks while observing community structure.

CONCLUSIONS

In this paper, we compared the information conveyed by syntactic and semantic WS descriptions, through the use of complex networks. For this purpose, we extracted a syntactic and a semantic dependency networks from one collection of descriptions, using two different matching functions. We processed, discussed and compared their topological properties. Both networks exhibit some properties observed in most real-world complex networks: small average distance, power law degree distribution, presence of a giant component and community structure. However, unlike most real-world network, our dependency networks are not highly transitive.

When comparing syntactic and semantic networks, we observed a greater proportion of nodes and links are included in the semantic giant component. Consequently, the number and the size of small components decrease as well as the number of isolated nodes. The semantic giant component interconnection structure leads to smaller average distance and diameter. This means one needs to chain fewer operations to produce a given parameter. We can conclude the introduction of semantics in WS description allows a more accurate representation of their dependency relations. Dependency network-based representations of WS collections were used before in various contexts. [START_REF] Hashemian | A Graph-Based Approach to Web Services Composition Symposium on Applications and the Internet (SAINT'05)[END_REF] used them for composition mining, i.e. finding the best composition relatively to some criteria of interest. Oh et al. elaborate a benchmark dedicated to WS discovery and composition [START_REF] Oh | WSBen: A Web Services Discovery and Composition Benchmark Toolkit[END_REF] and developed a WS composition algorithm [START_REF] Oh | Effective Web Services Composition in Diverse and Large-Scale Services Networks[END_REF]. The two latter works are based on a study of networks topological properties [START_REF] Kil | On the Topological Landscape of Semantic Web services Matchmaking[END_REF]. However, this study focused only on syntactic descriptions, and neither the directed nature of the links nor the community structure are considered, Yet, this is of great importance in the context of parameters production and operation composition. Additionally, this is the first time, to our knowledge, an analysis is conducted on the topology of semantic networks, and consequently on the comparison with syntactic networks.

The study presented in this paper can be improved according to two directions. First, the collection we used is based on a set of real-world WS descriptions, but part of them was generated through resampling, so it cannot be considered as perfectly realistic. As a matter of fact, no other publicly available collection provides both syntactic and semantic descriptions for the same services [START_REF] Cherifi | Looking for a Web Services Collection?[END_REF], which is an indispensable prerequisite to a consistent comparison. The only solution we can see is to constitute our own collection, by semantically annotating a set of real syntactic descriptions. Second, we used a selected set of matching functions to extract the dependency networks. Many other functions exist, in particular more flexible syntactic distances [START_REF] Cohen | A Comparison of String Distance Metrics for Name-Matching Tasks[END_REF] can be used to perform less strict comparisons of the parameters names. This could have significant implications on the resulting network properties, since it is directly related to the amount of false positives (nodes irrelevantly connected) and false negatives (nodes irrelevantly disconnected).

Besides those improvements on data and matching functions, we plan to extend our work in two ways. First, we want to analyze in greater details the partial overlapping observed between communities and domains. A related point is to test whether properties observed for the whole network are also valid for domains or sets of domains. Second, to compare the observations we made in this work on the parameters networks and in a parallel work on operations networks [START_REF] Cherifi | Benefits of Semantics on Web Service Composition from a Complex Network Perspective[END_REF], we will extract and study networks at the service granularity level.

Figure 1 :

 1 Figure 1: Example of dependency network extraction.

  , generally under the form of a value in b Operations Parameters Dependency Network

Figure 2 :

 2 Figure 2: Trimmed exact semantic network . The giant component is located in the middle.

Table 1 :

 1 Properties of the giant components.

	Property
	Nodes
	Links
	Average distance (directed / undirected)	/		/
	Diameter (directed / undirected)	/		/
	Transitivity
	Degree correlation
	Average degree		/	/	/	/
	(in / out / all)
	p-value		/		/
	(in / out / all)	/		/
	Communities
	Modularity
	4.4 Transitivity
	Unlike most real-world networks, the measured
	transitivity is relatively low for both syntactic and
	semantic networks, with values around		. By
	comparison, this transitivity coefficient is
	approximately	in both ER random networks of
	comparable size, known to have very low
	transitivity. As shown in Figure