Hervé Herbin

Robert Farrenq

Guy Guelachvili
email: guy.guelachvili@u-psud.fr

Nathalie Picqué
email: nathalie.picque@u-psud.fr

Concentration modulation Fourier transform spectroscopy Concentration-modulation FT emission spectroscopy of TiCl 4 /He plasma. Analysis of the C 4 ∆-X 4 Φ ∆v=0 perturbed transitions of TiCl

Keywords:

A TiCl 4 /He plasma is observed by high resolution double-modulation FTS using concentration-modulation as a selective detection method. Analysis of the C 4 ∆-X 4 Φ ∆v=0 transitions of 48 Ti 35 Cl reveals perturbations affecting the C 4 ∆ 1/2 sub-state. ©2005 Optical Society of America

One major step in the development of infrared spectroscopy of unstable species has been the introduction of selective detection techniques like concentration modulation [START_REF] Amano | The ν 1 fundamental band of HCO + by difference frequency laser spectroscopy[END_REF] or velocity modulation [START_REF] Gudeman | Velocity modulated infrared laser spectroscopy of molecular ions: the ν 1 band of HCO +[END_REF] laser spectroscopies of discharge plasmas. These methods exploit physical properties of unstable molecules like lifetime, electrical charge, paramagnetism to distinguish them against the more abundant precursors. Selective detection techniques in FTS have already been demonstrated with step-scan interferometers [START_REF] Guelachvili | Selective detection of paramagnetic species by high-information Fourier-transform spectrometry[END_REF][START_REF] Picqué | High resolution multi-modulation Fourier transform spectroscopy[END_REF][START_REF] Picqué | Fast phase-selective detection of transient species with step-scan Fourier transform spectroscopy[END_REF] and with rapid-scan interferometers [START_REF] Imajo | Interferogram amplitude modulation technique for selective detection of transient species with a continuous-scan Fourier-transform spectrometer[END_REF][START_REF] Hong | Velocity modulated Fourier transform emission as a plasma diagnostic and a spectroscopic tool[END_REF]. They have never been extensively taken advantage of, most likely because of their instrumental complexity. In this paper, high resolution double modulation Fourier-transform spectroscopy has been applied to selective detection of short-lived species by concentration modulation in a TiCl 4 /He plasma. Emission spectra of the C 4 ∆-X 4 Φ ∆v=0 rovibronic transitions have been recorded and analyzed.

The principle of double-modulation FTS has been described elsewhere [START_REF] Picqué | High resolution multi-modulation Fourier transform spectroscopy[END_REF][START_REF] Herbin | Doppler-shifted transitions of a neutral molecule revealed by velocity modulation FTS[END_REF]. Here it is demonstrated using concentration modulation as a selective modulation. The modulation originates from an alternating current discharge, with synchronous detection of the interferogram at twice the discharge frequency. Two selective interferograms are detected in quadrature on two separate channels. The spectra are obtained from the emission of a glow discharge through a continuous flow of TiCl 4 and He. The discharge tube is 25 cm long and has an inner diameter of 0.7 cm. Helium (1300 Pa) and traces of TiCl 4 vapor enter the cell by a central inlet. A (800 V,0.3 A,10 kHz) ac power supply, driven by an audio amplifier connected to a transformer, is connected to the two electrodes of the tube. The emission of the source is sent into a stepping-mode Fourier transform interferometer. The interferometer is equipped with a CaF 2 beamsplitter and two liquid-nitrogen cooled InSb detectors. The recording spectral range is about 1800-4000 cm -1 and the non-apodized resolution equal to about 10 10 -3 cm -1 .

Fig. 1 and Fig. 2 display small portions of spectrum (nb.3525). Fig. 1 illustrates that the technique provides effective selectivity against stable precursor emission, since the lines of HCl, which is present as an impurity, are removed from the selective spectra. Fig. 2 shows that the thermal background and the superimposed absorption lines due to atmospheric CO 2 present in the nonselective spectrum are also eliminated from the selective spectra.

The spectra have also allowed the observation of the C 4 ∆-X 4 Φ ∆v=0 transitions of 48 Ti 35 Cl, with v = 0,1,2,3, around 3 µm. The 0-0 rovibronic transitions had already been investigated by Ram and Bernath [START_REF] Ram | Fourier transform infrared emission spectroscopy of the C 4 ∆-X 4 Φ, G 4 Φ-X 4 Φ, and G 4 Φ-C 4 ∆ systems of TiCl[END_REF]. In the work reported here, the rovibronic transitions 1-1, 2-2, 3-3 and weak perturbations occurring in the 0-0, 1-1, and 2-2 C 4 ∆ 1/2 -X 4 Φ 3/2 subbands, that had not been reported previously, have been identified and analyzed. The perturbations are due to Λ-doubling and avoided crossing phenomena induced by the never observed B 4 Π and A 4 Σ -states. Effective analysis of the unperturbed and perturbed bands has been performed, as well as a global analysis, including the accurate submillimeter lines measured by Ref. [START_REF] Maeda | Submillimeter-Wave Spectroscopy of TiCl in the ground electronic state[END_REF].

The equilibrium structure of 48 Ti 35 Cl in the C 4 ∆ state is determined from the resulting molecular constants [START_REF] Herbin | Perturbation analysis in the X 4 Φ -C 4 ∆ rovibronic transitions of 48 Ti 35 Cl at 3 µm[END_REF].

This research was financially supported by the Programme National de Physique et Chimie du Milieu Interstellaire (PCMI) du Centre National de la Recherche Scientifique (CNRS).

Fig. 1 :Fig. 2 :

 12 Fig. 1: Nonselective and selective portions of a high-resolution spectrum: lines from HCl stable molecule are removed from the selective components. The two selective spectra result from the synchronous detection of the selective interferogram, which is made in phase and in quadrature on two separate channels


H Brunel

D Ambard

H Dufour

P H Roche

V Costalat

F Jourdan

Rupture limit evaluation of human cerebral aneurysms wall: experimental study

Keywords: Intracranial aneurysm, Soft tissue, Hyperelastic material, Rupture limit, human specimen, Uniaxial traction test

Background and purpose

Rupture risk of intracranial aneurysms is a major issue for public healthcare. A way to obtain an individual rupture risk assessment is a main objective of many research teams in the world. For many years, we have investigated the relationship between the mechanical properties of aneurysm wall tissues and the rupture risk. In this work, we try to go further and investigate rupture limit values.

Methods

Following surgical clipping, a specific conservation protocol was applied to aneurysmal tissues in order to preserve their mechanical properties. Thirty-nine intracranial aneurysms (27 females, 12 males) were tested using a uniaxial tensile test machine under physiological conditions, temperature, and saline isotonic solution. These represented 24 unruptured and 15 ruptured aneurysms. Stress/strain curves were then obtained for each sample, and a fitting algorithm was applied following a Yeoh hyperelastic model with 2 parameters. Moreover, uniaxial tensile tests were conducted until rupture of samples to obtain values of stress and strain rupture limit.

Results

The significant parameter a 𝐶 2 of the hyperelastic Yeoh model, allowed us to classify samples' rigidity following the terminology we adopted in previous papers (Costalat 2011, Sanchez 2013): Soft, Stiff and Intermediate. Moreover, strain/stress rupture limit values were gathered and analyzed thanks to the tissue rigidity, the status of the aneurysm (initially ruptured or unruptured) and the gender of the patient.

Conclusion

Strain rupture limit was found quite stable around 20% and seems not to be correlated with the status of the aneurysm (initially ruptured or unruptured), neither with the gender of the patient. However, stretch and stress rupture limit seems not to be independent on the rigidity.

The study confirms that ruptured aneurysms mainly present a soft tissue and unruptured aneurysms present a stiff material.

Introduction

Hemorrhage stroke is a major issue for public healthcare. Approximately 12% of patients die before receiving medical attention, 40% of hospitalized patients die within one month after the accident, and more than one third of those who survive have major neurological deficits. The prevalence of intracranial aneurysms in the general population, as reported by [START_REF] Wardlaw | The detection and management of unruptured intracranial aneurysms[END_REF], ranges between 3% and 6.6%. Fortunately, very few aneurysms rupture. The incidence of ruptured aneurysms is approximately 0.5% per year. However, when a patient knows he has a "bomb" in his head, he is stressed and wants to be treated. But endovascular treatment of unruptured aneurysms, which is the safest treatment, is not without risk and has about 1% mortality rate [START_REF] Sluzewski | Rupture of intracranial aneurysms during treatment with guglielmi detachable coils: Incidence, outcome, and risk factors[END_REF]. Unruptured intracranial aneurysms therefore represent a dilemma for the physicians. The risk of aneurysm rupture with respect to its natural history has to be carefully balanced against the risk of morbidity and mortality from an endovascular or surgical repair.

Many factors are available to help decisions for surgery, like the size of the sac and its location in the Circle of Willis (ISUIAI, 1998). Other parameters, such as irregular aneurysm shape and, in particular, the presence of blebs [START_REF] Asari | Natural history and risk factors of unruptured cerebral aneurysms[END_REF][START_REF] Asari | Natural history and risk factors of unruptured cerebral aneurysms[END_REF][START_REF] Wiebers | The significance of unruptured intracranial saccular aneurysms[END_REF] are recognized as high risk factors.

In 2011 [START_REF] Costalat | Biomechanical wall properties of human intracranial aneurysms resected following surgical clipping[END_REF], we proposed an additional indicator base on the relationship between the mechanical properties of aneurysm wall tissues and the rupture risk. First conclusions suggested that the softer tissue is, the higher rupture risk is.

The present study aims to confirm these results, initiates with only 16 samples, studying 24 more aneurysms, and tries to go further investigating rupture limit values of aneurysmal tissues.

Materials and methods

Patients and clinical data

A total of sixty-three patients treated for ruptured or unruptured aneurysms by surgical clipping were recruited by four French neurosurgical teams. The research study protocol was approved by the local ethical committee in each center. A consent form was signed by patients with normal neurological status, or by the relatives in all other cases. For each patient, clinical, and radiological informations were collected concerning age, gender, aneurysm status (ruptured/unruptured), size (measured from the dome of the aneurysm and the neck represented by the communication of the aneurysm with the parent artery), the "dome to neck" ratio (aneurysm size/neck length), location on the Willis circle, morphological evaluation (classified "simple shaped" for a regular unilobulated aneurysm and "complex shaped" for a multilobulated and irregular aneurysm), as well as documented rupture risk factors; multiple aneurysms, previous ruptured aneurysm, positive family history of ruptured intracranial aneurysm, autosomal dominant polycystic kidney disease, hypertension, alcohol and tobacco. A possible mycotic intracranial aneurysm etiology was considered an exclusion criterion, as well as any previous history of endocarditis or inflammatory disease. All documented risk factors were then recorded in order to be related to the biomechanical behavior of each aneurysm.

Biomechanical testing methodology

Immediately after resection, each aneurysm wall was progressively frozen respecting a procedure described in [START_REF] Costalat | Biomechanical wall properties of human intracranial aneurysms resected following surgical clipping[END_REF]. The resected aneurysm was initially inserted in a tube containing a Ringer lactate, 10% DMSO solution. This first tube was then placed in a larger second one containing isopropanol. This combination of the two tubes was placed in a freezer (-80°C). The sample was progressively frozen due to the surrounding isopropanol solution in order to maintain its biomechanical properties.

One hour before mechanical testing, aneurysm samples were thawed at ambient temperature. Then a regular rectangular piece was dissected from the aneurysm preserving maximum length. This was a delicate point due to the small size of the samples. Many of them were lost during this cutting.

Once cut, a uniaxial stretch test was carried out on the sample within a warmed (37 °C) physiological liquid in order to simulate in vivo conditions. This testing device was composed of a Texture Analyzer (TA-XT2, Stable Microsystems, UK) with a 50 N load cell. Aneurysm strips were fixed by glue on each extremity to aluminum grips.

Two digital video cameras, placed in two orthogonal directions, recorded the mechanical tests (see Figure 1).

The uniaxial stretch tests consisted of a series of increasing load steps until rupture of the sample.

-For the first load step the sample was stretched by about 10% of its initial length, then unloaded and that was repeated five times. This 10% value was initially calculated by [START_REF] Karmonik | In-vivo quantification of wall motion in cerebral aneurysms from 2d cine phase contrast magnetic resonance images[END_REF]) in an in vivo wall motion MRI study of 7 aneurysms. In accordance to standard mechanical testing protocol for biological tissue, the specimens were first preconditioned [START_REF] Holzapfel | International Study of Unruptured Intracranial Aneurysms Investigators, Unruptured intracranial aneurysms--risk of rupture and risks of surgical intervention. International study of unruptured intracranial aneurysms investigators[END_REF] during the first four cycles. The extension rate was 0.01 mm/s and the tension load was recorded every 0.01 s. Velocity of the solicitations was small enough to not consider viscous phenomena. A baseline tension of 0 N was applied to the strip before starting each test.

-For the next load steps, the stretch was increased by 10% with the same number of loading/unloading cycles, under the same velocity conditions.

-Loads were increased until rupture of the sample occurred.

During the tests, the two subset cameras were used to record the displacement of the sample. These images (resolution of 4μm/pixel) allowed to determine the exact dimensions (width, length and thickness) of the strips. A Force/Displacement graph was obtained from each sample tested (see Figure 2).

Data post-processing

Mechanical parameter identifications

Only the measurements from the last elongation of load step 1 were used to obtain more realistic mechanical characterization of the data. Note, however, that except for the very first cycle, the force/displacement graph was roughly cycle independent. In order to identify the parameters of an equivalent hyperelastic model, the force/displacement graph described above was converted into a strain/stress graph. For this calculation, the length, thickness and width of each strip were considered. A baseline for this aneurysmal strip dimensions were obtained at 0-Newton traction in the third cycle of each test. Using the assumption that the specimen was subjected to a uniform traction, the engineering stress, noted Π, was computed, and the engineering strain, noted 𝜀, registered.

During the cycles, some permanent deformations in the traction phase were observed causing slight compressions of the sample in the rest phase. This is reflected by the negative values of the curve origin and is in accordance to the elasto-plastic behavior of the tissue. Since we consider a hyperelastic model to represent the tissue, the plastic effects cannot be represented and only the positive part of the curve was used to identify material behavior.

Once the strain/stress graph was obtained, we proceeded to a curve fitting using a Sequential Least Squares Programming algorithm in order to determine the corresponding hyperelastic model and its coefficients. In our cases, the best match was obtained with an incompressible Yeoh model with 2 parameters. Let F be the measured force, 𝑆 0 the initial section and 𝜆 = 1 + 𝜀 the elongation of the sample, the behavior law is given by the following equation:

Π = 𝐹 𝑆 0 = 2(𝜆 -𝜆 -2 )(𝐶 1 + 2𝐶 2 (𝜆 2 + 2𝜆 -1 -3))
where mechanical parameters 𝐶 1 and 𝐶 2 have to be identified.

Rupture limit identifications

Hemorrhage occurs when the aneurysmal wall ruptures, but this rupture is rather a local cracking than a total and entire tearing. In majority of the ruptures we observed in-vitro, the sample was not cut into two distinct pieces, but there was an abrupt loss of resistance. The specimens stayed in one piece held by undamaged fibers. Hence, we decided that the rupture limit would be given by the point of maximum stress.

An example of strain/stress curve, with visible rupture, is given in Figure 3. This figure represents the last load for the aneurysm of Figure 2, but expressed with stress and strain variables.

For each sample the limit of stress Π 𝑟 and elongation λ 𝑟 values were collected.

Results

We initially collected 63 specimens, but were able to test only 39. It was mainly because of the cutting of the samples due to their small size. The second cause of the losses is related to the positioning of the samples in the tensile machine. We broke a few cases during the procedure. All results are gathered in table 1. It summarizes clinical, anatomical and biomechanical data of the 39 cases tested (A tiny fraction of the information is missing).

Material parameters values

For this type of behavior law (Yeoh hyperelastic model), parameter 𝐶 1 is related to the Young modulus 𝐸, which is the slope at origin (𝜆 = 1) of the function Π(𝜆). It could be easily demonstrated that 𝐶 1 is equal to 𝐸/6. Parameter 𝐶 2 is representative of the curvature of the function Π(𝜆). The larger the parameter 𝐶 2 is, the higher rigidity of the material is.

After the identification procedure, we gathered the values of parameters 𝐶 1 and 𝐶 2 respectively in the graphs Figure 4 and Figure 5. These values are sorted according to the aneurysm status before surgery (ruptured (red points) or unruptured (green points)) and to the gender (female (circles) or male (squares)). On average, unruptured aneurysms are stiffer than ruptured ones. Indeed, the average value for 𝐶 1 is 0.07 MPa for ruptured and 0.12 MPa for unruptured aneurysms. The same trend is observed for parameter C2: the average value is 2.1 MPa for ruptured and 6.8 MPa for unruptured. But, the results show also that some unruptured aneurysms have soft material properties. For both parameters, there seems to be an upper limit for ruptured aneurysms: 0.2 MPa for 𝐶 1 and 10 MPa for 𝐶 2 .

Moreover, we noticed that all ruptured aneurysms have a 𝐶 2 lower than 3.7 MPa, except one which is equal to 7 MPa. Then we stratified aneurysms in 3 categories:

-Soft tissues are those having a 𝐶 2 lower than 3.7 MPa -Intermediates tissues are those having a 𝐶 2 between 3.7 MPa and 7 MPa -Stiff tissues are those having a 𝐶 2 greater than 7 MPa

Concerning the influence of the gender on material parameters, the graphs in Figure 4 and Figure 5 do not show a significant dependence. Aneurysmal tissues from males seem not to be stiffer than those of females and vice versa (p-value = 0.79 for 𝐶 1 and 0.39 for 𝐶 2 ).

Rupture limit values

For each specimen, the stress and stretch rupture limit have been identified and sorted by different criteria: gender, status (ruptured/unruptured) and rigidity (soft/intermediate/stiff). Results are presented in the graphs Figure 6 and Figure 7. Symbols and color codes are the same as in the previous paragraph. Neither correlation between the previous specific criteria and the stress was found, nor with the stretch. Stretch rupture limit does not depend on the gender (p-value = 0.22), nor on the status (p-value = 0.78). Stress rupture limit also does not depend on gender (p-value = 0.38). However, stretch and stress rupture limit seems not to be independent on the rigidity (respectively p-value = 0.006 and p-value = 0.04).

Furthermore, when reporting the engineering stress and the engineering strain (instead of the stretch) limit in a same graph (Figure 8) we clearly see that a notion of strain limit is more relevant than one of stress limit. The average value for the strain limit is about 0.2 (20% of deformation), with a standard deviation of 0.06 while the average value for the engineering stress limit Π 𝑟 is about 0.38 MPa, with a standard deviation of 0.24 MPa.

Limitations and discussion

Several points about experimental results and limitations must be discussed.

Result analysis

Almost certainly the physicians made the right choice in treating the unruptured aneurysms having a 𝐶 2 lower than 3.7 MPa (soft tissues). In a general manner, both soft and intermediate tissues had a risk of rupture, because we found a ruptured aneurysm with a 𝐶 2 equals to 7 MPa. At this time, only stiff tissues ( 𝐶 2 > 7 MPa) seem to be safe.

In terms of comparisons, we are able to compare parameter 𝐶 1 with those obtained in the literature. In [START_REF] Valencia | Mechanical test of human cerebral aneurysm specimens obtained from surgical clipping[END_REF], they extracted from a Mooney-Rivlin hyperelastic model, an average value of parameters 𝐶 01 + 𝐶 10 , equivalent to our 𝐶 1 , of about 0.17 MPa, while in [START_REF] Toth | Laboratory tests for strength paramaters of brain aneurysms[END_REF]) they found 0.12 MPa. They are of same order than our average value for 𝐶 1 : 0.11 MPa. Unfortunately, we cannot compare 𝐶 2 , which is however the most significant.

Beyond that, our results highlighted an average rupture limit for strain of about 20%. Already, in 1989 Steiger and co-authors [START_REF] Steiger | Strength, elasticity and viscoelastic properties of cerebral aneurysms[END_REF], presented rupture limit value for strain. Only 7 aneurysms were tested for an average strain rupture limit of 37%. More recently, works of [START_REF] Cebral | Wall Mechanical Properties and Hemodynamics of Unruptured Intracranial Aneurysms[END_REF] and [START_REF] Valencia | Mechanical test of human cerebral aneurysm specimens obtained from surgical clipping[END_REF], extracted also rupture limit values from their uniaxial traction tests. In both articles few samples were tested, due to the difficulty of getting aneurysms from surgery. Cebral and co-authors found an average value of 37% for a total of 8 unruptured aneurysms and compared the rupture limit values to those of control arteries in [START_REF] Robertson | Diversity in the Strength and Structure of Unruptured Cerebral Aneurysms[END_REF]. Valencia and co-authors, found an average value of 38% for 6 samples. The difference between our results and those of the literature may come from several reasons. First, the testing protocol is mainly different, because in our study, the sample is immersed in a physiological liquid controlled in temperature at 37 °C. The effect of the temperature is probably not negligible. Secondly, all our samples were initially frozen (as reported in [START_REF] Masson | Mechanical properties of arteries cryopreserved at -80 °C and -150 °C[END_REF]), while in the other studies, they were tested without freezing in a maximum of 48h after surgery. Finally, the total number of tested samples could also explain this difference. In a general manner, we think that in vivo rupture limit may be lower than 38%, because a simple mechanical analysis on a hemispherical model of aneurysm, using the Laplace theory and assuming standard material thickness and radius values, shows that strains are about 2 or 3 % in physiological conditions of pressure. Then, the pressure should be increased by almost a factor of 10 for reaching the values of measured rupture limits.

On the other hand, several authors [START_REF] García-Herrera | Mechanical behaviour and rupture of normal and pathological human ascending aortic wall[END_REF][START_REF] Pichamuthu | Differential tensile strength and collagen composition in ascending aortic aneurysms by aortic valve phenotype[END_REF][START_REF] Pham | Biomechanical characterization of ascending aortic aneurysm with concomitant bicuspid aortic valve and bovine aortic arch[END_REF][START_REF] Forsell | Biomechanical properties of the thoracic aneurysmal wall: differences between bicuspid aortic valve and tricuspid aortic valve patients[END_REF] have investigated the uniaxial rupture properties of ATAAs (Ascending Thoracic Aortic Aneurysms). Works of Duprey and co-authors [START_REF] Duprey | Biaxial rupture properties of ascending thoracic aortic aneurysms[END_REF] on ATAAs exhibit an average rupture limit value for strain of 69%, which is different than our. This mismatch could be explained because pathologies are really different. ATAAs mainly concern old persons with stiff arterial properties due to the atheromatous process, while cerebral aneurysms occur more often in younger people.

All these observations go in the same direction. For a given arterial pressure, a soft tissue has a higher risk of rupture than a stiff one, because it has a larger deformation which tends to come closer to the strain rupture limit (20%). Our study also confirms observations suggesting that big aneurysms are dangerous [START_REF] Wiebers | International Study of Unruptured Intracranial Aneurysms Investigators. Unruptured intracranial aneurysms: natural history, clinical outcome, and risks of surgical and endovascular treatment[END_REF]. Indeed, thanks to the Laplace theory, the larger the bending radius is, the higher the stress in the wall of the membrane (aneurysm wall) is. In big aneurysms, the strain would then be higher and closer to the rupture limit, than in a small aneurysm for a same arterial pressure. This phenomenon is worsened for soft tissues.

Finally, from a mechanical point of view, consideration of only the wall stiffness is not sufficient, the strain rupture limit must be taken into account too. Indeed, for the ruptured aneurysm having a 𝐶 2 equals to 7 MPa, we found a strain rupture limit equal to 10% (two times inferior of the average limit value). This aneurysm was the unique specimen from the posterior vertebrobasilar circulation we tested and it is well known that they are more dangerous at the same size than those located in other places [START_REF] Wiebers | International Study of Unruptured Intracranial Aneurysms Investigators. Unruptured intracranial aneurysms: natural history, clinical outcome, and risks of surgical and endovascular treatment[END_REF]. So, wall stiffness has to be considered in the light of the strain rupture limit. Moreover, with the same tissue (material properties) and at the same arterial pressure, the strain is higher if the thickness of the wall is smaller, which increases the risk of rupture.

Of course, all these values have to be considered carefully, even though we paid a lot of attention to measurements and in the control of results. Despite this, a security coefficient should be applied to these bounded values.

Limitations

Because of the surgical resection necessary to obtain these samples, only aneurysms easily and safely accessible were selected, introducing a potential selection bias. Hence, the MCA (Middle Cerebral Artery) location was over-represented in this study. There may be also a bias induced by the differences between cases with indication of surgical treatment compared to those indicated for interventional therapy.

Uniaxial strain/stress testing is not representative of the anisotropic behavior of the aneurysm wall in vivo.

Although, bi-axial testing was contemplated in this study, it is technically challenging to carry out. In our experiments the main limitation was the small size of the strips ranging from 1.3 to 8 mm. Work by [START_REF] Toth | Laboratory tests for strength paramaters of brain aneurysms[END_REF] also confirms that bi-axial testing does not generate reliable and reproducible results. MacDonald and coauthors [START_REF] Macdonald | Directional wall strength in saccular brain aneurysms from polarized light microscopy[END_REF] investigated the molecular strength of the collagen fibers layers in four aneurysms. When comparing directional tissue strength, an anisotropy was demonstrated by a factor of two. In our study, aneurysm samples were selected in a meridional direction (which means for us the greater length of the sample and not a direction of anisotropy) in both groups demonstrating a difference between ruptured and unruptured aneurysms in this direction. A bi-axial testing of the sample may have addressed this high level of anisotropy, this was not possible due to physical limitations as mentioned above.

Furthermore, the probability of consistently and systematically slicing the aneurysm sample in the weakest direction in the ruptured aneurysms, and the strongest direction in unruptured aneurysms was unlikely.

The taking of measures of the size of samples such as the thickness and the width, was difficult to carry out because they are relatively irregular. We took average values, which can skew the estimation of the stresses and explain a higher dispersal of the values compared with those of strains. This also has consequences for the estimation of parameters 𝐶 1 and 𝐶 2 of the hyperelastic model.

Because of the small size of samples, in several cases the hypothesis of pure traction was not insured (the proportion of the length over the width was too low). This is an additional factor of error to be considered.

Aneurysm strips were fixed by glue on each extremity to aluminum grips. This procedure is delicate. It allows to block the sample on the grips, but can sometimes damage it when the glue is badly arranged and diffused in the sample, which is very small.

Conclusion

With a total of 39 analyzed aneurysms, our study confirms that ruptured aneurysms mainly present a soft tissue and unruptured aneurysms present a stiff material. But, contrary to what we observed in the previous study [START_REF] Costalat | Biomechanical wall properties of human intracranial aneurysms resected following surgical clipping[END_REF], including only 16 samples, gender has no influence on material parameters. Aneurysmal tissues from males seem not to be stiffer than those of females and vice versa.

Secondly, the strain rupture limit is quite stable around 20 % and seems not to be correlated with the status of the aneurysm (initially ruptured or unruptured), neither with the gender of the patient. However, stretch and stress rupture limit seems not to be independent on the rigidity. The existence of a fixed rupture limit for strain confirms that a soft tissue has a higher risk of rupture than a stiff one. That also confirms the observations suggesting that big aneurysms are dangerous.

Finally, from a mechanical point of view, consideration of the wall stiffness alone is not sufficient, the strain rupture limit must be taken into account too.

All these experimental results have to be considered carefully, a security coefficient should be applied on the bounded values.

Further studies investigating the biomechanical properties of cerebral aneurysms should be led to validate these observations and conclusions.

Figure 1 :Figure 2 :Figure 4 :Figure 3 :Figure 6 :

 12436 Figure 1 : Tensile test device

Figure 7 :

 7 Figure 7 : Stress (PK1) rupture limit versus status (ruptured (red)/ unruptured (green)), gender (female (circle)/ male (square)) and rigidity (soft/intermediate/stiff) of the aneurysm

Table 1 :

 1 Summary of clinical, anatomical and biomechanical data of the 39 cases tested.

	(U=Unruptured, R=Ruptured, U-

Conflict of interest


Günce Keziban

Vincent Labatut
email: vlabatut@gsu.edu.tr

A Modification to Improve the Realism of Networks Generated with the LFR Model

Keywords: Complex Networks, Community Detection, Random Networks, Networks Generation, Networks Properties

Community detection consists in searching cohesive subgroups of nodes in complex networks. It has recently become one of the domain pivotal questions for scientists in many different fields where networks are used as modeling tools. Algorithms performing community detection are usually tested on real, but also on artificial networks, the former being costly and difficult to obtain. In this context, being able to generate networks with realistic properties is crucial for the reliability of algorithms testing. Recently, Lancichinetti et al.

[1] designed a method called LFR, able to produce realistic networks with a community structure, and power law distributed degrees and community sizes. However, other realistic properties such as degree correlation and transitivity are missing. In this work, we propose a modification of the LFR model, based on the preferential attachment model, in order to remedy this limitation. We analyze the properties of the generated networks and compare them to the original approach. We then apply different community detection algorithms and observe significant changes in their performances when compared to results on networks generated with the original approach.

Proposition d'une modification du modèle LFR pour un meilleur réalisme des réseaux générés

Résumé. La détection de communauté consiste à rechercher des sous-ensembles de noeuds densément connectés dans des réseaux complexes. Il s'agit d'une problématique centrale pour des chercheurs issus des nombreux domaines différents dans lesquels les réseaux complexes sont utilisés comme outil de modélisation. Les algorithmes de détection de communauté sont généralement testés à la fois sur des réseaux réels et artificiels. Les premiers sont plus difficiles et coûteux à obtenir, tandis que le niveau de réalisme des derniers a un effet direct sur la fiabilité des tests. Récemment, Lancichinetti et al. [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF] ont défini un modèle appelé LFR, qui permet de produire des réseaux possédant une structure de communauté, ainsi que des degrés et des tailles de communautés distribués selon une loi de puissance. Cependant, d'autres propriétés observées dans un grand nombre de réseaux réels sont manquantes, telles qu'une corrélation de degré non-nulle et une transitivité élevée. Dans cet article, nous proposons une modification du modèle LFR basée sur le modèle d'attachement préférentiel, afin de résoudre cette limitation. Nous analysons les propriétés des réseaux générés et les comparons à ceux obtenus avec la méthode originale. Nous appliquons ensuite différents algorithmes de détection de communauté à ces réseaux et observons des modifications significatives dans leurs performances, comparées à celles obtenus sur les réseaux issus de LFR.

Mots-clés:

Réseaux complexes, Détection de communauté, Réseaux aléatoires, Génération de réseaux, Propriétés des réseaux.

Introduction

Complex networks constitute a powerful modeling tool, able to represent most realworld systems. The objects composing the system are represented under the form of nodes while their interactions correspond to links. Thanks to this versatility, they became very popular during the last decades, attracting the attention of scientists from many different domains, and are now used to model and analyze complex systems in areas like physics, biology, social sciences or computer science [START_REF] Newman | The structure and function of complex networks[END_REF][START_REF] Fred | Robust Data Clustering[END_REF].

Among the various approaches used to study complex networks properties, community detection has become one of the most popular ones. A community in a complex network is a cohesive subset of nodes with denser inner links, relatively to the rest of the network [START_REF] Newman | Finding and evaluating community structure in networks[END_REF]. Because of the spread of complex networks, the community detection problem has been studied in many different areas too, resulting in tens of algorithms based on a whole range of principles: hierarchical clustering, optimization methods, graph partitioning, spectral properties of the network, etc. [START_REF] Fortunato | Community detection in graphs[END_REF] Until recently, those algorithms were tested on a few real and/or artificial networks. Limiting these tests to real networks can be considered as an issue for several reasons. First, building such networks is a costly and difficult task, and determining reference communities can only be done by experts. This leads to small networks, where actual communities are not always defined objectively, or even known. Second, a complex network is characterized by various properties like its average degree, degree distribution, shortest average path, etc. By definition, it is not possible to control these features in a real network. This means the algorithm is tested only on a very specific and limited set of features. Hopefully, artificial networks allow overcoming these limitations. First, by using some generative model, it is possible to automatically build many of them. Moreover, since they are generated, objective reference communities are defined. Finally, depending on the selected generative model, various properties can be directly controlled. The only point of concern with artificial networks is their level of realism. Indeed, for algorithm testing to be relevant, the generated networks must exhibit realistic properties. For this purpose, generative models are generally defined in order to mimic known real networks properties. Of course, current knowledge regarding these properties may not be exhaustive, and we can consequently never be completely assured the generated networks are perfectly realistic. For this reason, tests on artificial networks should be seen as complementary to tests on real networks.

A few methods have been designed to generate networks with a community structure. The most popular one is certainly the model by Girvan and Newman (GN) [START_REF] Girvan | Community structure in social and biological networks[END_REF]. First, an empty (i.e. without any link) network is created, then nodes are randomly assigned equal-sized virtual communities, and finally links are randomly drawn with probabilities and for intra-and inter-community links, respectively. This results in a set of interconnected Erdős-Rényi networks [START_REF] Erdos | On random graphs[END_REF]. Several extensions were derived to generate weighted or oriented links, and hierarchical or overlapping communities (see [START_REF] Fortunato | Community detection in graphs[END_REF] for more details). However, in this paper, we focus on non-oriented unweighted networks with non-overlapping communities, because almost all existing community detection algorithms are dedicated to this type of networks. Although widely used to test and compare these algorithms [START_REF] Girvan | Community structure in social and biological networks[END_REF][START_REF] Donetti | Detecting network communities: a new systematic and efficient algorithm[END_REF][START_REF] Duch | Community detection in complex networks using extremal optimization[END_REF][START_REF] Radicchi | Defining and identifying communities in networks[END_REF], the GN method is limited in terms of realism [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF]: the generated networks are rather small compared to most real world networks studied in the literature [START_REF] Da Fontura Costa | Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications[END_REF]; all nodes have roughly the same degree; and all communities have the same size. Yet, it is well known real networks exhibit power law or exponential degree distribution [START_REF] Newman | The structure and function of complex networks[END_REF][START_REF] Da Fontura Costa | Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications[END_REF], and community sizes generally follow a power law [START_REF] Da Fontura Costa | Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications[END_REF][START_REF] Guimerà | Self-similar community structure in a network of human interactions[END_REF]. To tackle this problem, several GN variants were defined, producing bigger networks, and communities with heterogeneous sizes [START_REF] Fortunato | Community detection in graphs[END_REF][START_REF] Danon | The effect of size heterogeneity on community identification in complex networks[END_REF][START_REF] Pons | Computing communities in large networks using random walks[END_REF]. More recently, a different approach appeared, based on rewiring [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF][START_REF] Bagrow | Evaluating local community methods in networks[END_REF]. First an initial network with desired properties (but no community structure) is randomly generated, then virtual communities are drawn, and finally some links are rewired so that these communities appear in the network, but without changing the existing degree sequence. The method described by Bagrow [START_REF] Bagrow | Evaluating local community methods in networks[END_REF] uses the Barabási-Albert [START_REF] Barabasi | Emergence of scaling in random networks[END_REF] model to generate the initial network, resulting in a power law degree distribution, but produces small networks with equal-sized communities. The method by Lancichinetti et al. (LFR) [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF] is based on the configuration model [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF], which generates networks with power law degree distribution, too. However, unlike Bagrow's method, LFR generates power law distributed community sizes, and the network size is not constrained. In other terms, LFR exhibits the most realistic properties among the presented generative approaches. Nevertheless, it also has some noticeable limitations regarding the low transitivity and close to zero degree correlation measured in the generated networks [START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF]. According to Newman [START_REF] Newman | The structure and function of complex networks[END_REF], real world networks usually have a clearly non-zero degree correlation, and their transitivity, or clustering coefficient, is relatively high.

Interestingly, improvement on the realistic aspect of the generated networks has a noticeable effect on most community detection algorithms. A performance drop was observed when authors switched from equal-sized communities to heterogeneous distributions [START_REF] Danon | The effect of size heterogeneity on community identification in complex networks[END_REF][START_REF] Pons | Computing communities in large networks using random walks[END_REF]. This allowed to show one algorithm may not perform on the same level depending on the size of the considered communities. The introduction of a power law degree distribution also made the benchmarks more discriminatory, allowing to highlight differences between algorithms whose performances were considered similar before [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF]. The fact Lancichinetti et al.'s method, which is the most realistic one to date, still has room for improvement naturally raises two questions, which we will try to answer in this work: 1) how is it possible to produce more realistic networks, and 2) will this have an effect on community detection algorithms. In section 2, we describe briefly the LFR method, its characteristics and the improvements we proposed. We also describe a few community detection algorithms, to be used to test the effect of network realism on community detection. In section 3, we present the properties of the networks generated with the modified method, and use them to compare the community detection algorithms performances. Finally, in section 4 we comment these results and propose some further improvements.

A Modification to Improve the Realism of Networks Generated with the LFR Model 4/16

2 Methods

LFR Generative Method

The LFR method was proposed by Lancichinetti et al. [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF] to randomly generate undirected and unweighted networks with mutually exclusive communities. Nodes degrees and community sizes are both distributed according to a power law. The method was subsequently extended to generate weighted and/or oriented networks, with possibly overlapping communities [START_REF] Lancichinetti | Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities[END_REF][START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF], but here we focus on the first version. This method allows to control directly the following parameters: number of nodes , desired average and maximum degrees, exponent for the degree distribution, exponent for the community size distribution, and mixing coefficient . The latter represents the desired average proportion of links between a node and nodes located outside its community, called inter-community links. Consequently, the proportion of intra-community links is -. It is generally not possible to meet this constraint exactly, and the mixing coefficient is therefore only approximated in practice. It is an important parameter, because it determines how clearly the communities are defined in terms of structure. For small values, the communities are distinctly separated, whereas for high values, the network has almost no community structure, making community identification a difficult task. Intuitively, one may think the community structure remains clear for , because this value corresponds to a network where, in average, nodes have the same number of inter-and intra-community links. In practice though, some algorithms are able to successfully detect communities even when approaches [START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF][START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF]. Let us note (resp.

) the ratio of existing to possible intra-(resp. inter-) community links, or in other terms: the probability to have a link between two nodes belonging to the same community (resp. different communities). Then the communities are welldefined when , which translates to , where and are the number of nodes in the network and in the biggest community, respectively [START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF].

The LFR method is three-stepped. First, it uses the configuration model [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF] to generate a network with average degree , maximum degree and power law degree distribution with exponent . Second, virtual communities are defined, so that their sizes follow a power law distribution with exponent . Third, an iterative process takes place to rewire certain links, so that is approximated, but without changing the degree distribution.

LFR Properties

By construction, the LFR method guaranties to obtain several realistic properties: size of the network, power law distributed degrees and community sizes. Moreover, some parameters give the user a direct control on these properties: network size ( ), A Modification to Improve the Realism of Networks Generated with the LFR Model 5/16 degree distribution ( , , ), community structure ( , ). For these reason, we call them controlled properties.

However, real world networks are known to exhibit additional properties. They have the small world property [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF], which states that for a fixed average degree, the average distance (i.e. the length of the shortest path) between pairs of nodes increases logarithmically (or slower) with the number of nodes [START_REF] Newman | The structure and function of complex networks[END_REF]. This property is important, because it is related to the network efficiency to propagate information. Another property of interest is related to the transitivity coefficient (also called clustering coefficient [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]), which assesses the density of triangles (three completely connected nodes) in the network. The higher the transitivity, the more probable it is to observe a link between two nodes both connected to a third one. A real network is supposed to have a higher transitivity than a random Erdős-Rényi network [START_REF] Erdos | On random graphs[END_REF] possessing the same number of nodes and links, by a factor of order [START_REF] Newman | The structure and function of complex networks[END_REF]. Finally, degree correlation constitutes another interesting property, by describing how a node degree is related to its neighbors. Real networks usually show a non-zero degree correlation. If it is positive (resp. negative), the network is said to be assortatively (resp. 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. Degree correlation is related to the concepts of hubs and authorities, which are specific nodes with central positions.

In the context of this work, the purpose of network generation is to compare algorithms which completely rely on the networks structure to identify communities. These three additional properties are hence particularly important, in the sense they are all related to the network structure. The LFR method does not allow controlling them directly, but these uncontrolled properties were analyzed on a wide range of parameters values [START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF]. At this occasion, it was shown LFR generates small world networks, with relatively high transitivity and degree correlation, but only under certain circumstances. Indeed, uncontrolled properties are affected by changes in certain parameters, and this is especially true for the mixing coefficient. All three properties exhibit realistic values when is almost zero, but when it gets closer to , one can observe strong decreases, resulting in unrealistic transitivity and degree correlation values. In all studies evaluating community detection algorithms, performances are assessed in function of some index representing the communities level of separation [1, 5, 6, 8-10, 13-15, 18]. In the case of the LFR method, this index is , so its influence on some structural property can be a serious limitation. Indeed, when comparing some algorithm performances on networks generated with different , one generally considers the only difference between the networks is how much communities are separated. But also affects other properties, possibly influencing the observed performance.

Proposed Modifications

One of the possible causes for the observed unrealistic properties is the use of the configuration model (CM) [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF] to generate the initial network during the LFR first A Modification to Improve the Realism of Networks Generated with the LFR Model 6/16

step. On the one hand, the CM is very flexible in the sense it is able to produce networks with any size and degree distribution, but on the other hand it is known these networks have zero correlation [START_REF] Serrano | Weighted Configuration Model[END_REF] and low transitivity (when degrees are power law distributed) [START_REF] Newman | The structure and function of complex networks[END_REF]. We propose to use a different generative model, with more realistic properties. We considered the Barabási-Albert preferential attachment model [START_REF] Barabasi | Statistical mechanics of complex networks[END_REF] and one of its variants called evolutionary preferential attachment [START_REF] Poncela | Complex Cooperative Networks from Evolutionary Preferential Attachment[END_REF]. The rest of the method is not modified: community sizes are still drawn from a power law distribution, and the rewiring process must be applied to make the community structure appear. The Barabási-Albert preferential attachment model (BA) [START_REF] Barabasi | Emergence of scaling in random networks[END_REF] was designed as an attempt to explain the power law degree distribution observed in real networks by the building process of these networks. Starting from an initial network containing connected nodes, a realistic iterative process is applied to simulate growth. At each iteration, one node is added to the network, and is randomly connected to m existing nodes (

). These nodes are selected with a probability which is a function of their current degree :

. In other terms: the higher a node degree, the higher its chances of being selected. This so-called preferential attachment mechanism results in a power law degree distribution, since degree increases faster for nodes with higher degree, as new nodes are added to the network. The exponent cannot be controlled though, and tends towards [START_REF] Barabasi | Emergence of scaling in random networks[END_REF]. The average distance is always less than in same-sized Erdős-Rényi networks, so it has the small world property [START_REF] Barabasi | Statistical mechanics of complex networks[END_REF]. The average degree depends directly on the parameter: [START_REF] Newman | The structure and function of complex networks[END_REF]. Transitivity is greater than in Erdős-Rényi networks, but nevertheless decreases with network size following a power law [START_REF] Barabasi | Statistical mechanics of complex networks[END_REF]. The evolutionary preferential attachment (EV) [START_REF] Poncela | Complex Cooperative Networks from Evolutionary Preferential Attachment[END_REF] model is a variant of the BA model. It also uses the preferential attachment and growth mechanisms, except the attachment probabilities are not based on some topological properties, like the current degree in the case of BA, but on some nodal dynamic property, updated using the prisoner's dilemma game. Every few iterations, each node plays either cooperation or defection against all its neighbors. It gets a total score depending on the individual results:

for unilateral cooperation or bilateral defection, for bilateral cooperation, and for unilateral defection, with . The first move is randomly chosen, whereas the next one depends on the respective results of the considered node and a randomly picked neighbor. If the neighbor's score is better, the node might switch to its strategy, with a probability depending on the difference between their scores. Nodes with higher scores are more attractive to a node added to the network, because by being connected to them, it may use a strategy which proved to be successful. According to its authors, this process is more realistic and leads to networks with high transitivity and degree correlation. Besides the parameters already needed by BA ( , and ), EV uses (points scored for unilateral cooperation) and ε (selection pressure). The latter allows to modulate the influence of the preferential attachment mechanism: all nodes are equiprobable when , whereas the nodes scores are fully considered for .

Community Detection Algorithms

To study the effects of the networks realism on the community detection process, we applied four popular algorithms: Newman et al.'s Fast Greedy algorithm (FG) [START_REF] Clauset | Finding community structure in very large networks[END_REF] relies on a modularity-based agglomerative hierarchical approach. Its name is due to the use of a standard greedy method, making it relatively faster than earlier algorithms, and allowing it to process large networks. Pons and Latapy's Walktrap algorithm (WT) [START_REF] Pons | Computing communities in large networks using random walks[END_REF] follows another agglomerative hierarchical method, in which the distance between two nodes is defined in terms of random walk processes.

Raghavan et al's Label Propagation algorithm (LA) [START_REF] Raghavan | Near linear time algorithm to detect community structures in large-scale networks[END_REF] analyzes information diffusion to identify communities. Each node is initially labeled with a unique value.

Then, an iterative process takes place, where each node takes the label which is the most spread in its neighborhood. When the process ends, communities correspond to sets of nodes with identical labels. Blondel et al.'s Louvain algorithm (LV) [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] is the most recent of the considered algorithms. It relies on a two-stepped hierarchical modularity optimization method. The first step consists in detecting small communities by performing a greedy optimization on modularity. In the second step, the algorithm creates a network whose nodes represent the communities identified during at step one. These two steps are repeated to build the complete hierarchy of communities.

Results and Discussion

Generated Networks Properties

The networks were generated by applying first one of the three previously presented methods (CM, BA, EV) to produce initial networks, and then using the LFR approach to generate the communities sizes and perform rewiring. In other terms, the generating processes differ only in their first step. For simplicity matters, we will thereafter refer to them by using only the name of the model employed during their first step. Consequently, CM will correspond to the original LFR method, whereas BA and EV are modified versions based on the corresponding model. We selected our parameters values based on previous experiments in artificial networks generation [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF][START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF] and descriptions of real world networks measurement from the literature [START_REF] Newman | The structure and function of complex networks[END_REF][START_REF] Da Fontura Costa | Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications[END_REF][START_REF] Barabasi | Statistical mechanics of complex networks[END_REF][START_REF] Boccaletti | Complex networks: structure and dynamics[END_REF]. Some parameters are common to all three processes: we fixed the size and the power law exponent for the community sizes distribution ; and made the mixing coefficient range from to with a step. Other parameters are model-dependent. In particular, with the original LFR method based on CM, it is possible to specify the desired power law exponent for the degree distribution, and average and maximal degrees . We used the values , and . The alternative models do not allow as much control as the CM, and we had to adjust their A Modification to Improve the Realism of Networks Generated with the LFR Model 8/16 parameters so that the resulting networks had approximately the same degreerelated properties. Preferential attachment does not give any control on , which tends towards by construction. To control the average degree, we used for both BA and EV. The maximal degree is not controlled, but the values measured in the resulting networks are of the same order than the values specified for CM. EV additionally allows controlling transitivity, and we found out score and selection pressure gave the best results.

We produced networks for each combination of parameters, and averaged the measured properties. Figure 1 shows the results for average distance, degree correlation and transitivity. Results were very similar for and , so we only present the latter here, but comments apply to both. The largest communities in the generated networks have around nodes, so communities are supposed to be structurally well-defined (cf. section 2) for . This mixing limit is represented on the plots under the form of a vertical line.

The average distance is rather similar for all three models, both in terms of absolute value and sensitivity to . It ranges approximately from to , and is relatively stable, especially for . On the one hand, the stability of this property is a good point, since it means networks with much separated communities (small ) and networks with very mixed communities (high ) have comparable average distances. Consequently, the effect of this property can be considered as negligible when comparing algorithms performances on networks generated with various values. But on the other hand, since all three models lead to very close average distances, this property cannot be used to compare them in terms of realism of the generated networks.

CM has the highest transitivity, with values around (the theoretical minimum and maximum being and , respectively) for , but it also has almost zero transitivity for , exhibiting a serious sensitiveness to . Other methods also show a decreasing transitivity when increases, but the range is much smaller, mainly because their values for are significantly smaller: around and for BA and EV, respectively. Like CM, they reach almost zero value when . So contrarily to what we expected, networks generated with EV do not have a higher transitivity than CM, at least for small . However, thanks to its lesser sensitivity to , EV has a better transitivity for . Note that in the literature, real world networks with a transitivity greater than are considered highly transitive [START_REF] Da Fontura Costa | Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications[END_REF], so we can state all three models exhibit realistic transitivity for small . The issue is more about their sensitivity to , leading to non realistic values for high . This nonlinear decrease in transitivity observed for all three models could be linked to the rewiring process performed by the LFR method. In this case, the final transitivity would never be stable, whichever model is used to generate the initial networks. But testing this hypothesis would require an exhaustive analysis of the side-effects of rewiring on networks, which is out of the scope of this work. Another explanation would be that network transitivity is directly related to the nature of community structure itself, independently of the way the network is created. Testing this hypothesis would require being able to quantify the separation level of communities in real world networks (using Newman's modularity [START_REF] Newman | Finding and evaluating community structure in networks[END_REF], for instance), in order to compare it to the measured transitivity. The authors are not aware of any work of this kind, which is also out of this paper scope.

Considering the degree correlation, there is a clear difference between CM and the other two models. CM degree correlation has acceptable values for small ( ), but it decreases rapidly and oscillates around zero for . EV shows the highest degree correlation, with values greater than for . It also decreases when increases, resulting in values close to for . Finally, unlike other models, BA degree correlation slightly increases with , ranging approximately from ( ) to ( ). Although its values are lower than for EV, it is also A Modification to Improve the Realism of Networks Generated with the LFR Model 10/16 more stable both in terms of sensitivity to and low standard-deviation (especially for ). In conclusion to this section, we can state EV and BA are slightly above CM in terms of realism. All three of them have extremely similar results on the average distance. They have significantly different transitivity, but all three are realistic, at least for small values. Concerning the degree correlation, both BA and EV exhibit realistic values for any , whereas CM is realistic only for . The main difference between the reviewed generative models is related to their sensitivity to . CM is clearly the most sensitive, showing the largest range of values for both transitivity and degree correlation, whereas BA is the most stable. However, EV generally has highest values than BA, so it is difficult to decide which one is the most adapted. The next subsection will be dedicated to study how these differences in stability and realism translate in terms of community detection performances.

Community Detection Performances

We applied the four community detection algorithms presented in section 2 on all the networks we generated. These algorithms do not need the user to specify any parameter, except for WT, for which we used the default value to define the length of the random walks. We compared the performances using the normalized mutual information (NMI). This measure was defined in the context of conventional clustering to compare two different partitions of one data set [START_REF] Fred | Robust Data Clustering[END_REF][START_REF] Kuncheva | Using diversity in cluster ensembles[END_REF] and was later used to assess community detection performance [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF][START_REF] Lancichinetti | Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities[END_REF][START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF][START_REF] Danon | Comparing community structure identification[END_REF].

Figure 2 shows the results for all four algorithms, in function of the mixing coefficient . Although we applied the algorithms on networks with average degree and , there was no relevant difference between the results: the performances were uniformly slightly better for than for . Consequently, our plots show only the former. Generally, as expected from previous studies [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF][START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF][START_REF] Lancichinetti | Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities[END_REF][START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF], the accuracy of all algorithms decreases along increases, i.e. communities become more mixed and difficult to distinguish. When , all algorithms manage to successfully identify communities, whereas when the mixing limit of is reached, they all perform badly. The way the performance evolves in function of depends on the algorithm, though. It is almost linear for FG, which has poor performances even for values of far from the mixing limit. For the other algorithms, the performance stays close to the maximum until some individual limit is reached, at which point a sudden drop occurs. This individual limit is very close to for LV and WT, whereas it is around for LP. The main differences between LV and WT are the former's performance slightly decreases before suddenly dropping off, whereas the latter's stays maximal; and LV performance are bellow WT's when . So a clear hierarchy appears between algorithms, in terms of general accuracy: FG<LP<LV<WT.

The effect of the generative model on community detection performance depends strongly on the considered algorithm. FG does not seem to be sensitive at all, since its performances for all three models are not significantly different. This suggests the information it uses to identify communities is not related at all to transitivity nor degree correlation. FG is essentially based on a modularity optimization approach, so on the one hand, this raises a question regarding the sensitivity of modularity to these properties. On the other hand, FG is not the best algorithm for modularity optimization, plus LV, which is also modularity-based, shows signs of sensitivity to the model. More efficient modularity optimization approaches such as Spinglass [START_REF] Reichardt | Statistical mechanics of community detection[END_REF] could be applied on the same networks to verify this hypothesis (the authors could not achieve this task by lack of time, Spinglass being extremely long compared to other algorithms, especially those presented here). LP, which is not modularity-based, is much more sensitive to the generative model. EV and BA have close low drop-off limits, around and , whereas it is approximately for CM. However, note these values may not precisely represent the actual performance, due to the high variance observed in LP results. LP performance is far better for CM than for the other models, which could suggest it finds more realistic networks harder to process. More precisely, the way models are ordered in terms of performance is the exact opposite of their order in terms of degree correlation. This could mean LP does not handle well networks with positive degree correlation, maybe because such a property modifies the way labels spread in the network. But this observation is difficult to confirm here. Indeed, for BA has a higher correlation degree than EV, but the effect on performance cannot be discussed because NMI has already reached for these values of .

As stated before, LV is modularity-based but, unlike FG, it performs differently depending on the model. WT does not rely on modularity to identify communities, but it is generally used as a criterion to select the best partition (or cut) in the hierarchy it outputs (called dendrogram). Both algorithms do not show any modelsensitiveness until they reach their drop off limit. Then performances are clearly better for CM and EV than for BA. In the case of WT, CM leads to even higher performances than EV on the range . This order fits with the models transitivity, so we could assume LV performs better when this property is high enough. However, EV transitivity is higher than CM's for and this does not appear at all in the performance plot. On the contrary, the performance for CM stays above the other models until , whereas its transitivity is roughly the same. The compared algorithms use different principles and mechanisms to identify communities, which can explain why their performances are influenced in various ways by the studied generative models. However, if we do not take FG into account, it generally appears the Barabási-Albert model is the most difficult to process, whereas the configuration model is associated to the highest results. The evolutionary preferential attachment model lies somewhere in between (LV), sometimes closer to the former (LP) and sometimes closer to the latter (WT). It remains difficult to explain exactly why in general we observed differences in performances, because the data seem to be incomplete for that matter. Drawing more solid conclusions would necessitate first applying other algorithms, if possible using a wide range of principles to detect communities. Additionally, one could consider studying other network properties, possibly responsible for the changes in performance. However, for now, we would say results measured on the BA-based LFR method are the more reliable, because of the stability of generated networks properties to changes in the mixing coefficient .

Conclusion

In this paper, we proposed an improvement for the LFR method designed by Lancichinetti et al. [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF] to randomly produce realistic complex networks with community structure. LFR uses the configuration model (CM) [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF] to randomly generate a network with power law distributed degree, and then rewires it partially to make a community structure appear. An important parameter called mixing coefficient allows controlling the level of separation between communities. This type of method is used to benchmark community detection algorithms on huge collections of artificial networks, before testing them on real world ones [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF][START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF][START_REF] Lancichinetti | Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities[END_REF]. For this reason, their realistic aspect is of utmost importance and is known to affect the algorithms performances [START_REF] Danon | The effect of size heterogeneity on community identification in complex networks[END_REF]. To our knowledge, LFR is currently the method generating the most realistic networks, but it has some limitations regarding certain properties [START_REF] Orman | A Comparison of Community Detection Algorithms on Artificial Networks[END_REF]: degree correlation and transitivity (a.k.a. clustering) are not realistic and are not stable to changes in the mixing coefficient. Our improvement consists in replacing the configuration model by the Barabási-Albert (BA) [START_REF] Barabasi | Emergence of scaling in random networks[END_REF] and evolutionary preferential attachment (EV) [START_REF] Poncela | Complex Cooperative Networks from Evolutionary Preferential Attachment[END_REF] models, which are known to produce more realistic networks regarding these properties. We generated several collections of networks with the LFR method, based successively on CM, BA and EV. We used realistic values for all the directly controlled properties (number of nodes, average degree and power law exponent) and compared the values of the uncontrolled ones (average distance, transitivity and degree correlation). We found out our improvement allows producing networks with comparable average distance, more realistic and stable degree correlation and more stable transitivity, compared to the original LFR method (using CM). For these properties, EV exhibits better absolute values but BA is more stable.

In order to study the effect of our modification on the community detection process, we applied four different algorithms on the generated collections: Fast Greedy (FG) [START_REF] Clauset | Finding community structure in very large networks[END_REF], Label Propagation (LP) [START_REF] Raghavan | Near linear time algorithm to detect community structures in large-scale networks[END_REF], Louvain (LV) [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] and Walktrap (WT) [START_REF] Pons | Computing communities in large networks using random walks[END_REF]. We assessed their performances thanks to the normalized mutual information measure, already used for the same purpose before [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF][START_REF] Lancichinetti | Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities[END_REF][START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF][START_REF] Danon | Comparing community structure identification[END_REF]. For all algorithms and on all networks we observed the usual decrease in performance caused when increasing the mixing coefficient [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF][START_REF] Lancichinetti | Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities[END_REF][START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF]. Moreover, three algorithms out of four showed significant changes in their performances depending on the considered generative model (CM, BA or EV). FG is not sensitive at all, but has the poorest results and seems a bit out of date compared to more recent algorithms. Globally, for the three sensitive algorithms, the highest performances are obtained when applied to CM, whereas the lowest correspond to BA. The results on EV networks depend on the considered algorithm: close to BA for LP, close to CM for WT, and somewhere in between for LV. We could not determine if the observed changes in performance were due to some property in particular, though. BA seems to be the most interesting model in terms of discrimination of the community detection algorithms, because its stability to changes in the mixing coefficient allows to consistently compare performances for different levels of separation of the communities.

Our goal was to improve the realism of the networks generated by the LFR method, and from this point of view the modifications we proposed were efficient. But they also resulted in a loss of control, since the replacement models (BA and EV) do not allow to specify directly as many properties as CM. For instance, in CM the exponent of the degree power law distribution is a parameter, whereas BA can generate only networks with an exponent (which, hopefully, is an extremely realistic value). We suppose it is one of the reasons why Lancichinetti et al. chose to base their approach on this model in the first place. Moreover, the improvement was not as strong as expected, especially concerning the transitivity, which is still very sensitive to changes in the mixing coefficient. Different ways can be explored to try to solve these limitations. First, it would be interesting to study the side effects of the rewiring process used in the LFR approach, by simply comparing the generated networks properties before and after the wiring step. This work is necessary to know if some properties observed in the final networks depend on the initial (pre-rewiring) network or on the rewiring process itself. Second, many other models exist to generate networks with power law distributed degree [START_REF] Chen | A highly clustered scale-free network evolved by random walking[END_REF][START_REF] Chuang | Prisoner's Dilemma Game on Clustered Scale-Free Networks under Different Initial Distributions[END_REF][START_REF] Dangalchev | Generation models for scale-free networks[END_REF][START_REF] Holme | Growing scale-free networks with tunable clustering[END_REF][START_REF] Klemm | Growing scale-free networks with small-world behavior[END_REF][START_REF] Saramäki | Scale-free networks generated by random walkers[END_REF][START_REF] Shi | A Simple Model of Scale-free Networks Driven by Both Randomness and Adaptability[END_REF][START_REF] Tam | Construction of Scale-Free Networks with Adjustable Clustering[END_REF]. A systematic review could allow detecting more flexible models, offering more control on the generated networks properties, and more realistic properties. It is a long and difficult task though, because source codes are rarely easily available.

Concerning the effect of realism on community detection algorithms, our work can be extended in two ways. First, other algorithms could be applied to networks generated with the modified LFR method. Indeed, the four algorithms we compared exhibit rather different reactions to the models we used, making it difficult to infer general remarks concerning their effects on community detection performance. For example, we observed FG, which is a modularity-based algorithm, was not sensitive to the selected generative model, whereas LV, which optimizes modularity too, is. By considering algorithms such as Spinglass [START_REF] Reichardt | Statistical mechanics of community detection[END_REF], which is known to be a good estimator of maximal modularity, we could study the influence of the generative model on modularity. Second, we could consider other properties to characterize networks. Maybe we did not find any strong relationships between the generated networks properties and the performance changes because we did not focus on the relevant properties. We chose the most widely used ones in the context of real networks analysis, but many others exist [START_REF] Da Fontura Costa | Characterization of complex networks: A survey of measurements[END_REF][START_REF] Milo | Network motifs: simple building blocks of complex networks[END_REF], even if they have not been so popular up to now.

Figure 1 .A

 1 Figure 1. Influence of the mixing coefficient on the measured properties: (a) average distance, (b) degree correlation and (c) transitivity. Networks were generated with parameters , , ,and using the LFR method on three different generative models: configuration model (CM), Barabási-Albert model (BA) and evolutionary preferential attachment model (EV). Each point corresponds to an average over generated networks. The vertical lines at represent the average limit above which communities stop being clearly defined.

Figure 2 .A

 2 Figure 2. Community detection performances in function of the mixing coefficient , for (a) Fast Greedy, (b) Label Propagation, (c) Louvain and (d) Walktrap algorithms. The networks are the same than in Fig.1 ( , , and). Each point corresponds to an average over processed networks. The vertical lines at represent the average limit above which communities stop being clearly defined. Performances are expressed in terms of normalized mutual information.