Hailong Wu
email: hailong.wu@utbm.fr

Daniel Depernet
email: daniel.depernet@utbm.fr

Vincent Lanfranchi
email: vincent.lanfranchi@utc.fr

Khadija El Kadri

Mohd Azri

Hizami Rasid

K E K Benkara

M A H Rasid

A novel and simple torque ripple minimization method of synchronous reluctance machine based on torque function method

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

A Novel and Simple Torque Ripple Minimization

Method of Synchronous Reluctance Machine Based on Torque Function Method

Hailong Wu , Daniel Depernet, Vincent Lanfranchi , Khadija Ei Kadri Benkara , and M. A. H. Rasid

Abstract-Synchronous reluctance machine (SynRM) has been studied widely in order to reduce its high torque ripple and improve its low power factor. This article proposes a torque ripple reduction method for SynRM by defining a novel parameter-torque function. With the help of torque function, the optimal currents aiming to compensate torque harmonics are obtained. The advantage of the proposed method is that torque function can be obtained easily and torque ripple reduction can be achieved by a simple control method. For the purpose of verifying the proposed method, both simulations and experiments are performed. Finally, the influence of power supply and parameter uncertainty on the proposed method has also been discussed.

Index Terms-Optimal current, synchronous reluctance machine (SynRM), torque function, torque ripple.

I. INTRODUCTION

T HE synchronous reluctance machine (SynRM) has been studied widely since the last decade because of its simple and robust structure and high speed operation [START_REF] Zhang | Overmodulation of constant-switchingfrequency-based DTC for reluctance synchronous motors incorporating field-weakening operation[END_REF]- [START_REF] Arafat | Optimal phase advance under fault-tolerant control of a five-phase permanent magnet assisted synchronous reluctance motor[END_REF]. It can be applied to various fields, such as traction drive [START_REF] Bianchi | Electric vehicle traction based on synchronous reluctance motors[END_REF]- [START_REF] Trancho | PM-assisted synchronous reluctance machine flux weakening control for EV and HEV applications[END_REF], automotive accessories [START_REF] Rasid | Preliminary thermal evaluation of actuator for Steerby-Wire vehicle[END_REF], household [START_REF] Payza | Investigation of losses for a concentrated winding high-speed permanent magnet-assisted synchronous reluctance motor for washing machine application[END_REF], pumps, and fans [START_REF] Palmieri | Design of a 50.000-r/min synchronous reluctance machine for an aeronautic diesel engine compressor[END_REF]. However, its salient rotor topology produces high torque ripple which restricts the application of SynRM.

In order to minimize the high torque ripple of SynRM, different methods have been put forward. The mostly applied approach is to optimize its structure such as flux barriers [START_REF] Moghaddam | Design optimization of transversely laminated synchronous reluctance machine for flywheel energy storage system using response surface methodology[END_REF]- [START_REF] Yan | Design and analysis for torque ripple reduction in synchronous reluctance machine[END_REF], rotor rib [START_REF] Babetto | Design methodology for highspeed synchronous reluctance machines[END_REF], [START_REF] Donaghy-Spargo | Electromagnetic-mechanical design of synchronous reluctance rotors with fine features[END_REF], rotor skewing [START_REF] Bianchi | Sensitivity analysis of torque ripple reduction of synchronous reluctance and interior PM motors[END_REF], [START_REF] Bonthu | Optimal torque ripple reduction technique for outer rotor permanent magnet synchronous reluctance motors[END_REF], and adding

The second idea to reduce torque ripple of SynRM is based on adding the selected current harmonics to the original sinusoidal stator currents. Yammine [START_REF] Yammine | Contribution to the synchronous reluctance machine performance improvement by design optimization and current harmonics injection[END_REF] analyzed the average torque of a two phase SynRM and defined the optimal current with several stator inductance harmonics. Decreasing each torque harmonic requires several current harmonics. So if several dominant torque harmonics are considered, the process of defining the relationship between each torque harmonic and the corresponding current harmonics could be time-consuming and complex. Besides, it is very difficult to define the optimal currents for a multiphase SynRM by the method presented in [START_REF] Yammine | Contribution to the synchronous reluctance machine performance improvement by design optimization and current harmonics injection[END_REF]. Truong et al. [START_REF] Truong | Torque ripple minimization in non-sinusoidal synchronous reluctance motors based on artificial neural networks[END_REF] measured stator inductances and their low order harmonics to calculate the optimal currents by electromagnetic torque equation. But, the accurate measurement of high stator inductance harmonics is very difficult. It means the optimal currents obtained by the measured inductances may not result in the optimal torque ripple reduction.

In this article, a novel parameter, torque function, is proposed for the purpose of decreasing torque ripple of SynRM easily by control method. With the help of torque function, all torque harmonics could be reduced theoretically by the calculated optimal currents. Moreover, torque function can be obtained easily by the finite-element method (FEM).

In addition, both simulations and experiments are performed and presented to validate the torque function and the torque ripple reduction method. The principles of analyzing the results are also put forward. During the experiment, the sensibility of measurement of the torque harmonics depends on the inertia of the load, which makes it difficult to determine the amplitude of harmonics. Hence, when the SynRM is controlled without load, a method is proposed to obtain torque harmonics by measuring rotor position.

A SynRM designed for an e-Clutch is manufactured and studied in this article. First, the manufacturing constraints limit the choice of rotor topology for this application. Besides, the minimum torque produced by the maximum available current is the first consideration for the studied e-Clutch. Compared with massive rotor and flux barriers rotor, the salient pole rotor topology provides better result [START_REF] Doc | Contribution à la conception et au dimensionnement d'un actionneur d'embrayage[END_REF]. Hence, the salient pole rotor is selected.

II. DEFINITION AND CALCULATION OF TORQUE FUNCTION

A. Electromagnetic Torque Equation

The electromagnetic torque of SynRM could be calculated by the co-energy W (I,θ m )

T e = ∂W (I,θ m ) ∂θ m = P 2 3 i=1 3 j=1 I i I j ∂L ij (θ) ∂θ (1)
where θ m is the mechanical rotor position and θ is the electrical position; P is the number of pole pairs; I is the stator current; i and j are the phase number of stator windings; L ij is the stator self-inductance or mutual inductance.

The stator currents which can be represented by the currents of dq-axis and the stator inductances L ij (θ) are introduced in the Appendix. Substituting the stator currents and stator inductances into (1), the torque (A3) containing all torque harmonics can be obtained as presented in the Appendix. More details about the derivation can be found in [START_REF] Wu | Analysis of torque ripple reduction in a segmented-rotor synchronous reluctance machine by optimal currents[END_REF].

Furthermore, the SynRM can be controlled for different objectives by defining different current angles β. It is the angle between the current vector and d-axis current. For example, when current angle β is 45°, the maximum torque per ampere (MTPA) strategy is achieved in linear condition. As a result, a parameter k dq can be defined to represent different current angles

k dq = I q I d =tanβ. (2)
Substituting (2) into the torque (A3) in the appendix and separating the current and the inductance harmonics, the torque function K t (θ, β, I) can be defined and calculated. It is introduced by (A4) in the Appendix.

Because the stator inductances contain many harmonics which are functions of rotor position and torque function contains stator inductances, torque function becomes also a function of rotor position. Besides, the function of current angle (2) is included in (A4), thus torque function is also a function of current angle to achieve different control objectives. When the magnetic saturation phenomenon appears in the SynRM, the stator inductances could be influenced. The saturation is linked to the stator current. Therefore, the stator inductances and torque function are also functions of stator current.

Based on torque function, the electromagnetic torque of SynRM can be represented

T e = K t (θ, β, I) I 2 ds23 (3)
where I ds23 is the current of d-axis obtained by the invariant Park transformation according to the stator currents.

B. Torque Ripple Reduction Method

The equation of torque function needs all stator inductance harmonics. It is obvious that based on stator inductances, the calculation of torque function is not convenient. According to (3), torque function can be calculated easily

K t (θ, β, I)= T e I 2 ds23
.

(4) For example, if the sinusoidal stator current is supplied to the finite element model of the SynRM, the electromagnetic torque can be obtained. The corresponding current of d-axis can be calculated. Consequently, the torque function could be calculated conveniently by [START_REF] Bianchi | Electric vehicle traction based on synchronous reluctance motors[END_REF].

The torque ripple reduction method is illustrated by Fig. 1 [START_REF] Wu | Comparison of torque ripple reductions and copper losses of three synchronous reluctance machines[END_REF]. It is introduced for a salient pole SynRM in Fig. 1, but this method is also applicable for other rotor topologies of SynRM. In this article, saturation is not considered. So torque function K t (θ, β, I) becomes K t (θ, β). At first, the sinusoidal stator currents are supplied into the machine in order to calculate torque function. Then, the optimal currents in dq-axis can be calculated by the constant reference torque and torque function. In addition, the optimal stator currents can be obtained by the inverse Park transformation. Finally, the torque ripple could be minimized by supplying the optimal stator currents into SynRM.

The torque ripple minimization approach without saturation and with saturation has been analyzed and verified by FEM in [START_REF] Wu | Analysis of torque ripple reduction in a segmented-rotor synchronous reluctance machine by optimal currents[END_REF]. When there is no saturation, the original currents and the corresponding optimal currents are introduced in Fig. 2.I t can be observed that the stator currents without compensation are sinusoidal while the optimal stator currents contain many harmonics. The current of d-axis without compensation is constant while the optimal currents of dq-axis are not constant. The uncompensated torque and compensated torque are presented in Fig. 3. It can be observed that the torque ripple of the studied SynRM is minimized greatly.

III. SIMULATION

Torque control simulation, as presented in Fig. 4, is first performed in Simulink in order to test the proposed torque ripple reduction method. All stator inductance harmonics are obtained by finite element method in order to build the model of SynRM which can depict its torque ripple.

The reference peak value of stator current is calculated by the reference torque. When torque ripple is not compensated, it is calculated by the mostly used average torque equation

I * m = 4T ref 3P (L d -L q )sin2β (5)
where L d and L q are the average values of dq-inductances respectively and they are given in Table I. The reference current with compensation is calculated based on torque function which is obtained by offline method

I * m = 2T ref 3K t (θ, β)cos 2 β . (6)
According to the reference current I * m and the power invariant Park transformation, the reference currents of dq-axis are

I qref = 3 2 I * m sinβ . ( 7
)
The reference current calculation in Fig. 4 is represented by ( 5)- [START_REF] Rasid | Preliminary thermal evaluation of actuator for Steerby-Wire vehicle[END_REF]. It can be applied to any value of β. In this article, the MTPA control strategy is applied during simulation and experiment. So the current angle β is 45°in ( 5)- [START_REF] Rasid | Preliminary thermal evaluation of actuator for Steerby-Wire vehicle[END_REF] The PI controllers are used to regulate the dq-currents and they are designed as

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ G d (s)=K d 1+ L d R s L d R s G q (s)=K q 1+ Lq R s L q R s ( 8
)
where s is the Laplace operator; R is the stator phase resistor, and K d/q is the gain of controllers as given in Table I.

According to these parameters, the closed loop bandwidths of d-axis and q-axis are calculated and both are 2360 Hz.

Fig. 5 presents torque function obtained by finite-element method for different current angles. It can be noticed that K t is different at different β. To reduce torque ripple for different control strategies (β), different K t should be applied. In this paper, K t (β = 45 • ) is used.

Besides, saturation is neglected during the simulation. Its influence on the proposed torque ripple reduction method has been analyzed in [START_REF] Wu | Analysis of torque ripple reduction in a segmented-rotor synchronous reluctance machine by optimal currents[END_REF]. The torque reference is defined as 0.15 N•m and the load torque is 0.1N•m. The inertia, the dry friction torque and the viscous friction coefficient are given in Table I.

The results in one electrical period of torque control simulation are introduced and studied. The rotor positions without compensation and with compensation are introduced in Fig. 6 and are applied to define one electrical period (180°mechanical rotor position). The corresponding rotor speeds are introduced by Fig. 7. It can be observed that the speed without compensation in one electrical period has six pulsations and the speed with compensation does not have noticeable pulsations.

Then, the accelerations of these two cases are calculated by the speeds and are presented in Fig. 8. In order to reduce the high noise in acceleration, an acceleration filter with the cutoff frequency 800 Hz is used. The design of this filter is based on the principles presented in Section IV-C.

It can be seen that the acceleration without torque ripple reduction has also six pulsations as illustrated in Fig. 8. Compared with the acceleration without compensation, the acceleration with compensation is minimized greatly. The amplitudes of these two acceleration pulsations, especially the acceleration without compensation are reduced when the simulation time is longer. Because torque control is applied in the simulation. When the speed is larger, the viscous friction becomes larger. So the acceleration will become smaller until to zero when the total friction torque and load torque equal the reference torque.

In the simulation, the electromagnetic torque could be obtained easily. The torques without compensation and with compensation are presented in Fig. 9. It can be observed that the compensated torque is nearly constant while the uncompensated torque has high torque ripple. The harmonics of these two torques are introduced by Fig. 10. Comparing with the torque harmonics without compensation in Fig. 10(a), the dominant torque harmonics (6th and 12th) in Fig. 10(b) are minimized by supplying the optimal currents. As a conclusion, based on torque function, the proposed torque ripple reduction method could decrease torque harmonics effectively.

IV. EXPERIMENT

A. Experimental Setup

Experiment is preferred to verify the torque function-based torque ripple reduction method. But, mechanical disturbances of the connection of torque meter and of hysteresis break with the studied SynRM in the experimental bench make it difficult to measure electromagnetic torque harmonic accurately. Besides, the electromagnetic torque produced by the studied machine is small. It means the measured electromagnetic torque could be affected by the disturbing torque. Therefore, the electromagnetic torque could not be measured accurately by torque meter. The SynRM can be only controlled without load to not disturb the electromagnetic torque harmonic.

The control diagram is introduced in Fig. 4. The torque control without load is applied. The torque function (β = 45°) presented in Fig. 5 is used to calculate the reference current and compensate torque ripple during the experiment. Fig. 11(a) illustrates the experimental bench. It includes several parts: host computer, dSPACE MicroAutoBox and power electronic (Rapidpro) system, power supply and the studied SynRM. The studied SynRM is presented in Fig. 11(b).

An incremental encoder is used to measure the rotor position. The rotor speed is calculated by the measured position and is filtered by a low pass filter. The acceleration is obtained according to the speed.

B. Validation Approach

In order to test the proposed torque ripple reduction method, the electromagnetic torque should be calculated by the measured data. The dynamic equation of SynRM without load is introduced

T e -T f -f ω ω m = J dω m dt (9)
where ω m is the speed calculated by position. Therefore, the electromagnetic torque could be calculated In [START_REF] Moghaddam | Design optimization of transversely laminated synchronous reluctance machine for flywheel energy storage system using response surface methodology[END_REF], the dry friction torque T f , the inertia J and the viscous friction coefficient f ω are constant and have been measured as given in Table I.

T e = J dω m dt + f ω ω m + T f . (10)

C. Analysis Principles

In order to analyze the results of experiment, several principles should be presented first because the measured position signal needs to be treated and processed before that the useful information can be extracted.

1) One Electrical Period: Because the validation method is based on the torque control, the speed will be increased gradually. In order to reduce the influence of variation of the speed, it is better to analyze the torque measurement in one electrical period which means a mechanical angle of 180°as shown in Fig. 12. This period is enough and convenient to determine the torque harmonics.

The applied incremental encoder has a resolution of 500 cycles per revolution. This resolution is multiplied by four by the quadrature encoder pulse module of the control system. The position is then measured with a resolution of 2000 cycles per revolution.

2) Selection of Speed Range: According to the measured position, the corresponding speed could be calculated. The unfiltered speed as introduced in Fig. 13 has many harmonics. It cannot be applied to control the SynRM. This is mainly due to the low resolution of encoder. Therefore, the speed should be filtered and the result is also presented in Fig. 13. The design of the digital filter is explained in the following section. Besides, during the experiment, the speed is increased gradually but the sampling frequency is constant. If the rotor speed is higher, it will contain fewer points than the lower speed during the same position length (180°). Then, the lower the speed during the measurement is, the greater the precision of the position measurement is. It is better to choose the rotor speed when it is low to analyze acceleration and torque.

3) Design of Speed Filter: The calculated speed contains many harmonics as presented in Fig. 13. In order to obtain useful information, it should be filtered by a low-pass filter. Besides, this filter could influence the speed harmonics and the calculated acceleration harmonics. Therefore, the frequency of the filter should be designed carefully.

The electrical frequency of the SynRM is

f e = Pf m = P ω m 2π . ( 11
)
The frequency of the m-th torque harmonic is

f me = mf e = mP ω m 2π . (12)
If the mth torque harmonic is considered and compensated, the cut-off frequency of the speed filter should be higher than the frequency of the mth torque harmonic

f c >f me . (13)
The available order of torque harmonic is

m< f c f e = 2πf c Pω m . (14)
Equation ( 14) presents the relationship among cut-off frequency, rotor speed and the order of torque harmonic. The available order of torque harmonic is proportional to the frequency f c ; but it is inversely proportional to the rotor speed. To analyze higher order torque harmonic, two methods are applicable. If the cut-off frequency of speed filter is fixed, the torque harmonics should be studied at low speed. If the selected speed range (in one electrical period) is fixed, the cut-off frequency of speed filter should be increased.

In the experiment, the cut-off frequency of the speed filter is fixed as f c . The order of the dominant torque harmonic of the studied SynRM is m (m = 6, 12 in this article). The maximal speed can be calculated

ω m < 2πf c mP . ( 15
)
In conclusion, the measured speed should be defined according to the electrical period, proper speed range and speed filter design.

D. Analysis of Results

Considering the speed limit and the limit of torque calculation, the torque reference is defined as 0.005 N•m. The rotor positions without compensation and with compensation are applied to present the electrical period and are shown in Fig. 14(a) and (b), respectively. The 6th and the 12th torque harmonics of the studied SynRM are dominant as introduced in Fig. 10(a). Thus, the maximum order of the dominant torque harmonic m is 12. The cut-off frequency of speed filter is defined as 250 Hz in experiment. Therefore, according to [START_REF] Yan | Design and analysis for torque ripple reduction in synchronous reluctance machine[END_REF], the maximum speed during the analysis is ω m < 65.45 rad/s = 625 r/min.

Then, the stator currents without compensation and with compensation are presented in Fig. 15(a) and (b) respectively. The rotor speeds without compensation and with compensation are introduced in Fig. 16(a) and (b), respectively. It can be noticed that the speed without compensation has six pulsations in one electrical period while the speed with compensation is smoother. In order to verify the proposed torque ripple reduction method, the accelerations of these two cases are calculated and presented in Fig. 17 According to [START_REF] Moghaddam | Design optimization of transversely laminated synchronous reluctance machine for flywheel energy storage system using response surface methodology[END_REF], the electromagnetic torques without compensation and with compensation is calculated by the corresponding accelerations and are presented in Fig. 18. It can be seen that the uncompensated torque in Fig. 18(a) contains six pulsations. On the other side, the amplitudes of the compensated torque are decreased. The high-frequency torque harmonics in Fig. 18 are produced by the limitation of the low-pass filters.

In addition, the torque ripple reduction simulation in Flux 2-D is also performed in order to confirm the calculated electromagnetic torque by the experimental data. The produced torques in Flux 2-D without compensation and with compensation are also introduced in Fig. 18(a) and (b), respectively. It can be observed that the calculated torques of experiment is very close to those obtained by Flux 2-D except that the experimental torque contains more high frequency harmonics. Besides, the torque ripple is reduced from 109 to 4.9% which validates the proposed torque ripple reduction method.

Then, these two torques of experiment are analyzed and their harmonics are illustrated in Fig. 19. Compared with the torque harmonics in Fig. 19(a), it can be observed that the 6th and the 12th torque harmonics in Fig. 19(b) are decreased greatly by the proposed torque ripple reduction approach. According to the results of simulation and the experiment, it can be confirmed that the torque ripple of the studied SynRM is minimized by the proposed torque ripple reduction method.

Finally, the torque function is also obtained during experiment and is presented in Fig. 20. According to (4), it is calculated by the uncompensated torque in Fig. 18(a) and the corresponding current I d calculated by the stator currents in Fig. 15(a). Compared with the torque function obtained by finite-element simulation, it can be noticed that they are nearly identical except for the high harmonics of the measured torque function.

E. Discussion About the Proposed Method

1) Influence of Power Supply:

The proposed torque ripple reduction method can compensate torque ripple effectively. On the other hand, it can be noticed in Figs. 2(a) and 15(b) the optimal currents are larger than the original sinusoidal stator currents. So, the stator voltages could also be changed when torque ripple is decreased which means the required power will be changed. It is better to analyze the influence of the increased power on the proposed method.

At first, the stator voltages corresponding to the simulation in Figs. 2 and3 are presented by Fig. 21. The stator voltages without compensation in Fig. 21 are also not sinusoidal. Because the sinusoidal stator currents are defined as 30 A and the speed is kept as 1500 r/min. The high torque ripple leads to nonsinusoidal voltage. On the other side, the stator voltages with compensation are also nonsinusoidal. They are much larger than the voltages when torque ripple is not minimized. According to the corresponding optimal currents in Fig. 2(a), it is obvious that the required power should be increased to produce additional torque harmonics which are applied to compensate the original high torque harmonics.

When the sinusoidal stator current is kept as 30 A and current angle β is 45°, the produced average electromagnetic torque is 0.205 N•m. Then, the load torque is kept as 0.205 N•m and the maximum available speed without torque ripple reduction is 4300 r/min with the current limit and voltage limit as given in Table I. If the torque ripple is minimized for this load torque at higher speed (>1500 r/min), the increased power will limit the maximum available speed.

Then finite element simulation is performed to study the maximum available speed. The results show when the speed is 2200 r/min, the voltage reaches its maximum available value. Its means if speed is lower than 2200 r/min, torque ripple can be minimized. If speed is larger than 2200 r/min, the proposed method will be limited by the available voltage which means power supply. It should be stated that the limit of power supply on the proposed method depends on the reduced torque ripple. Its influence could not be so noniterable for other SynRM.

2) Influence of Measurement Errors on Torque Function:

According to (4), two methods can be proposed to obtain torque function. The first one is based on FEM. This method is simple and efficient. But it requires the finite element model of machine which may be not available. The second approach is to measure K t by experiment. This method can provide the actual torque function. But, the electromagnetic torque should be measured by torque meter accurately or be calculated by the dynamic equation as presented by [START_REF] Moghaddam | Design optimization of transversely laminated synchronous reluctance machine for flywheel energy storage system using response surface methodology[END_REF]. In Section IV, the torque function obtained by FEM is applied to confirm the proposed method and is also compared with the measured torque function. It illustrates that if there is no finite element model of a SynRM, torque function can also be measured accurately.

The dynamic equation method requires several mechanical parameters, such as inertia and viscous friction coefficient. If there exist errors in these mechanical parameters, the torque function could be influenced. Therefore, the influence of measurement errors of inertia and viscous friction coefficient (J and f ω ) on the proposed method are analyzed. These two parameters have been given in Table I.Itisassumed that there exist 10% and -10% errors from either J or f ω . The simulation conditions (reference torque 0.15 N•m and load torque 0.1 N•m) are same as Section III. When torque ripple is not compensated, according to (4) the torque functions are calculated by the estimated torques with added errors of J and f ω . They are presented by Figs. 22 and23, respectively.

It can be seen that torque function with 10% inertia error K t (10%) in Fig. 22 becomes larger at peaks and becomes smaller at valleys than torque function without error K t (0%). On the contrary, torque function with -10% inertia error K t (-10%) in Fig. 22 becomes smaller at peaks. Besides, torque function with viscous friction coefficient errors is hardly changed. Therefore, the inertia error has larger impact on torque function measurement for the studied machine. It will be further studied by compensating torque ripple.

Then, torque functions with inertia errors are applied to reduce torque ripple. The results are presented in Fig. 24.I t can be observed that compared with the uncompensated torque T e_uncom , torque ripple is also reduced by applying K t (10%) and K t (-10%). Besides, the compensated torques T e (-10%), T e (0%) and T e (10%) obtained by using K t (10%), K t (-10%) and K t (0%), respectively are analyzed by fast Fourier transformation. The results are presented in Fig. 25. It indicates that the 6th and 12th torque harmonics are decreased greatly. Besides, the difference among these compensated torques is very small. As a result, inertia error could influence torque function, but can hardly affect the compensated results for the studied SynRM. The measurement of torque function is robust to mechanical parameter uncertainties.

V. C ONCLUSION

A novel parameter, torque function was proposed in this article. Based on torque function, a torque ripple reduction method was put forward. This approach was simple to be applied. Several analyzing principles, such as speed range and design of speed filter were also introduced to implement an accurate measuring method of electromagnetic torque harmonic. The results of simulation and experiment showed that the torque ripple of the studied SynRM was decreased by more than 95%. On the other hand, the proposed method required more power to reduce torque ripple. The increased power requirement could limit the method. The analysis showed that the torque function method was robust to mechanical parameter uncertainties.

In the future, the proposed method will be applied for high power SynRM at high speed with different loads. Besides, the limit of PWM frequency on the proposed method should also be studied further.

APPENDIX

T e = P -k 2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ k=m 1 L k +2M k 2 sin (k -2) θ + k=m 2 L k +2M k 2 sin (k +2)θ + k=m 3 (L k -M k )sinkθ I 2 ds23 + -k=m 1 L k +2M k 2 sin (k -2) θ -k=m 2 L k +2M k 2 sin (k +2)θ + k=m 3 (L k -M k )sinkθ I 2 qs23 + -k=m 1 (L k +2M k )cos(k -2) θ + k=m 2 L k +2M k 2 cos (k +2)θ I ds23 I qs23 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (A3) K t (θ, β, I) = P -k 2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ k=m 1 L k +2M k 2 sin (k -2) θ + k=m 2 L k +2M k 2 sin (k +2)θ + k=m 3 (L k -M k )sinkθ + -k=m 1 L k +2M k 2 sin (k -2) θ -k=m 2 L k +2M k 2 sin (k +2)θ + k=m 3 (L k -M k )sinkθ k 2 dq + -k=m 1 (L k +2M k )cos(k -2) θ + k=m 2 L k +2M k 2 cos (k +2)θ k dq ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (A4)
The stator inductances are defined as in (A2) shown at the top of the page, where L k and M k are the amplitudes of kth self-inductance and kth mutual-inductance.

The torque equation containing all torque harmonics is obtained in (A3) and (A4) shown at the top of the page, where the orders of torque harmonics are

Fig. 1 .

 1 Fig. 1. Torque ripple reduction based on torque function.

Fig. 2 .

 2 Fig. 2. Original currents and optimal currents (β = 45°, I m = 30 A) of FEM. (a) Stator currents. (b) Currents of dq-axis.

Fig. 3 .

 3 Fig. 3. Uncompensated torque and compensated torque of FEM (β = 45°).

Fig. 4 .

 4 Fig. 4. Control diagram of torque ripple reduction based on torque function.

Fig. 5 .

 5 Fig. 5. Torque function for different β (I m = 30A).

Fig. 6 .

 6 Fig. 6. Rotor position without compensation and with compensation.

Fig. 7 .

 7 Fig. 7. Rotor speed without compensation and with compensation.

Fig. 8 .

 8 Fig. 8. Acceleration obtained by speed without compensation and with compensation.

Fig. 9 .

 9 Fig. 9. Torque without compensation and with compensation.

Fig. 10 .

 10 Fig. 10. Torque harmonics. (a) Without compensation. (b) With compensation.

Fig. 11 .

 11 Fig. 11. Experimental platform. (a) dSPACE test bench. (b) Studied SynRM.

Fig. 12 .

 12 Fig. 12. Selected position data in one electrical period.

Fig. 13 .

 13 Fig. 13. Filtered and unfiltered speed in experiment.

  (a) and (b), respectively. The acceleration without compensation has also six obvious pulsations. Compared with the acceleration without compensation, the pulsations of the acceleration with compensation are reduced greatly.

Fig. 14 .

 14 Fig. 14. Measured rotor position in experiment. (a) Without compensation. (b) With compensation.

Fig. 15 .

 15 Fig. 15. Stator current in experiment. (a) Without compensation. (b) With compensation.

Fig. 16 .

 16 Fig. 16. Rotor speed calculated by position in experiment. (a) Without compensation. (b) With compensation.

Fig. 17 .

 17 Fig. 17. Acceleration obtained by speed in experiment. (a) Without compensation. (b) With compensation.

Fig. 18 .

 18 Fig. 18. Electromagnetic torque calculated in experiment. (a) Without compensation. (b) With compensation.

Fig. 19 .

 19 Fig. 19. Amplitudes of torque harmoics in experiment. (a) Without compensation. (b) With compensation.

Fig. 20 .

 20 Fig. 20. Torque functions obtained by experiment and FEM.

Fig. 21 .

 21 Fig. 21. Stator voltages without compensation and with compensation (β = 45°, I m = 30 A, ω = 1500 r/min).

Fig. 22 .

 22 Fig. 22. Torque function with inertia error (10% and -10%).

Fig. 23 .

 23 Fig. 23. Torque function with viscous coefficient error (10% and -10%).

Fig. 24 .

 24 Fig. 24. Uncompensated torque and compensated torques by K t with inertia error.

Fig. 25 .

 25 Fig. 25. Torque harmonics of compensated torques.

  The stator currents can be calculated by the power invariant Park transformation 4... L k cos (kθ) k=0,2,4... M k cos k θ+ 2π/3 k=0,2,4... M k cos k θ-2π/3 k=0,2,4... M k cos k θ+ 2π/3 k=0,2,4... L k cos k θ-2π/3 k=0,2,4... M k cos (kθ) k=0,2,4... M k cos k θ-2π/3 k=0,2,4... M k cos (kθ) k=0,2,4... L k cos k θ+ 2π/3

1

 1 =2, 8, 14, 20,... m 2 =4, 10, 16, 22,... m 3 =6, 12, 18, 24,... . (A5)


Mathieu Mével

François Lamarche

Jean-Claude Clément

Pascale Laurent

Jean-Jacques Yaouanc

Laure Burel-Deschamps

Philippe Giamarchi

Tristan Montier

Pascal Delépine

Pierre Lehn

Mathieu M Ével

Franc ¸ois Lamarche

Jean-Claude Cl Ément

Laure Burel

Pascal Del

Paul-Alain Jaffr

Claude F Érec

New Polyamines Phosphoramidate Vectors for Gene Therapy

Keywords: Cationic lipid, phosphoramide, polyamine, transfection

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

DNA delivery into cells can be achieved by using synthetic vectors that compact DNA forming a cationic lipoplex that can interact with cell membrane by electrostatics interactions. The next step will consists to liberate the plasmid DNA into the cyctoplasm followed by its migration up to perinuclear region and finally across the nuclear membrane. Finally, it migrates to the nucleus where it penetrates via the nuclear pores or during the mitosis. The chemical structure of the synthetic vectors can be classified into two categories: cationic polymers (PEI . . . ) and cationic lipids. In our group, cationic lipids possessing structure inspired from phospholipids present into membranes have been synthesized and tested for in vivo and in vitro essays. 1,2,3 Of note, many cationic polymers (e.g., PEI) or cationic lipids (DOGS, DPPES, DOSPA), identified by their transfection efficiency, are characterised by the presence of several amino functional groups on their backbone. We attempt the synthesis of cationic lipids having a phosphoramidate lipidic moiety and a polyamino polar headgroup. For this study, the polar headgroup is formed by the spermine as depicted below.

Address correspondence to Mathieu Mével CEMCA, UMR CNRS 6521, Faculté des Sciences et Techniques, UniversitédeBretagne Occidentale, 6 avenue Le Gorgeu, 29238 Brest, France. E-mail: mathieu.mevel@univ-brest.fr


Pedro Ramaciotti Morales

Jean-Philippe Cointet
email: jeanphilippe.cointet@sciencespo.fr

Julio Laborde

Your most telling friends: Propagating latent ideological features on Twitter using neighborhood coherence

Keywords: Multidimensional scaling, ideological scaling, latent features in social networks, political ideology, propagation in social networks

Multidimensional scaling in networks allows for the discovery of latent information about their structure by embedding nodes in some feature space. Ideological scaling for users in social networks such as Twitter is an example, but similar settings can include diverse applications in other networks and even media platforms or e-commerce. A growing literature of ideology scaling methods in social networks restricts the scaling procedure to nodes that provide interpretability of the feature space: on Twitter, it is common to consider the sub-network of parliamentarians and their followers. This allows to interpret inferred latent features as indices for ideology-related concepts inspecting the position of members of parliament. While effective in inferring meaningful features, this is generally restrained to these sub-networks, limiting interesting applications such as country-wide measurement of polarization and its evolution. We propose two methods to propagate ideological features beyond these sub-networks: one based on homophily (linked users have similar ideology), and the other on structural similarity (nodes with similar neighborhoods have similar ideologies). In our methods, we leverage the concept of neighborhood ideological coherence as a parameter for propagation. Using Twitter data, we produce an ideological scaling for 370K users, and analyze the two families of propagation methods on a population of 6.5M users. We find that, when coherence is considered, the ideology of a user is better estimated from those with similar neighborhoods, than from their immediate neighbors.

I. INTRODUCTION

Methods for embedding networks have become ubiquitous tools for analysis [START_REF] Goyal | Graph embedding techniques, applications, and performance: A survey[END_REF], [START_REF] Cai | A comprehensive survey of graph embedding: Problems, techniques, and applications[END_REF]. These methods exploit the geometrical representation of networks in a feature space, which can be used in tasks such as compression [START_REF] Wang | Structural deep network embedding[END_REF], clustering [START_REF] White | A spectral clustering approach to finding communities in graphs[END_REF], visualization [START_REF] Pearson | Liii. on lines and planes of closest fit to systems of points in space[END_REF], link prediction [START_REF] Wang | Structural deep network embedding[END_REF], node classification [START_REF] Tang | Line: Large-scale information network embedding[END_REF], or scaling of a latent property that is determinant in the structure of networks. Examples of this latter application can be found in the use of multidimensional scaling of networks for the retrieval of latent features in music for artist recommendation [START_REF] Platt | Fast embedding of sparse similarity graphs[END_REF], or in the use of Expectation Maximization algorithms for the estimation of ideological positions of voters, legislators, This work has been funded by the French National Agency for Research under grant ANR-19-CE38-0006: Geometry of Public Issues (GOPI).

or online users [START_REF] Imai | Fast estimation of ideal points with massive data[END_REF]. A growing domain of research has sought to exploit these scaling methods to infer ideological feature spaces in which to embed users of social networks. Taking inspiration in ideological scaling of voting data, pioneered by Poole et al. in the 1980s [START_REF] Poole | A spatial model for legislative roll call analysis[END_REF], and developed in the 1990s into the widely-used NOMINATE method [START_REF]Patterns of congressional voting[END_REF], some adaptations to social network data have been proposed. The first such adaptation was proposed by Bond et al. [START_REF] Bond | Quantifying social media's political space: Estimating ideology from publicly revealed preferences on facebook[END_REF], then popularized by Barbera in the mid-to late 2010s [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF], [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF]. These new scaling methods have been successful in identifying latent features in social networks related to the ideology of users. They are, however, often limited to small subsets of the whole network of users. The reason for this may be found in computational limitations in computing scaling for large networks, but mostly in the need for interpretability for the found features. On Twitter, traditionally, an ideological scaling is limited to a sub-graph of parliamentarians and their followers, using the embedded features of the first ones to provide an interpretation for the found features [START_REF] Barberá | Understanding the political representativeness of twitter users[END_REF]. This raises the question: How to compute features for portions of the network outside this seed sub-graph? The setting of this problem is similar to that of problems such as latent feature or link prediction, that accounts for a wealth of works to be discussed in Section II. But crucially, it deviates from these settings in that a minority of nodes have known features (estimated via scaling), while those of a comparatively large set of nodes must be estimated from this initial seed set. This marks a difference with statistical learning methods, where a flexible model is learned on a majority of nodes, and then applied to a minority of nodes with missing values. The setting of ideological scaling invites the use of less flexible models, including strong assumptions about the link between ideology and the structure of the network.

This article takes inspiration in methods for information propagation in social networks and message coherence in telecommunications networks to propose a method for the propagation of scaled ideological features, from a seed set of nodes, to larger parts of a network. While this problem arises in the case of ideological scaling in social networks in particular, it can also suit a more general class of problems in arXiv:2103.07250v1 [cs.SI] 12 Mar 2021 which n-dimensional features are only known for a subset of nodes in a network. One may consider that latent ideological features are defined only for users that follow members of parliament (MPs). However, our method supposes that a larger set of nodes could be positioned in the same latent feature space. This underlying hypothesis is leveraged by our method exploiting two strong assumptions: 1) the ideology of a node is similar to that of its neighbors if this neighborhood is ideologically coherent, and 2) the ideology of a node is similar to other structurally similar nodes, that follow the same users for example, if these followed users are ideologically coherent.

After providing a review of the relevant related work in Section II, we lay out the definitions and the notation needed for the treatment of the problem of coherent feature propagation in networks in Section III. Using this framework, we define the proposed propagation methods in Section IV. To analyze these methods, we develop an application case based on the ideology scaling on Twitter. Section V presents the case of a seed set of Twitter accounts, followers of MPs in France, with known ideological scaling computed using procedures well-established in the literature. Finally, in Section VI, we devise and execute experimental protocols to assess 1) the prediction capabilities using both of our methods following the two hypotheses in estimating ideological features, and 2) the limits of the estimation in terms of the trade-off between precision of the estimation, and its reach as parts of the whole network.

II. RELATED WORK

The methods proposed in this article are based in the propagation of features, from nodes for which they are known, to nodes for which they are not. This is mainly related to three different domains of research here reviewed. We intentionally omit the -vast-scientific literature regarding the identification of important and central nodes in networks, outside the scope of these selected domains.

1) Latent ideological features in social networks: The first category of related works include those that compute latent features in social networks. While several such methods exist, this article is concerned by those that leverage the interpretability of a subset of nodes to extract meaningful feature spaces. This stands in contrast to works that produce feature embeddings on which to perform statistical learning such as deep learning, without necessarily providing an interpretation for the embedded features (cf. [START_REF] Yan | Graph embedding and extensions: A general framework for dimensionality reduction[END_REF]). A numerous family of methods, connected with link prediction, learn latent features in social networks, but specifically in relation with the probability of existence of a link, to be used in the propagation of local features into other parts of the time-evolving network [START_REF] Wang | Structural deep network embedding[END_REF], [START_REF] Heaukulani | Dynamic probabilistic models for latent feature propagation in social networks[END_REF], or the retrieval of missing features for some nodes [START_REF] Tang | Line: Large-scale information network embedding[END_REF]. The most relevant works for this article, are those that use multidimensional scaling to extract ideological features [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF], [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF]. Other works seek to predict ideology for large networks, linking prediction to homophily, but use ideological labels and not a -continuous-ideological feature space [START_REF] Colleoni | Echo chamber or public sphere? predicting political orientation and measuring political homophily in twitter using big data[END_REF]- [START_REF] Xiao | Timme: Twitter ideologydetection via multi-task multi-relational embedding[END_REF].

2) Propagation in social networks: A second category of works pertinent for this study concerns the propagation in social networks, accounting for numerous results in a long tradition, but organized here in propagation of information and of properties. Propagation of information counts important cases such as the spread of rumors and misinformation [START_REF] Bao | A new rumor propagation model and control strategy on social networks[END_REF], with connection to epidemiological models [START_REF] Sahneh | Competitive epidemic spreading over arbitrary multilayer networks[END_REF], [START_REF] Chakrabarti | Epidemic thresholds in real networks[END_REF]. This setting is fundamentally different from the one of this article in that it considers the network as the support for the flow of messages that have specific positions in time and space (as opposed to permanent latent features or properties). However, there are possible connections between network structure and homophily on the one hand, and structure in networks and flow of information on the other [START_REF] Aral | Distinguishing influencebased contagion from homophily-driven diffusion in dynamic networks[END_REF], [START_REF] Mcpherson | Birds of a feather: Homophily in social networks[END_REF]. This connection is also related, for example, to types of users in rumor propagation cascades in social networks [START_REF] Friggeri | Rumor cascades[END_REF].

Finally, propagation in social networks has connections with label propagation on graphs, used for example in heuristic procedures in community detection [START_REF] Zhuγ | Learning from labeled and unlabeled data with label propagation[END_REF], [START_REF] Cheng | On modeling malware propagation in generalized social networks[END_REF]. However, again, this differs from the setting of this article in that a label can be equated to discrete classification, whereas features can provide (n-dimensional) continuous indices for nodes.

3) Node coherence: Node coherence is a concept that can be encountered in domains such as neuroscience [START_REF] Thagard | Coherence as constraint satisfaction[END_REF], artificial intelligence [START_REF] Mackworth | Consistency in networks of relations[END_REF], or communications networks [START_REF] Larsson | Large-scale cooperative relaying network with optimal coherent combining under aggregate relay power constraints[END_REF], and often addresses the degree of dissimilarity of signals coming from different sources. This concept is of particular importance in multi-path networks. Depending on the type of network, coherence can be taken into account for node properties rather than relayed messages. The most relevant domain of works along this line, is found in node coherence in opinion dynamics, where an opinion is often modeled as a continuous feature [START_REF] Edmonds | Modeling belief change in a population using explanatory coherence[END_REF]. In this domain, the concept of opinion coherence is sometimes considered in dynamic models [START_REF] Rodriguez | Collective dynamics of belief evolution under cognitive coherence and social conformity[END_REF]. In opinion dynamics, however, the focus is put in measuring different indices for node coherence to then be used in simulations to study evolution of features in systems.

III. PRELIMINARIES

This section proposes the notation and definitions required to treat the concept of node and neighborhood coherence in networks, before proposing feature propagation methods in Section IV.

A. The universe network

Let us consider a large universe network as a directed graph G = (V, E), for some non-empty set V with directed edges E ⊆ V × V. In social networks, nodes typically represent user accounts, and directed edges represent a relation of following between them. Information flows downstream, contrary to the direction of the edges: if a user shares information, it will be received by her followers and not by her followees 1 .

For a node v ∈ V, we consider the set of its downstream or in-neighbors n down (v) = {u ∈ V : (u, v) ∈ E} and its upstream or out-neighbors

n up (v) = {u ∈ V : (v, u) ∈ E}.
Abusing notation, we define the downstream neighborhood of a set of nodes V ⊂ V as n down (V ) = {u ∈ V : ∃v ∈ V ((u, v) ∈ E)}, and its upstream neighborhood as

n up (V ) = {u ∈ V : ∃v ∈ V ((v, u) ∈ E)}. It is worth noticing that, in general, we cannot assure V ∩ n d (V ) = ∅, for d ∈ {up, down}.
Given a direction d, we will denote its opposite direction with a bar, as d.

B. Features of nodes and estimation error

Let us suppose that we can attribute N -dimensional features for some limited set of nodes V ⊂ V. When available for a node v ∈ V, we denote their known N -dimensional features by e(v) ∈ R N , and its i-th component by e i (v). When initially unavailable for a node v ∈ V, N -dimensional features may be attributed through feature propagation from nodes for which they are known. Methods for doing so are at the core of this article, and will be proposed in Section IV. We denote estimated features for a node v ∈ V by ê(v) ∈ R N , and by êi (v) its i-th component.

Whenever we can estimate features ê(v) through propagation methods for a node v ∈ V for which we also know its true features e(v), we can compute the estimation error E(ê(v), e(v)). We measure this estimation error as the p-norm in the feature space:

E(ê(v), e(v)) = ê(v) -e(v) p . (1)
For a set of nodes V , the mean estimation error will be denoted by

E(V ) = (1/|V |) v∈V E(ê(v), e(v)).

C. Node coherence

Node coherence is a key concept in this article, and seeks to capture the degree of similarity of a set of nodes in the feature space. Given V ⊂ V for which features are known or estimated, we denote by I(V ) the incoherence of V . Several incoherence metrics are possible, of which we propose to use the standard deviation of the distance to the centroid of V in the feature space:

I(V ) = 1 |V | v∈V e(v) -c V 2 p , (2)
where

c V = (1/|V |) v∈V e(v) is the centroid of V .
We prefer the definition of Equation ( 2) over other intuitive common metrics such as entropy [START_REF] Onesto | Relating the small world coefficient to the entropy of 2d networks and applications in neuromorphic engineering[END_REF]: a set V can have minimal entropy and high values of deviation I(V ) if the underlying probability distribution of V is multi-modal in the feature space. D. ε-coherent neighborhoods of a set of nodes Whenever we have a set of nodes V ⊂ V, we can consider the up-or downstream neighborhoods of nodes that are also coherent: i.e., nodes that are linked to a set, and whose incoherence with respect to this set is below a given threshold ε. We formalize this notion, for both directions, as the εcoherent upstream and downstream neighborhood of set V :

n ε d (V ) = u ∈ n d (V ) : I(n d (u) ∩ V ) ≤ ε
for d ∈ {up, down}, and d the direction opposite to d.

0 1-dimensional feature V v 1 e(v 1 ) v 2 v 3 v 4 v 5 v 6 e(v 4 ) e(v 5 ) e(v 6 ) e(v 2 ) e(v 3 ) I({v 4 , v 5 , v 6 }) ≤ ε I({v 2 , v 3 }) ≤ ε n down (V ) n ε down (V ) u 1 u 2 u 3
Fig. 2. Illustration of the construction of a ε-coherent downstream neighborhood n ε down (V ) of a set V embedded in a 1-dimensional feature space.

IV. PROPAGATION METHODS FOR LATENT FEATURES IN SOCIAL NETWORKS

We consider two different approaches for propagating ideological features in a larger set of the network: Method A) directed sequences of ε-coherent neighborhoods, and Method B) sequences of projections using ε-coherent neighborhoods. Both families of methods are based on the concept of coherent neighborhoods, but motivated by distinct intuitions. Method A assumes that the features attached to nodes are "diffusing" along the social network formed by follower/followee relationships. This method is rooted in the larger homophily hypothesis "that birds of a feather flock together" [START_REF] Mcpherson | Birds of a feather: Homophily in social networks[END_REF], meaning that two connected users are likely to demonstrate homophilic behavior, sharing a similar ideological position. Method B has different premises. Now, if unknown, ideological features of a node are estimated using those of other nodes that occupy a similar position in the network. This method is based on the structural equivalence hypothesis, which posits that two nodes sharing the same neighborhood are similar [START_REF] Sailer | Structural equivalence: Meaning and definition, computation and application[END_REF].

The approach of this second family of methods is, for example, at the core collaborative filtering approaches in algorithmic recommendation. Here we will follow the same principle, estimating the ideological features of a user on Twitter, as the aggregation of features of users following, or being followed by the same users.

A. Directed sequences of ε-coherent neighborhoods

This method generates two sequences of sets of nodes: one of coherent nodes {V i } i≥0 , for which we estimate the ideology, and one of incoherent nodes {V i } i≥0 , that we avoid using in ideology estimation.

Let us consider a seed set V 0 ⊂ V for which features e(v) for v ∈ V 0 are known, and a direction d ∈ {up, down}.

Starting at V 0 , a directed sequence of ε-coherent sets of nodes V 0 , V 1 , V 2 , . . . is computed as V i+1 = V i ∪ ∆V i for i = 0, 1, 2, . . ., with ∆V i = v ∈ n ε d (V i ) : v / ∈ V i ∪ V i , (3)
where V i+1 = V i ∪ ∆V i for i = 0, 1, 2, . . ., with V 0 = ∅ and

∆V i = v ∈ n d (V i ) : v / ∈ V i ∪ ∆V i . (4)
By definition,

{V i } i≥0 is an incremental (V i ⊂ V i+1 ) by disjoint additions (V i ∩ ∆V i = ∅) sequence of ε-coherent neighbors. Sequence {V i } i≥0 is also increasing (V i ⊂ V i+1 ) by disjoint additions (V i ∩ ∆V i = ∅), but of incoherent neighbors.
Nodes deemed incoherent are stored to avoid coherence collisions: an incoherent node cannot become coherent by virtue of new nodes who had been attributed features by propagation. Whenever new ε-coherent neighbors ∆V i are discovered at the i-th step, the features of their nodes are estimated as

ê(v) = 1 n d (v) ∩ V i u∈n d (v)∩Vi ê(u), for v ∈ ∆V i , (5)
setting

ê(v) = e(v) for v ∈ V 0 . V i V i ∆V i ∆V i ∆V i I ≤ ε
Fig. 3. Schematic representation of the computation of ∆V i and ∆V i on the i-th step of an upstream directed sequence of ε-coherent neighborhoods using Method A. In the upstream variant, users from V i follow users from ∆V i .

B. Sequences of projections using ε-coherent neighborhoods

Inspired by the way in which features are computed in multi-dimensional scaling for users depending who they follow, we propose a second family of methods, Method B, for feature propagation using ε-coherent neighborhoods. Most ideology scaling methods exploit the collective structural relations between nodes that are important for the latent features of a network (typically politicians) and their followers [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF], [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF]. While intuitive, the first family of methods proposed in Section IV-A overlooks this aspect. This can have disadvantages: for example, some nodes can attract followers without themselves following many nodes coherently. This could arguably be the case of Twitter accounts of media outlets, institutions, or public figures. Accordingly, we propose a second family of methods centered around the notion of structural similarity: nodes that are similar because they follow (upstream) or are followed (downstream) by the same users, hereafter called pivots. As with the previous method, Method B generates two sequences of sets of nodes: one of coherent nodes {V i } i≥0 , for which we estimate the ideology, and one of incoherent nodes {V i } i≥0 that we now avoid using as pivots.

Let us consider a seed set V 0 ⊂ V for which features are known and a direction d ∈ {up, down}. Starting at V 0 , a sequence of projected ε-coherent sets of nodes V 0 , V 1 , V 2 , . . . is computed as V i+1 = V i ∪ ∆V i for i = 0, 1, 2, . . ., for which we consider the set P ε i of ε-coherent pivots:

P ε i = n ε d (V i )\V i , (6)
where

V i+1 = V i ∪ ∆V i for i = 0, 1, 2, . . ., with V 0 = ∅, and
∆V i = v ∈ n d (V i ) : v / ∈ n ε d (V i ) ∪ V i . (7)
As with the previous family of Method A, the sets V i store the nodes deemed incoherent and that cannot be used, but now as pivots. The sets P ε i are used at each iteration to compute additions ∆V i according to coherent structural similarity:

∆V i = v ∈ n ε d (P ε i ) : v / ∈ V i . (8)
Again by definition, sequences {V i } i≥0 and {V i } i≥0 are incremental by disjoint additions. In contrast with the first family of Method A from Section IV-A, now it is the coherence of the pivot nodes in sets P ε i that is assured. Whenever new neighbors ∆V i are discovered at the i-th iteration, their features are estimated as

ê(v) = 1 |C i (v, P ε i , V i )| u∈Ci(v,P ε i ,Vi) ê(u), for v ∈ ∆V i , (9)
where

C i (v, P ε i , V i ) is the set of co-neighbors of v in V i through pivot P ε i : C i (v, P ε i , V i ) = V i ∩ n d (n d (v) ∩ P ε i ) . ( 10
) V i V i ∆V i ∆V i P ε i I ≤ ε ∆V i P ε i Fig. 4
. Schematic representation of the computation of ∆V i and ∆V i using pivot P ε i on the i-th step of an upstream (co-followers-based) sequence of projected ε-coherent neighborhoods using Method B. In the upstream variant, users from V i and ∆V i co-follow users from pivot set P ε i .

V. FRENCH POLITICAL TWITTER DATASET

In order to analyze the feature propagation methods described in the previous section on real data, here we present a pertinent part of the Twitter network2 and an ideological scaling producing an interpretable ideological feature space.

A. French MPs and their followers

Our data collection starts with the set P of the 831 (out of 925) French MPs present on Twitter3 , belonging to 10 main different parties or groups/alliances of parties. We then proceeded to collect all the followers of the accounts of MPs in P. This collection was conducted on May 2019, and resulted in the constitution of the set F = n down (P) of followers of P, which amounts to |F| = 4.487.430 unique Twitter accounts.

B. An ideological inference procedure via multidimensional scaling

We follow the methodology described by Barbera et al. [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF] for inferring ideological features through multidimensional scaling of the sub-graph of the MPs and their followers. After removing from this sub-graph the followers that follow less than 3 MPs, and then removing users that had a repeated set of followed MPs (to assure full rank of the adjacency matrix), we obtained a set F of 368.831 accounts. We represent this subgraph as a {0, 1} | F |×|P| adjacency matrix , where a relation of following is encoded with the value 1, and its absence with the value 0. Next, we produce a reduced-dimensionality representation of these 368.831 observations using a Correspondence Analysis (CA) [START_REF] Greenacre | Correspondence analysis in practice[END_REF]. The first 2 principal components of the reduced-dimensionality space, PC1 and PC2, explain 0,97% and 1,07% of the inertia. This might seem as little explanation of the variability in the observed ways in which users from F follow MPs. However, the experience in several countries has revealed that the first components offer useful interpretations in the Twitter MPs' network. Examples include the UK, Spain, Italy, and Netherlands [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF]). Similar results have also been found in France [START_REF] Briatte | Recovering the french party space from twitter data[END_REF], [START_REF] Cointet | What color are the yellow vests ? : Ideological scaling of facebook groups[END_REF]. In accordance with analyses made for the cases of other countries, those of France suggest feature interpretation where the first component is identified with leftright ideology.

C. Interpreting the ideological feature space

To provide and interpretation of PC1 and PC2 for our sets P and F, we consider exogenous variables for the parties of the MPs. We use the 2019 Chapel Hill Expert Survey (CHES) data [START_REF] Bakker | 2019 chapel hill expert survey[END_REF]: an estimation on ideology and policy position of for accounts in P, 8 are also present in the CHES data. We computed the correlations between the positions of MPs on PC1 and PC2 of the feature space with all the 51 criteria were CHES data produces estimation for their political parties. This allowed to identify the two most relevant criteria related to our axes. Fig. 5 shows the ordering of these eight parties according to the two different criteria: 1) parties' economic views, from left to right, and 2) parties' attitudes towards European integration, from opposed to favorable. For each party in Fig. 5, we provide the embedded features of PC1 per MP for criterion 1 (left-right ideology), and the embedded features of PC2 per MP for criterion 2 (attitudes towards European integration). While it is not the objective of this article, Figure 5 illustrates the potential applicability of the multidimensional scaling procedure.

Under this interpretation, the French Twitter network of MPs and their followers (the sets P and F) may be represented in a 2-dimensional latent ideological feature space. In this feature space, the first dimension (PC1) provides an index for the concept of left and right positions, and the second dimension (PC2) provides an index for attitudes towards European integration, which is in line with observations that have been made in the case of other countries 4 . Fig. 6 illustrates the positions of the sets P of parliamentarians and F of their followers in this bidimensional ideological feature space.

D. Datasets for experimentation

We are now concerned with the problem of using the set F with 368.831 accounts with known features to establish datasets for the testing and evaluation. We cannot set V 0 = F for testing and evaluation. This is because when producing the next set V 1 in the sequence (with either method) the new added nodes would not have known true features e(v) (as computed with ideological scaling) to test the estimated ones ê(v), for v ∈ ∆V 0 . To circumvent this difficulty, we take a subset A ⊂ F, and then collect its upstream and downstream neighborhoods to use in the described methods, allowing for some elements of these new neighborhoods to be also in F. To account for the possible specificities of the different regions of the ideological feature, we sample 4.4835 nodes uniformly in space from F to produce set A (see Fig. 6). Had we not sampled with spatial uniformity, the sampling would have produced a majority of nodes near the origin of the ideological feature space (see the concentration of nodes near the origin in Fig. 6). Next, we collect the followees/friends of A as B = n up (A) (obtaining |B| = 1.304.812), and the followers of set A as C = n down (A) (obtaining |C| = 5.528.716). This sub-sampling and collection operations achieve sets for our test evaluations that are such that |B ∩ F| = 138.424, and |C ∩ F| = 231.035.

VI. NUMERICAL EXPERIMENTS

In this section we analyze some results of the methods proposed in Section IV for estimating ideological features on Twitter data. For both methods, we center the analysis around the set A built for this purpose, and described in the previous section. We will be interested in analyzing, for different values of coherence ε, 1) the accuracy (the degree of error in estimating ideological features), and 2) the coverage (the size of the disjoint additions with estimated features) of the proposed methods. The main aspect we seek to investigate in this section is the relation between the coherence threshold parameter ε and the trade-off between estimation error and size of a disjoint addition ∆V i at a step i. Throughout this section, we compute error E (cf. Equation ( 1)) and coherence I (cf. Equation ( 2)) using p = 2.

A. Directed sequences of ε-coherent neighborhoods

Because we have collected the upstream and downstream neighborhoods of set A (sets B and C), we can readily compute the first step for Method A in both directions. While intuitive, the estimation of ideological features through directed propagation of Method A in either direction performs poorly. Imposing higher levels of coherence (lower thresholds ε for incoherence) does improve the mean estimation error of nodes in ∆V 0 . However, this improvement is marginal and, most importantly, limited. Upstream neighbors (friends/followees) of users in A that are followed by highly coherent users, have their ideology estimated with a mean error no lower than E(∆V 0 ) =0.6, which amounts to 12.5% of the diameter of F in the feature space. Similarly, downstream neighbors (followers) of users in A that follow highly coherent users, have their ideology estimated with a mean error no lower than E(∆V 0 ) =0.575 (12.0% of the diameter of F). Had set ∆V 1 be computed (upstream or downstream), its ideological features would be estimated only from those already -poorly-estimated for set ∆V 0 (disjoint additions, assured by Equation (3), imply that n d (V 0 ) ∩ ∆V 1 = ∅).

B. Sequences of projections using ε-coherent neighborhoods

The setting of Method B is similar to that of collaborative filtering (CF) [START_REF] Schafer | Collaborative filtering recommender systems[END_REF] in Recommender Systems. In user-based CF, the similarity of users is computed according to structural similarity on a bipartite graph: users are similar if they have chosen similar items. Items can then be proposed to a user among those already chosen by similar users (structural similarity is related to preference). In the setting of our Method B, users are deemed similar if they follow, or are followed by (depending if the selected direction is upstream or downstream) similar users of a so-called pivot set for a predetermined coherence ε. Taking on known evaluation protocols for CF [START_REF] Bobadilla | A collaborative filtering approach to mitigate the new user cold start problem[END_REF], we propose a method for assessing the accuracy and coverage of Method B. Given a positive integer K, we perform a K-fold bipartite cross-validation. We divide A in K parts, taking one as A K test and the rest as A K train . We set V 0 = A K train , we compute P ε 0 using B and C for upstream and downstream directions, and then set R = A K test ∩ ∆V 0 . To analyze the coverage, we examine the quantity cov(R) = |R|/|A K test |, ranging from 0 (no coverage) to 1 (total coverage). We choose K = 20 and report the results for varying values of incoherence ε, providing, for the ensemble of the 20 folds, the median, the maximum, and the minimum value in Fig. 8. test ∩ ∆V 0 and P 0 for the Method B in a K-fold cross-validation retrieval of ideologies of A (K =20). The distribution of errors and coverage over the folds is reported with the median, the minimum, and the maximum values .

Results obtained with Method B, reported in Fig. 8, are more satisfactory than those obtained with Method A. Thresholds for incoherence below ε = 1 (approximately) already allow for less error in estimation in comparison to Method A. In fact, under our setting, estimation errors for Method B can be made as low as (approximately) 0.1 in the feature space by sacrificing coverage of new nodes added in the following step of the sequence. With a small value of threshold for (lower than 0.1) the estimation error is around 0.1 in distance in the feature space (2.1% of the diameter of F). This trade-off in coverage does not come at great expense, as suggested by our proposed metric cov(R): at least 50% of nodes left in the test set A K test can be recovered, independently of the threshold for incoherence. The amount of nodes in the set of pivots P ε 0 is that of ∆V 0 for Method A, and follows intuition in that, when forcing less incoherence, less nodes are available for the search of co-neighbors (cf. Equation ( 10)).

While it was possible to evaluate the quality of Method A precisely, the same evaluation would have required the collection of the downstream neighbors of B (i.e. n down (n up (A))) and the upstream neighbors of C (i.e. n up (n down (A))), which would amount to a mixed-direction snowball-sampling of the Twitter network from A. Thus the need for a different evaluation protocol for both methods. While there is a slight risk of over-estimating the coverage capacity of Method B using our testing protocol (due to the fact that R is restricted to A) it has the great advantage of allowing for experimentation with large sets: the (up-and downstream) neighbors of A, i.e., the total population for feature propagation, account for nearly 6.5M users.

VII. CONCLUSIONS

We proposed two methods (A and B) for the propagation of latent ideological features on Twitter. These methods use propagation for the estimation of ideology of users, represented in a continuous n-dimensional features space. Method A is based on the homophily hypothesis: if users are connected (one follows the other) their ideology is similar. Method B is based on the structural similarity hypothesis: if users are connected to the same neighborhoods their ideology is similar. In these methods, we proposed the consideration and formalization of these notions when depending on ideological coherence of neighbors, which we model with a single parameter.

To analyze our methods, we collected Twitter data using the accounts of French MPs. We were able to produce a 2dimensional ideological embedding for a sub-graph of nearly 370K users. With the help of the Chapel Hill Expert Survey, we validate the interpretation of the two emerging dimensions as related to left-right ideology, and attitudes toward European integration. Collecting again neighborhoods of some of these users we used our methods to propagate these ideological features with predefined degrees of accuracy within a potential population of nearly 6.5M users. To total number of users for which ideology is propagated, depends on the error to be accepted, which is determined by a coherence threshold ε. The protocols established for evaluation allow us to analyze the relation between estimation accuracy for ideology in propagation, and the coverage of the method (i.e., the number of users to which we can propagate). The trade-off between these two competing and desirable properties was modulated by our coherence parameter. Analyzing Methods A and B, we find that, when coherence is considered, the ideology of a user on Twitter is better estimated using other users that are structurally similar (Method B), than using other users that might be directly connected (Method A). These results lend support to the structural similarity hypothesis over the homophily hypothesis. Even more: while the difference is small, the ideology of a user is better estimated using users that follow the same accounts, than using users that are followed the same Twitter accounts.

The formalism used to treat coherence in social networks, and the methods proposed for propagation, can be extended to any other network were directed edges model the fact a user can receive information from another. On Facebook, for example, the fact that user a and user b are friends, could be represented by two directed edges between both users, and in different directions. In other networks, such as Instagram or YouTube (using the subscription relation) the formalism and methods can be directly applied.

Fig. 1 .

 1 Fig. 1. Convention used for upstream and downstream directions on directed social networks, where edges indicate following/friendship.

Fig. 5 .

 5 Fig. 5. Distribution of the embedded ideological features PC1 and PC2 for the French MPs on Twitter per party. Parties are ordered according to the two criteria using the 2019 Chapel Hill Expert Survey data: left-right economics for feature PC1 (top), and attitudes toward European integration for feature PC2 (bottom).

Fig. 6 .

 6 Fig. 6. Position of P French MPs in the first two embedded dimensions, PC1 (left to right ideology) and PC2 (pro-and against European integration), histogram of the positions of their followers F , and positions of sampled accounts in set A.

Fig. 7 .

 7 Fig.7. Mean estimation error E(∆V 0 ) and size of ∆V 0 for Method A in the upstream and downstream directions, using V 0 = A.

Fig. 8 .

 8 Fig.8. Estimation error and coverage of set R = A K test ∩ ∆V 0 and P 0 for the Method B in a K-fold cross-validation retrieval of ideologies of A (K =20). The distribution of errors and coverage over the folds is reported with the median, the minimum, and the maximum values .

In conformity with the General Data Protection Regulation 2016/679, the project called Ideology scaling Twitter France, whose data we exploit, has been declared the 19 Mars 2020 at the registry of data processing at the Fondation Nationale de Sciences Politiques (Sciences Po), and respects Twitter's data use policies.

The list of French MPs present on social networks is provided by the French chambers of parliament at http://www2.assemblee-nationale.fr/ deputes/liste/reseaux-sociaux, for deputies, and http://www.senat.fr/espace presse/actualites/201402/les senateurs sur twitter.html for senators.

Research results in several countries have also identified this second axis as related with issues dividing public along pro-and anti-establishment positions[START_REF] Schmidt | Measuring social response to different journalistic techniques on facebook[END_REF],[START_REF] Chiche | L'espace politique des électeurs franc ¸ais à la fin des années 1990: nouveaux et anciens clivages, hétérogénéité des électorats[END_REF].

Originally, we sampled 5.000 nodes, but the later collection of followers and followees for 517 of these nodes was not possible because these accounts were either closed between collections, or private and information could not be collected.

ACKNOWLEDGMENTS

We thank Justin Clarke and Benjamin Ooghe-Tabanou for their help in the collection of Twitter data.