Ming Cheng
email: cheng@insa-toulouse.fr

Claude Le Men

Alain Line

Philippe Schmitz

Luc Fillaudeau

Methodology to investigate instantaneous and local transmembrane pressure within Rotating and Vibrating Filtration (RVF) module

Keywords: Dynamic filtration, hydrodynamics, power consumption, core velocity coefficient, periodic and random pressure fluctuation. Highlights: Semi-empirical correlations to estimate mixing and pumping powers, Determination of local core velocity coefficients and mixing pressure

Dynamic filtration exhibits high performances by generating wall shear stress (tangential to membrane) and pressure stress (normal to membrane) by the mechanical movements, such as rotating, oscillating or vibrating systems. Rotating and Vibrating Filtration (RVF) module includes rotating flat blades impellers which generate high and fluctuating shear stress and pressure at the membrane surface. To understand performances in turbulent regime and to optimise the operating conditions, global parameters (power consumption, pressure drop) and driving forces (mean, instantaneous and local pressure at the membrane surface) were characterised. For global approach, friction and mixing power in the RVF module were described by semi-empirical correlations. Euler number correlations were integrated based on feeding and mixing conditions. The balance between nominal power and thermal dissipation was reported. On the other hand, the mechanical power calculated with the empirical correlation of local shear stress was underestimated. For semi-local and local approaches, the local pressure at the membrane surface was measured with a specially designed and instrumented porous substrate. Mean radial pressure and core velocity coefficients were quantified versus flowrate and mixing rate. The core velocity coefficient decreases with mixing rate and radius up to a plateau value close to 0.6. For fluctuating component, pressure oscillation and its amplitude were treated by statistical analysis, probability distribution function and Fast Fourier transform. These methods show similar results with maximum fluctuating intensity between 15 and 30 Hz, which increase with radius. The maximum value can be obtained at the outer edge of the impeller with a relative standard deviation of over 25%. It indicates that the influence of pressure fluctuations should be carefully considered to enhance filtration performances. Pressure fluctuation distributions were accurately modelled by the convolution of periodic (sinusoid wave) and random (normal) functions. The area of intensive fluctuation was identified, in which periodic component accounts for 60% up to 97% of total energy input.

Table of content

Introduction

Membrane separation technologies have attracted great attention and have been applied in a wide range of industrial applications: water treatment (drinking water and wastewater), food industry and biotechnology [START_REF] Pendergast | A review of water treatment membrane nanotechnologies[END_REF][START_REF] Lipnizki | Cross-flow membrane applications in the food industry[END_REF][START_REF] Selvamuthukumaran | Applications of Membrane Technology for Food Processing Industries[END_REF][START_REF] Drioli | Biocatalytic membrane reactors: applications in biotechnology and the pharmaceutical industry[END_REF]. Traditional dead-end filtration is limited by cake formation at the membrane surface. A parallel flow relative to the membrane in cross-flow filtration mitigates the concentration polarisation and fouling by modifying the feeding flow direction. However, the increase of local shear stress is affected by the feeding flowrate, leading to an increase in pumping power. In comparison, dynamic filtration enhances both local shear (tangential) stress and pressure (normal stress) by the mechanical movement, such as rotating, oscillating or vibrating systems, which produce complex perturbations in the filtration system [START_REF] Jaffrin | Dynamic filtration with rotating disks, and rotating and vibrating membranes: an update[END_REF][START_REF] Ding | Dynamic Filtration with Rotating Disks, and Rotating or Vibrating Membranes[END_REF]. Therefore, the hydrodynamics within the filtration unit needs to be identified in order to understand then to estimate the filtration performance [START_REF] Bouzerar | Influence of Geometry and Angular Velocity on Performance of a Rotating Disk Filter[END_REF][START_REF] Fillaudeau | Investigation of rotating and vibrating filtration for clarification of rough beer[END_REF][START_REF] Fillaudeau | Etude hydrodynamique d'un module de filtration dynamique[END_REF][START_REF] Akoum | An hydrodynamic investigation of microfiltration and ultrafiltration in a vibrating membrane module[END_REF][START_REF] Xie | Local hydrodynamic investigation by PIV and CFD within a Dynamic filtration unit under laminar flow[END_REF].

High and stable permeate flux (J) is linked with the local shear rate at the membrane surface, and the empirical equation was promoted as = [START_REF] Akoum | An hydrodynamic investigation of microfiltration and ultrafiltration in a vibrating membrane module[END_REF][START_REF] Jaffrin | A hydrodynamic comparison between rotating disk and vibratory dynamic filtration systems[END_REF][START_REF] Frappart | Effect of vibration frequency and membrane shear rate on nanofiltration of diluted milk, using a vibratory dynamic filtration system[END_REF][START_REF] Ding | Milk protein concentration by ultrafiltration with rotating disk modules[END_REF][START_REF] Li | Recovery of linseed oil dispersed within an oil-in-water emulsion using hydrophilic membrane by rotating disk filtration system[END_REF][START_REF] Brou | Dynamic microfiltration of yeast suspensions using rotating disks equipped with vanes[END_REF][START_REF] Al-Akoum | Microfiltration and ultrafiltration of UHT skim milk with a vibrating membrane module[END_REF]. There is a break between two different regimes, above which the shear rate becomes more important as if the switch from laminar to turbulent regime [START_REF] Akoum | An hydrodynamic investigation of microfiltration and ultrafiltration in a vibrating membrane module[END_REF][START_REF] Jaffrin | A hydrodynamic comparison between rotating disk and vibratory dynamic filtration systems[END_REF]. Among the rotating system, four different flow patterns were proposed due to the rotating disk [START_REF] Cheng | Extensive review about industrial and laboratory dynamic filtration modules: scientific production, configurations and performances[END_REF]. When there is a narrow gap s between the rotor and the membrane, the boundary layers are merged together so that a continuous variation of tangential velocity in the gap. The limiting layer comes to separate in a larger gap, in which the fluid core is rotating at angular velocity k•2πN, k is the core velocity coefficient. Considering the mixing Reynolds number Remixing, the merged and separate boundary layers will result in different local shear rate expression at the membrane surface in the laminar or turbulent flow [START_REF] Bouzerar | Influence of Geometry and Angular Velocity on Performance of a Rotating Disk Filter[END_REF][START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF][START_REF] Ketola | Pressure, frictional resistance, and flow characteristics of the partially wetted rotating disk[END_REF][START_REF] Rudniak | Influence of hydrodynamics of rotary dynamic filters on separation processes[END_REF][START_REF] Bouzerar | Concentration of ferric hydroxide suspensions in saline medium by dynamic cross-flow filtration[END_REF].

There has been a lot of attention to improving k by increasing the roughness of the rotor or the modification of the rotating system in order to intensify shear-enhanced filtration. The flow between a stationary and a rotating disk system was early studied by Wilson et al., who indicated a k value of 0.31 for the two infinite disks [START_REF] Wilson | Flow between a stationary and a rotating disk with suction[END_REF]. A similar result obtained by Bouzerar et al. [START_REF] Bouzerar | Influence of Geometry and Angular Velocity on Performance of a Rotating Disk Filter[END_REF] showed a k value is 0.32 with a 3 mm gap of Plexiglas disk in the turbulent regime. By changing the gap, number and/or width of vans, k increases from 0.44 for a smooth disk to 0.84 for an 8 mm gap and equipped with eight pairs of 6 mm vans [START_REF] Bouzerar | Influence of Geometry and Angular Velocity on Performance of a Rotating Disk Filter[END_REF][START_REF] Jaffrin | A hydrodynamic comparison between rotating disk and vibratory dynamic filtration systems[END_REF][START_REF] Brou | Dynamic microfiltration of yeast suspensions using rotating disks equipped with vanes[END_REF][START_REF] Ding | High shear skim milk ultrafiltration using rotating disk filtration systems[END_REF][START_REF] Chen | Removal of Cd (II) from dilute aqueous solutions by complexation-ultrafiltration using rotating disk membrane and the shear stability of PAA-Cd complex[END_REF]. With the same theory, a three-blade impeller at a 3 mm gap in the RVF module gave a k equal to 0.71 [START_REF] Fillaudeau | Investigation of rotating and vibrating filtration for clarification of rough beer[END_REF].

The contribution of rotation is not only the high shear rate to limit the cake layer, but also the increased transmembrane pressure. Fillaudeau et al. [START_REF] Fillaudeau | Etude hydrodynamique d'un module de filtration dynamique[END_REF] found at a 50 Hz mixing rate, the additional pressure generated by mixing could reach up to 900 mbar with the RVF module. However, this mean pressure fluctuates on the time scale, limited knowledge about that instantaneous pressure is reported in the literature [START_REF] Cheng | Extensive review about industrial and laboratory dynamic filtration modules: scientific production, configurations and performances[END_REF]. Therefore, it is necessary to investigate the fluctuating pressure at the membrane surface during the rotor rotation, which requires fast and accurate measurements. The physical signal of the unsteady pressure contains enough information about the fluid flow with respect to different operating conditions [START_REF] Meng | Experimental study on instantaneous pressure fluctuation time series in the novel tank agitated by multiple horizontal jets[END_REF][START_REF] Hirata | Instantaneous pressure measurement on a rotating blade of a cross-flow impeller[END_REF][START_REF] Wang | Effects of flow rate and rotational speed on pressure fluctuations in a double-suction centrifugal pump[END_REF][START_REF] Okhovat-Alavian | Investigating the flow structures in semi-cylindrical bubbling fluidized bed using pressure fluctuation signals[END_REF][START_REF] Kang | Pressure fluctuation and surface morphology induced by the high-pressure submerged waterjet confined in a square duct[END_REF].

Stress-enhanced filtration is defined as the mechanical movement to cause a high shear rate at the membrane surface. This technology will generate unsteady flow and result in the fluctuations of transmembrane pressure. In the present study, the RVF module equipped with a specially designed cell to achieve accurate measurements of the instantaneous pressure at the membrane surface. The characteristics of fluctuating pressure were analysed on time and frequency domain for different operating conditions under turbulent flow, then compared with the model established by a sinusoid wave and random component. Another focus of this work is the power consumption of the loop. The empirical relationships about the global pressure drop of RVF and the net power of the rotor were established, respectively, allow an easy estimation for the energy demand for the processing.

Materials and methods

Experimental set-up and instrumentation

RVF module

The lab-scale RVF module [START_REF] Fillaudeau | Investigation of rotating and vibrating filtration for clarification of rough beer[END_REF][START_REF] Xie | Local hydrodynamic investigation by PIV and CFD within a Dynamic filtration unit under laminar flow[END_REF] (Fig. 1a, and b) consists of two filtration cells with a volume of 0.2 L. Four disk membranes (0.048 m 2 filtration area per membrane) can be mounted on the porous substrates, which collect permeate drained to lateral ducts. Each cell includes two crown membranes with a gap of 14 mm, between which are located a three-blade impeller rotates with the central shaft (Fig. 1e). And it is driven by a motor that can be operated up to 50 Hz. The feeding fluid comes inside RVF from the inlet at the bottom and flows through the module along with the central shaft, finally leaves from the retentate outlet at the top. In Fig. 1c andd, a home designed and instrumented porous substrate was used to measure the local pressure at the membrane surface. Eight pressure taps (2 mm) were connected to stainless tubes, with one extremity welded to the porous support and the other extremity located on the outer cell wall. The pressure taps were distributed between 26.2 mm (R1) up to 64.9 mm (R8), as indicated in Table 1. The experimental set-up is displayed in Fig. 2; it includes a feed tank, a circulation loop and the RVF module Fig. 1. In the circulation loop, water was pumped from a double-jacket tank (2L), including thermal regulation at flowrate ranging from 50 to 300 L/h. Flowrate was controlled by a volumetric pump (Pump, TUTHILL, PM8014, 3800 rpm, 169 W) and acquired with a mass flowmeter (MF, KROHNE, Optimass 70000 S06), it enabled the measurement of mass flowrate (0/900 kg/h, ±0.1% liquid, ±0.1% gas), density (500/2000 kg/m 3 , ±2 kg/m 3 ) and temperature (0/130 ℃, ±1 ℃, was associated with outlet temperature). The inlet temperature was recorded from the conductivity sensor, Cond (Conductimetre Conducell 4USF PG 325, -20/150 ℃, ±0.5 ℃), in the feeding tank. The back pressure in RVF was adjusted by a counter-pressure valve coupled with a pressure gauge (PG, 0/4 bar) and a relative pressure sensor (PR1, Bourdon-Haenni Y913, 0/6 bar, ±0.2% full scale) located in the outlet. Before the experiments, the back pressure was maintained at 300 mbar to avoid cavitation caused by the high mixing rate. A Tachymeter (Tachymeter Testo 460, 100-30000 rpm) was used to adjust the mixing rate (N) from 0 to 50 Hz. Both current (CU, Ammeter, LEM Co. AC current transducer AT-B10, 0/20 A) and tension (TEN, Voltmeter, SINEAX U 504 31 LD, 0/250 V) of the motor were recorded for all the conditions. The differential pressure sensors (DP, HONEYWELL-STD 120, 0/1 bar, ±0.0375% full scale) were used to determine the pressure drop and local pressure of RVF. The relative pressure (PR2, Killer, -1/+1 bar, ±0.2% full scale, maximum acquisition frequency 5 kHz) was measured at the membrane surface. Global measurements along the circulation loop correspond to differential pressure (DP1, located between the inlet and outlet; DP2, installed between the local radius and outlet), flowrate, temperature, current and tension. All the sensors were connected to a data acquisition system (Agilent 34972A, Agilent Technologies, Loveland, USA) with a multiplexer acquisition card (34901A, 20 channels). After 2 min stabilisation, all electrical signals were recorded at 5 s intervals for 3 min.

Instantaneous pressure was measured with PR2 (shown in the blue line in Fig. 2) at 1000 Hz, connected to the local radius. The pressure signal was recorded with NI USB-6009 (National Instruments, USA, 1 kHz) at a sampling frequency of 1000 Hz for more than 40 s.

Global and semi-local analysis: pressure drop, mixing pressure and core velocity coefficient

For the dynamic filtration devices, global and mean local values of pressure, velocity, and shear rate in the filtration cell have been described previously [START_REF] Fillaudeau | Investigation of rotating and vibrating filtration for clarification of rough beer[END_REF][START_REF] Fillaudeau | Etude hydrodynamique d'un module de filtration dynamique[END_REF][START_REF] Xie | Local hydrodynamic investigation by PIV and CFD within a Dynamic filtration unit under laminar flow[END_REF][START_REF] Xie | Investigation of Local and Global Hydrodynamics of a Dynamic Filtration Module (RVF Technology) for Intensification of Industrial Bioprocess[END_REF]. From a global standpoint, RVF modules can be assimilated to a hydraulic singularity generating a pressure drop and as a mixing device. Linear pressure drops in the RVF module (ΔPRVF) is attributed to the friction loss, which can be expressed by Euler dimensionless number, Eu, given by Eq. ( 1).

= ∆ 1/2 (1)
where ΔPRVF was measured by DP1, ρ is the fluid density, u is the velocity at the inlet of RVF module. The Reynolds number for feeding (Refeeding) and mixing (Remixing) are defined as follows:

= (2) = ( 3
)
where d is the inlet diameter (12 mm), di is the diameter of the rotating impeller, μ is dynamic viscosity, N represents the mixing rate of the impeller. Two methods were used to measure power consumption. The first case is given by the current and tension of the motor. The electrical power needs to drive the shaft were measured without fluid, which is considered shaft loss. Thus, the net power ΦN consumed by the rotating impeller is equal to the difference between total power and shaft losses [START_REF] Brou | Dynamic microfiltration of yeast suspensions using rotating disks equipped with vanes[END_REF]. It can be described by Power number Np in Eq. ( 4). Another case is achieved by the thermal dissipation ΦT of water in the filtration module and given by Eq. ( 5).

= ! " # $ (4)
! % = & ' ( )* + , -* ./,+ , 0 (5)
where Tinlet and Toutlet are the inlet and outlet temperature, respectively, QF is the feeding flowrate, and CP is the specific heat capacity of water.

Since the impeller is installed with a narrow gap to the membrane, assuming the inviscid core layer rotates at an angular velocity of k•2πN. According to Bernoulli's equation in Eq. ( 6) [START_REF] Bouzerar | Influence of Geometry and Angular Velocity on Performance of a Rotating Disk Filter[END_REF], the mean local pressure 1 ) , 30 at the membrane surface equals the sum of P0 and ΔPmixing. P0 is the pressure at the centre of membrane or be given by the pressure when the absence of mixing, while ΔPmixing is the additional pressure driven by the rotating impeller.

1 ) , 30 = 4 5 ∆ = 4 5 1 2 )6 • 28 0 3 ( 6
)
where r is the radius at the membrane surface.

Local analysis: instantaneous pressure

Three blades impeller was used in the experiments, which result in a 150 Hz signal when rotating at 50 Hz. The instantaneous pressure has been measured with a 1000 Hz sampling frequency. Considering the pressure fluctuations as a result of vibrating flow, the instantaneous pressure can be expressed as Eq. ( 7), with 1 ) , 30 and 9 ) , 3, :0 represent the mean and fluctuating pressure, respectively. Fig. 3a and b illustrate the time evolution of absolute pressure and its fluctuating component.

Pressure oscillation and its amplitude can be treated by (i) statistical analysis, SA; (ii) probability distribution function, PDF and (iii) Fast Fourier transform, FFT.

) , 3, :0 = 1 ) , 30 5 9 ) , 3, :0

Statistical analysis

The standard deviation of the signal has been widely accepted to quantify fluctuating intensity [START_REF] Okhovat-Alavian | Investigating the flow structures in semi-cylindrical bubbling fluidized bed using pressure fluctuation signals[END_REF][START_REF] Johnsson | Characterization of fluidization regimes by time-series analysis of pressure fluctuations[END_REF][START_REF] Wassie | Detecting densified zone formation in membrane-assisted fluidized bed reactors through pressure measurements[END_REF]. In our condition, a total number of sampling (m=2 15 ) is acquired at a constant time interval (1 ms), as illustrated in Fig. 3b. The mean pressure 1 ) , 30 and standard deviation ; ( are expressed as the moment of first order and the square root of the central moment of second order, respectively. Their mathematical definitions for continuous and discrete functions are given in Eq. ( 8) and [START_REF] Fillaudeau | Etude hydrodynamique d'un module de filtration dynamique[END_REF]. The coefficient of variation, < is defined as the ratio between ; ( and 1 ) , 30, and give the relative standard deviation.

The normalised central moment of the third order, known as skewness (S), determines the symmetry of signal in the probability distribution, and zero means a symmetrical distribution.

F = 1 *; ( # = ) ) , 3, :0 -1 ) , 300 # : % 4 > 1 ?; ( # @) ) , 3, : 0 -1 ) , 300 # AB ( 1 1)
The flatness (F) is represented by the normalised central moment of the fourth order, indicates the sharpness of distribution.

G = 1 *; ( H = ) ) , 3, :0 -1 ) , 300 H : % 4 > 1 ?; ( H @) ) , 3, : 0 -1 ) , 300 H AB (12)

Probability distribution function

Another method for obtaining the intensity of fluctuating pressure can be carried out with probability analysis [START_REF] Meng | Experimental study on instantaneous pressure fluctuation time series in the novel tank agitated by multiple horizontal jets[END_REF]. By subtracting the average pressure, the deviation signal is divided into 100 classes considering 2 15 of raw data. The probability of each class can be calculated based on its occurrence. In Fig. 3c, the probability is illustrated against the fluctuating pressure. The unimodal (dominant random contribution, cf.

§3.3.4) distribution of probability is observed at 5 Hz while extended to bimodal (dominant periodic contribution) at 20 Hz. These PDF widths are defined by peak-to-peak differences divided by 2, (P2-P1)/2, finally resulting in the pressure intensity of 55.5 mbar at 20 Hz.

Fast Fourier transform (FFT)

The Fast Fourier Transform decomposes a signal into a series of sinusoid waves to be analysed and given by the frequency domain signal. It has been used to extract information about fluctuation (amplitude and frequency) from the time-series signal. Continuous and discrete Fourier transform can be represented by:

)I0 = = 9 ) , 3, :0 JK L , : MN JN > @ 9 ) , 3, : 0 J K L JB A4 , I = 0,1, … , ? -1 (1 3)
where 9 ) , 3, : 0 is the deviation of pressure at point i, f is the frequency. This formula is associated with the complex plane, composed of real and imaginary parts, and its amplitude (A) can be expressed as follows:

P = 2 ? Q )I0 (14)
The time-dependent fluctuating pressure can be converted into the frequency domain with FFT [START_REF] Hirata | Instantaneous pressure measurement on a rotating blade of a cross-flow impeller[END_REF][START_REF] Wang | Effects of flow rate and rotational speed on pressure fluctuations in a double-suction centrifugal pump[END_REF][START_REF] Okhovat-Alavian | Investigating the flow structures in semi-cylindrical bubbling fluidized bed using pressure fluctuation signals[END_REF][START_REF] Kang | Pressure fluctuation and surface morphology induced by the high-pressure submerged waterjet confined in a square duct[END_REF][START_REF] Johnsson | Characterization of fluidization regimes by time-series analysis of pressure fluctuations[END_REF]]. 2 15 sample points were chosen to get a sufficient precision of the fluctuated signal. The amplitude and frequency, driven by the rotating impeller, are discussed in the next part. Fig. 3d shows the typical result of FFT at 20 Hz. The peak amplitude (A) is defined as the intensity of fluctuation pressure.

Reconstruction of PDF with periodic and random functions

The fluctuating signal consists of two parts: periodic and random contributions (Table 2). The periodic signal can be simplified as a single sinusoidal fluctuation with f=3N, whose intensity is determined by the amplitude A. Since only one peak is considered, the A presented here is higher than the FFT amplitudes at the same frequency. The random signal conforms to a normal distribution with a zero mean, and σ denotes the standard deviation. The energy input in each contribution is given by their root mean square (RMS).

):0 = PRST)28I: 5 U0 A, f=3N, φ ( )V0 = 1 8√P -V P √2 ⁄ Random ):0~ )V̅ , ; 0 V̅ = 0, σ )V0 = 1 ;√28 J B ) J ̅ [ 0 \ σ
In theory, the continuous model of pressure fluctuation distribution results from the convolution of EP(x) with ER(x) as reported in Eq. [START_REF] Li | Recovery of linseed oil dispersed within an oil-in-water emulsion using hydrophilic membrane by rotating disk filtration system[END_REF]. Comparison between experimental data and model is realised by identifying amplitude, A for periodic component and standard deviation, σ for random term. Considering the statistical convergence, phase lag has no effect on PDF building (cf. §3.3.1). Both optimal parameters, A and σ are obtained by minimising the cumulative error function, Δ with Eq.( 16), thanks to Excel solver (Suite Office Microsoft 2013, Excel, GRG non-linear) for each operating condition. Fig. 4 illustrated the experimental and simulated PDF at R6 and 20 Hz.

]G = ( )V0 * )V0 (15)
_ = `ST a@ b) V 3S? T: -`c d0² B44 A4 f (16)
Fig. 4 PDF at 20 Hz. Model reconstruction of PDF with periodic and random functions.

Results and discussion

Global approach

3.1.1 Pressure drops (Eu vs Re) At the process scale, the RVF module can be identified as a hydraulic singularity (generating pressure drop) or a confined mixing device (power consumption). The evolution of Euler number versus feeding (800<Refeeding<9000) and mixing (10 3 <Remixing<10 6 ) Reynolds numbers are shown in Fig. 5a. In the absence of rotation, Eu(N=0) shows a decrease with increasing Refeeding from 41 to 34, which tends to be negligible at a higher mixing rate. That is consistent with the friction curve established by Fillaudeau et al. [START_REF] Fillaudeau | Etude hydrodynamique d'un module de filtration dynamique[END_REF] in the turbulent flow regime. By increasing the mixing rate, Eu rises dramatically with Remixing; the magnitude of Eu increase is inversely proportional to the flowrate. For process engineer and scaling, it appears useful to establish simple semi-empirical correlations to estimate the pressure drop and power consumption in such a module. Subtracting Eu(N=0), the Euler number increment is plotted as a function of Remixing/Refeeding in Fig. 5b. Two domains are found by the regression of data, giving two semi-empirical correlations as:

F =50 L/h Q F =100 L/h Eu(N) (/)
)"0 -)"A40 = 0.021 C ) hijikl mnnoikl 0 B.pq hijikl mnnoikl < 100 ( 17
)
)"0 -)"A40 = 0.222 C ) hijikl mnnoikl 0 B.#$ hijikl mnnoikl ≥ 100 (18)
These correlations have been validated for the feeding and mixing condition (10 3 <Refeeding<1.2×10 4 , and 0<Remixing<10 6 ).

Power consumption (Np vs Remixing)

Power consumption is a critical issue in evaluating the overall performance of a rotating dynamic filtration device. For RVF module, the total energy consumption includes the pumping and mixing power. Assuming the highest feeding flowrate at 300 L/h and back pressure of 300 mbar, the pumping power can be estimated to be 2.5 W. This can be negligibly in comparison to the mixing power for N≥20 Hz (ΦN≈400 W at 50 Hz).

In the small-scale filtration module, the mixing power without load (Pf) cannot be neglected due to the friction of the shaft. In the present case, the contribution of the power consumed by the impeller is inferior to Pf at the mixing rate below 20 Hz. As shown in the power consumption curve (Fig. 6a), the grey area indicates the excessive mechanical losses in the rotating shaft. Np is independent of Remixing when N≥20 Hz, and remains around 0.1.

In Fig. 6b, the mixing power was estimated thanks to 3 approaches: (i) power consumption curve (ΦN), (ii) thermal balance (ΦT) and (iii) investigation of local shear stress (ΦM). Power consumption and thermal balance show a good agreement, and the thermal dissipation of fluid constitutes 87.73% of the electrical power consumed by the mixing, the rest of which may be attributed to the dissipation of the cell wall. In a turbulent regime, the boundary layers merge together at a narrow gap (3 mm), and the local shear stress τ on the rotating disk in Eq. ( 19) illustrates the linear relation with N 1.75 [START_REF] Bouzerar | Concentration of ferric hydroxide suspensions in saline medium by dynamic cross-flow filtration[END_REF][START_REF] Bouzerar | Filtration dynamique dans un module plan à disque rotatif : Application à des suspensions minérales[END_REF]. As mentioned by Brou et al. [START_REF] Brou | Dynamic microfiltration of yeast suspensions using rotating disks equipped with vanes[END_REF], the mechanical power ΦM generated by the friction force on the plate disk is calculated by Eq. ( 20). In our case, ΦM varies linearly with the nominal power and is underestimated (24.06%). If the impeller surface area is considered, this ratio reduces to 12%. It demonstrates that the local shear stress on the blades only corresponds to a minor fraction of the torque. The major contribution of driving force contributes to the pressure difference between the leading and trailing edge of blades, which need to be further investigated.

t = 0.008 4.v$ )28 • 30 B.v$ ) R 0 4. $ (19)
! w = 2 = 2t i x )28 • 30)2830 3 = 6.632 4.v$ { R | 4. $ ) H.v$ -4 H.v$ 0 .v$ (20)

Semi-local approach: analysis of core velocity coefficient

The local pressure at the membrane surface was measured at different radius (26.2 to 64.9 mm), mixing rates (0 to 50 Hz) and flowrates (50 to 300 L/h). According to Eq. ( 6), the mixing pressure is plotted in Fig. 7, and parabolic variation with the mixing rate can be observed. At 50 L/h, the mixing pressure rises as the radius increases in terms of impeller tangential velocity. By increasing the feeding flowrate, a small increase in pressure can be observed due to the velocity generated by feeding. Fig. 8a shows the regression of mixing pressure as a function of N 2 r 2 from 50 to 300 L/h. The great correlation indicates a core velocity coefficient equals to 0.65, which is slightly lower than 0.71, reported by Fillaudeau et al. [START_REF] Fillaudeau | Etude hydrodynamique d'un module de filtration dynamique[END_REF]. Previously, the mixing pressure at a defined radius was measured with the permeable crowns. With large gaps (5 mm) in the annular cavity, the k value was overestimated using the mean radius. Fig. 8b demonstrates the k values vary with radius, flowrate and mixing rate.

The highest k values are observed at R1. They increase by 21% with flowrate between 50 and 300 L/h, while only 3% at the boundary (R8). The flowrate is more important at the inlet due to the small cross-section related to high radial velocity. Similar to the rotating disk module [START_REF] Bouzerar | Influence of Geometry and Angular Velocity on Performance of a Rotating Disk Filter[END_REF], the k value decreases with the increment of radius and further improves when reaching the edge of the disk. These variations with impeller are higher than the full disk, reaching 10% at 300 L/h, which may be explained by the shape of the blades.

The ideal conditions based on a full disk system assumes that mean local velocity (horizontal) results from radial and tangential velocities. Ur(r) is determined by feeding flowrate and local cross-section, and Uθ(r) by mixing rate and core velocity coefficient (Eq. ( 21)).

} ~• )30 = )6 ,€ . 28 30 5 • & 283R ' = ƒ6 " 28 3… (21)
For example, the radial velocity at R1 is around 0.028 m/s for 50 L/h, and the impeller angular velocity is equal to 0.823 m/s at 5 Hz. Under this condition, the ratio between radial and angular velocities is limited to 3%. At the highest feeding flowrate (300 L/h), this ratio reaches more than 20%. It indicates that the contribution of feeding is more important with the lowest radii and mixing rates, as described by Eq. ( 21). As shown in Fig. 8b, the k values tend to decrease with the rise of mixing rate and radius. In our conditions, a plateau value (∼0.6) appears for a radius superior to 45 mm. However, the slight k increase is not clear yet. Local approach: Instantaneous pressure at the membrane surface

Statistical analysis

As preliminary verification, the evolutions of 1 ) , 30, ; ( , S and F as a function of sampling number (Fig. 9) demonstrate that statistical convergence of raw data is reached for m>1000. The sampling number m (2 15 ) may be sufficient to be analysed The standard deviation (σP) of fluctuating pressure versus mixing rate and radius is shown in Fig. 10a. It is interesting to find that the variation is highly dependent on the mixing rate. The maximum fluctuation occurs in the range of mixing rate from 15 to 30 Hz. And the fluctuating pressure strongly increases with a higher radius when r>54 mm. In contrast, at other mixing rates, these deviations are limited to less than 20 mbar.

The filtration performance can be described by Darcy's law, as displayed in Eq. [START_REF] Bouzerar | Concentration of ferric hydroxide suspensions in saline medium by dynamic cross-flow filtration[END_REF]. In dynamic filtration, membrane fouling is controlled by the local shear stress at the membrane surface, limiting the increase in total hydrodynamic resistance (Rh). The pursuit of higher permeate flux requires to apply optimal operating conditions by considering the continuous and fluctuating contributions of transmembrane pressure and their radial distribution over filtration surface. Most of the time, transmembrane pressure was given as the mean value. Another contribution of the driving force is the fluctuating pressure, which has not been reported in the literature, whereas its contribution cannot be neglected. In Fig. 10b, the fluctuating component (σP) constitutes more than 10% of transmembrane pressure at 20 Hz for most of the radius, even reaching 25.3% at R8. The grey area (15 to 30 Hz) will be a good choice to intensify the fluctuating magnitude. Thus, as the maximum σP reach 100 mbar, a minimum counter pressure of 300 mbar (3σP) will be requested to avoid membrane detachment. Similar to ΔPmixing, a semi-empirical correlation to describe σP will be useful to estimate local transmembrane pressure. ) , 3, :0 = 1 ) , 30 5 9 ) , 3, :0 The skewness distribution of fluctuating pressure at different conditions are given in Fig. 11a. The skewness number varies from -0.3 to 0.3, showing a good symmetry of the distribution. The flatness indicates the degree of peakedness of distribution and displayed in Fig. 11b. It is concluded that the flatness increases as the mixing rate applied below 7.5 Hz, and followed by a decrease until 20 Hz. The relative lower mixing rate results in flatness maintain in the range from 5 to 35, which means the excessive sharpness of PDF. Above 20 Hz, most cases (r>45.3 mm) give the flatness inferior to 3, which indicates the great extension of PDF to large fluctuating amplitude. However, a small increase of F can be observed at a lower radius (r<45.3 mm). Furthermore, the flatness decreases with increasing radius at the same mixing rate. At 20 Hz, F reaches its minimum value, corresponding to the maximum fluctuating pressure. Centred moments such as skewness and flatness are known for normal and sinusoid wave distributions but can hardly be compared with experimental data. Skewness is not significant due to symmetric distribution. Flatness appears as an appropriate qualitative criterion to discriminate the fluctuating contribution as a function of the mixing rate. Fig. 12a shows the probability distribution of fluctuating pressure from 0 to 50 Hz at R6. When the mixing rate is below 10 Hz, the spectrum displays as unimodal, confined to a narrow fluctuating signal. When N>10 Hz, the peak at the zero deviations sinks downward and expands to the sides, forming a bimodal and reaching the highest deviation at 20 Hz. Nevertheless, The PDF at 10, 40 and 45 Hz can be defined as the transition zone. In general, the probability distribution of a sinusoidal signal has a U-shaped structure, while a random signal tends to be normally distributed. The transition from unimodal to bimodal will occurs with the increased contributions of the sinusoidal wave relative to the random component.

Considering the peak-to-peak values in Fig. 12b, the fluctuating pressure evolves with the mixing rate and radius. The unimodal region occurs below 10 Hz, whose amplitude cannot be achieved due to only one peak value observed. The dash lines demonstrate the transition areas in the range from 10 to 17.5 Hz and 30 to 50 Hz. However, at a large radius, such as R8, the intensive fluctuation (bimodal) happens at N>10 Hz and reaches the maximum peak-to-peak value of 144 mbar at 20 Hz.

Fast Fourier transform

Statistical analysis and probability distribution function provides a global overview of fluctuating pressures in the time domain. However, the characterisation and decomposition of the signal primarily rely on frequency domain analysis. Empirical Mode Decomposition (EMD) decomposes the signal into Intrinsic Mode Functions (IMF). Then, via the Hilbert Transform, the instantaneous frequency of the data is properly identified, commonly used in the analysis of non-stationary and non-linear signals [START_REF] Meng | Experimental study on instantaneous pressure fluctuation time series in the novel tank agitated by multiple horizontal jets[END_REF][START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[END_REF]. For periodic signals, frequencies remain constant over the time range, and hence the FFT is preferred [START_REF] Wang | Effects of flow rate and rotational speed on pressure fluctuations in a double-suction centrifugal pump[END_REF][START_REF] Kang | Pressure fluctuation and surface morphology induced by the high-pressure submerged waterjet confined in a square duct[END_REF].

After FFT, the periodic signal generated by mixing with a three-blade impeller includes the main contributions of the sinusoidal wave at N, 2N, 3N, 4.25N and 6N. Fig. 13a shows that the amplitudes for each component vary as N increase. The cumulative amplitude at 20 Hz tends to be the highest and followed by 17.5 and 22.5 Hz, which constitute the most intensive fluctuation. Below 10 Hz, the amplitude Increasing the mixing rate, other fluctuating contributions can be observed, especially a small increase of amplitude at N after 20 Hz. It should be noted that 3N shows to be the most important wave from 60% up to 90% at 17.5 and 20 Hz. In Fig. 13b, the main contribution of amplitudes (f=3N) is plotted as a function of mixing rate. The sharp increase of amplitude is found below 20 Hz, then fall to a constant value between 20 and 30 Hz. Above 30 Hz, the increase of mixing rate has little effect. It is likely that the amplitude increases with the radius, especially at r>53.9 mm. However, the amplitude evolution almost follows the same curve for R7 and R8, which is different from σp (Fig. 10a) and PDF amplitudes (Fig. 12b).

Model of fluctuating component and associated energy input

Modelling and quantifying periodic and random contributions of fluctuating pressure have been determined by identifying amplitude, A and standard deviation, σ (cf. §2.3.4). The total energy input and the weight of the periodic part are given by the sum of periodic (P √2 ⁄ ) and random (σ) intensities, and the ratio between periodic and total energies (

(M = † √ ⁄ † √ ⁄ M[
0, respectively. In Fig. 14a, the total energy evolves with radius and mixing rate. It is interesting to find that a rectangular area (17.5<N<25 Hz and 34.8<r<64.9 mm) corresponds to the extensive energy input over 40 mbar. In addition, the same zone can be observed at r>53.9 mm and 17.5<N<45 Hz. It includes the maximum energy of more than 100 mbar at 20 Hz for R8.

For the random signal, the standard deviation varies in the range from 5 to 22 mbar. While the amplitude for the periodic signal changes from 1 to 107 mbar, and is highly dependent on the mixing rate and radius. Fig. 14b shows the map of the ratio between periodic and total energy input for the fluctuating pressure. It indicates that periodic signal is dominant for most cases and contributes to more than 60% in a trapezoidal area from 12.5 to 45 Hz. The maximum ratio happens between 15 and 30 Hz, which is the same as the intensive fluctuation area given by statistical analysis. The evolution of fluctuating pressure for three strategies and modelling are compared in Fig. 15. From a global perspective, the deviations of the curves almost share the same trends with increasing mixing rate. The results from PDF are higher than other methods, around 1.04 (±0.17) times of σp above 15 Hz. Nevertheless, the former case contains information about fluctuations in all ranges, which can be useful in signal reconstruction. Meantime, the amplitude from FFT at 3N indicates 0.72 (±0.18) times of σp. This ratio increases to 0.9 if the contribution of other peaks (N, 2N, 4.25N and 6N) are considered. When it comes to the model, fluctuation intensity is composed of the period and random contributions. The total energy input is higher than σp and reaching 126%. As shown as a grey zone in Fig. 15,the

(Hz) R m R 8 R 7 R 6 R 5 R 4 R 3 R 2 R 1 R 0 21
Fig. [START_REF] Li | Recovery of linseed oil dispersed within an oil-in-water emulsion using hydrophilic membrane by rotating disk filtration system[END_REF] The evolution of pressure deviations (standard deviation, peak-to-peak/2, amplitude from FFT at 3N and total energy for modelling) as a function of mixing rate at R6.

Conclusions

Dynamic filtration enables to reduce fouling and to enhance permeate flux by a mechanical movement with rotating oscillating and/or vibrating. These mechanical configurations generate complex flow pattern with local fluctuation at the membrane surface. Therefore, the hydrodynamics within the filtration unit needs to be investigated from a global up to a local and instantaneous standpoint in order to understand the filtration performances. The present work introduces the methodology to investigate instantaneous and local transmembrane pressure within Rotating and Vibrating Filtration (RVF) module under a turbulent regime. A preliminary global approach is based on classical mixing power and pressure drop measurements; then, semi-local and local approaches interpret and describe local and instantaneous pressure at the membrane surface using alternative strategies.

From global and semi-local approaches, friction and mixing power in the RVF module are described by semi-empirical correlations based on mixing and feeding conditions. The generalised correlations between Euler and mixing effect (Remixing/Refeeding) are proposed to estimate the pressure drop. Pumping power can be easily achieved with feeding and back pressure; it demonstrates that the power for feeding can be neglected compared with mixing over 20 Hz. The balance between heat dissipation and net mixing power exhibits an accurate linear relation (slope=0.88). However, the calculated power based on the shear of the impeller surface is strongly underestimated. The major effect of pressure driving force between the leading and trailing edge at the blades needs to be specified.

The semi-local approach contributes to estimate and model the continuous component of the filtration driving force ∆Pmixing. The evolution of mixing pressure depends mainly on radius and mixing rate, rarely on flowrate. Due to the radial velocities generated by feeding flowrate, the integration of mixing pressure with N 2 r 2 yields the experimental core velocity coefficient kexp superior to the theoretical value ktheo. The great contribution of radial velocity on the kexp can be observed at the lowest Q F =50 L/h R6=53.9 mm radius R1 and increase with flowrate, but decrease with mixing rate and radius. The ratio between radial and angular velocity is easily determined by simple correlations in the rotating disk system, which can be extended to the conditions of the rotating impeller. However, the reason for the small increase in k value at a large radius (r>45 mm) is not clear. Instantaneous and local pressure at the membrane surface was deeply scrutinised by comparing analytical methodology (SA, PDF, FFT and modelling) in order to understand and to estimate the fluctuating component of driving force. The intensive fluctuation area for three methods appears in the same range of mixing rate from 15 to 30 Hz, and increase with the radial position. The important contribution of fluctuating pressure shows great potential for large-scale application. On frequency domain, the prominent peaks of FFT include N, 2N, 3N, 4.25N and 6N, where the amplitude at 3N is indicated as the dominant position (from 60 up to 97%). The modelling can be established by the random signal and sinusoid wave (f=3N), which related to the PDF evolution from unimodal (random) to bimodal (periodic) with mixing rate. It can be concluded that the periodic component is more sensitive to the mixing rate relative to the random part.

Future works will apply this methodology to compare and to screen different impellers (design and selection). This strategy can be extended to the laminar regime with Newtonian and Non-Newtonian fluids. The measurement and analysis of instantaneous and local pressure stood as a knowledge gap to understand the performances of the dynamic filtration module. In this aim, the instantaneous and local wall shear rate could be investigated by local measurements at the membrane surface (electrochemical technique) and compared with velocity field (PIV, particle image velocimetry).

1

 1 Introduction .................................................................................................. 4 2 Materials and methods .................................................................................. 5 Experimental set-up and instrumentation .............................................. 5 2.1.1 RVF module.................................................................................. 5 2.1.2 Experimental set-up ...................................................................... 6 2.1.3 Experimental measurement and data acquisition ......................... 7 Global and semi-local analysis: pressure drop, mixing pressure and core velocity coefficient ................................................................................................. 7 Local analysis: instantaneous pressure .................................................. 8 2.3.1 Statistical analysis ........................................................................ 9 2.3.2 Probability distribution function ................................................ 10 2.3.3 Fast Fourier transform (FFT) ..................................................... 10 2.3.4 Reconstruction of PDF with periodic and random functions ..... 11 3 Results and discussion ................................................................................ 12 Global approach .................................................................................. 12 3.1.1 Pressure drops (Eu vs Re) .......................................................... 12 3.1.2 Power consumption (Np vs Remixing) ........................................... 13 Semi-local approach: analysis of core velocity coefficient ................. 14 Local approach: Instantaneous pressure at the membrane surface ...... 15 3.3.1 Statistical analysis ...................................................................... 15 3.3.2 Probability distribution function ................................................ 17 3.3.3 Fast Fourier transform ................................................................ 18 3.3.4 Model of fluctuating component and associated energy input ... 19 3.3.5 Comparison of fluctuating intensity ........................................... 20 4 Conclusions ................................................................................................ 21 Nomenclature ....................................................................................................... 23

Fig. 1

 1 Fig. 1 Schematic diagram of Rotating and Vibrating Filtration module. (a) RVF module; (b) one dismantled filtration cell; (c) and (d) home designed and instrumented porous substrate for local

Fig. 2

 2 Fig. 2 Experimental set-up. Mean and instantaneous pressure measurements are illustrated by green (differential pressure, DP1 and DP2) and blue (relative pressure, PR2) lines.

2. 1 . 3

 13 Experimental measurement and data acquisition Two types of measurements were performed without permeate: (i) global measurement and (ii) instantaneous and local pressure measurement. Tap water (25±5 ℃) was used as feed fluid with 4 flowrates ranging from 50 to 300 L/h. The instantaneous and local pressure was measured at 8 radii from R1 to R8, and the 15 mixing rates from 0 to 50 Hz.

Fig. 3

 3 Fig. 3 The spectrum of instantaneous pressure and data treatment. (a) instantaneous and mean pressure; (b) fluctuating pressure and its standard deviation; (c) probability analysis at 5 and 20 Hz; (d) fluctuating pressure on frequency domain at 20 Hz.

  σ P (20Hz)=42.2 mbar Q F =50 L/h R6=53.9 mm P(N,r,t) (mbar) Time (ms)

Fig. 5

 5 Fig. 5 Friction curve. (a) the evolution of Euler number versus feeding and mixing Reynolds number; (b) the regressions of Euler number increment in diverse mixing rate and flowrate.

Fig. 6

 6 Fig. 6 Power consumption curve. (a) the variation of Power number versus mixing Reynolds number; (b) thermal dissipation and mechanical power versus net electrical power.

Fig. 7 .

 7 Fig. 7. The evolution of mixing pressure as a function of radius and feeding flowrate.

Fig. 8

 8 Fig. 8 The determination of the core velocity coefficient. (a) the regression of mixing pressure at the flowrate ranging from 50 to 300L/h; (b) the evolution of core velocity coefficient versus radius, flowrate and mixing rate.

Fig. 9

 9 Fig. 9 Statistical convergence of raw data.

€Fig. 10

 10 Fig. 10 Statistic analysis. (a) the standard deviation of fluctuating pressure at different mixing rate and radius; (b) the coefficient of variation versus mixing rate.

Fig. 11

 11 Fig.11The skewness and flatness distribution at different radius3.3.2 Probability distribution function

Fig. 12

 12 Fig. 12 PDF analysis. (a) the probability distribution of fluctuating pressure at R6; (b) the evolution of PDF width (peak-to-peak/2) versus mixing rate.

  main parts: 3N and 6N, but each of them is lower than 3 mbar.

Fig. 13

 13 Fig. 13 Frequency domain analysis. (a) Evolution of cumulative amplitude from FFT versus mixing rate at R6; (b) the amplitude at 3N versus mixing rate and radial position.

20 Fig. 14

 2014 Fig. 14 Spectrum of total energy input and periodic contribution as a function of mixing rate and radius

  intensive fluctuation occurs in the range from 15 to 30 Hz.

Table 1

 1 Radial distribution of pressure taps at the membrane surface (porous substrate).

	Radial position	Radius (mm)	Radial position	Radius (mm)
	R1	26.2	R5	45.3
	R2	29.3	R6	53.9
	R3	34.8	R7	63.4
	R4	38.2	R8	64.9
	2.1.2 Experimental set-up

Table 2

 2 Signal decomposition into periodic and random contributions (PP(t): periodic signal, PR(t): random signal, A: amplitude, f: frequency, φ: phase, σ: standard deviation, EP(x) and ER(x) are the PDF.).

	Contribution	Function	Parameters	PDF (mbar -1 )	Energy input (mbar)
	Periodic	(

Table 3

 3 Information on the data treatment

	Method Mean pressure SA 1 ) , 30 PDF /	Fluctuating pressure Periodic + Random Periodic Random σp, S, F / / Peak-to-peak / /
	FFT	/	/	Ai, fi	/
	Model	/	/	A1, f1	σ

Acknowledgements .............................................................................................. 24 References ............................................................................................................ 24 Acknowledgements Financial support from the China Scholarship Council (CSC) for Ming CHENG (grant No. 201801810069) is gratefully acknowledged. Thanks to Pascal DEBREYNE and Jacky SIX (INRAE, Lille) for the realisation of the experimental device about the instantaneous and local pressure measurement.


F Deroian

Endogenous Link Strength in Directed Communication Networks

Keywords: JEL Classification Numbers: D85, C70 Directed Communication Network, Endogenous Link Strength, Efficiency, Stability

We present a model of communication network formation in which links' strengths are endogenously determined by individual incentives. Agents are endowed with a fixed amount of resource which they can distribute as they want in directed links. Individuals capture benefits from both direct and indirect access to others, in a way that takes into account the strength of each link. The wheel architecture is shown to be the unique efficient and stable architecture under mild restriction.

Introduction

The growth of communication good industries, such as the Internet, mobile phones and computers, and more generally all numeric goods, raises new economic issues * The author is at GREQAM, Université d'Aix-Marseille, 2 rue de la Charité 13002 Marseille, France. † I wish to thank F. Bloch for many valuable discussions and comments about this work.

1 related to the formation of communication networks. At least two major queries are unavoidable: how do communication networks evolve through time? Do they maximize some collective outcome? One key aspect is that network formation is not supervised by any formal institution; rather, individuals decide whether to contribute to the shaping of such networks. This is therefore important to understand the individual incentives to join such networks. A second key aspect is that in many cases agents choose how much effort to devote to communicating in the network, and the level of investment in two distinct connections is not necessarily homogenous. For instance, as visiting your favorite web sites, you may decide how much time to spend on each, and in many circumstances you would not choose an homogenous distribution of time. That is, the link strength (as measured by the level of time, effort, money) is often endogenous to individual decisions. Third, many networks are directed. For instance, this may fit situations like visiting a web site, but consider also the (directed) relationship in a firm between a manager and a subordinate. With regard to non directed networks, a generic difference govern the formation of connections: in the latter case link formation is a bilateral action, while in the former case it is a unilateral action. Put differently, the coordination problem inherent to the non directed case vanishes in the directed case.

We address the three points in a simple model of network formation. This model complements [START_REF] Bloch | Communication networks with Endogenous Link Strength[END_REF]'s work on non directed network, so our setting is close to theirs. First, agents have a fixed and infinitely separable amount of resource that they can distribute over links. This may represent some effort, time or money spent in each relationship. Second, agents capture indirect benefits from accessing others, and this benefit depends on the level of investments made by intermediaries. We suppose that if there is a (directed) path linking agent i to agent j, the value for agent i stemming from that path is a function of the investments made by all intermediaries on that path; and we assume that the benefit she obtains from agent j is the greatest value among all possible paths of finite length. In that context, we show that the wheel network is the unique efficient and stable (Nash) architecture. Our resource-constraint environment is therefore sufficient to generate a unique architecture making compatible private incentives and collective outcome. This remarkable coincidence between Nash and efficient networks is inherent to our resource-constraint environment. In our setting agents prefer to concentrate their allocation on the partners who provide them with the greatest indirect benefits. This results in the formation of the unique architecture such that agents access others through a chain of maximal length and composed of weights equal to amount X. We also present two basic extensions: we first propose a more general setting in which the results keep true, then we set up a simple dynamic process of network formation leading to the wheel architecture with probability 1. This work can be related to studies devoted to the formation of communication networks 1 , or to enlarged context of directed network formation 2 . Among them, we discuss the closest articles.

Our contribution can be seen as an extension of [START_REF] Bloch | Communication networks with Endogenous Link Strength[END_REF] to directed networks. Indeed, the authors study the formation of a non directed communication network with endogenous link strength. In their setting, agents have a budget constraint and choose how much to invest in each relationship. Also, players obtain indirect benefit by the path maximizing the product of link strength (in the main part of the paper). The difference between the models is the following: in Bloch and Dutta's work, links are non directed and the value of a link depends on the level of investment exerted by each partner (investments are either substitutable or complementary). That is, link formation exhibits an intrinsic coordination problem which vanishes in our directed context. The authors show that substitutable investments promote the efficiency and stability of the star architecture, while complementary investments promote that of connected architectures (especially, the circle architecture is always stable and efficient for small populations). The reason why connected networks 1 See for instance [START_REF] Jackson | A Strategic Model of Economic and Social Networks[END_REF], [START_REF] Bala | A Noncooperative Model of Network Formation[END_REF], [START_REF] Deroian | Farsighted strategies in the formation of a communication network[END_REF], [START_REF] Bloch | Communication networks with Endogenous Link Strength[END_REF], [START_REF] Kim | Network Formation and Stable Equilibrium, forthcoming in the[END_REF], [START_REF] Matsubayashi | A note on network formation with decay[END_REF].

2 See [START_REF] Dutta | The Stability and Efficiency of Directed Communication Networks[END_REF].

emerge in that latter context is that in a tree, any two end-node agents have always an incentive to coordinate and invest part of their resource (the part exceeding the investment of their current partner) in the formation of a new link.

The second related paper, the discussion of which is presumably of first interest, is the one-way flow model of [START_REF] Bala | A Noncooperative Model of Network Formation[END_REF]. In this model agents form directed connections and obtain benefit from both direct and indirect access to other agents.

But the resource scarcity inherent to our context induces a clear distinction: in the one-way flow model (without decay), any agent connected to a component receives a benefit written as a function of the number of agents in the component, whatever the structure of the component. This favors multiple individual best-responses and gives rise to multiple Nash architectures. In contrast, in our context the very structure in which agents are embedded matters and the structure of the game is sufficient to select a unique Nash architecture.

To finish, [START_REF] Kim | Network Formation and Stable Equilibrium, forthcoming in the[END_REF] propose a directed communication model in which agents get benefit from communicating with others. Two agents communicate with each other whenever there exists a cycle in which both are inserted. When agents are homogenous, the wheel architecture is uniquely determined by this solution concept.

Our model is very distinctive in its foundations: first we do not integrate link formation costs, but rather agents invest a fixed amount of resource in links. Second, the endogenous link strength is specific to our context. Last, in our model agents capture benefit from accessing other agents, independently of whether or not others access them, i.e. value creation is not related to path circularity.

The article is organized as follows. Section 2 introduces the model. The next two sections respectively present the three-player case and the general case. Section 5 presents two extensions and section 6 concludes. All proofs are presented at the end of the note.

The model

Let N = {1, ..., n} be a set of agents. Every agent is endowed with a finite amount X ∈ [0, 1[ of resources that can be devoted to establishing and maintaining links.

Agents derive benefit from accessing other agents, either directly or indirectly. The key distinction with [START_REF] Bloch | Communication networks with Endogenous Link Strength[END_REF] is that the return of the investment made by agent i on agent j does not depend on how much agent j has invested on agent i. Hence benefits are one-sided, in the sense that agents access others through valued directed paths. A valued directed network represents the structure of allocations.

Directed networks and individual strategies. A strategy of agent i is a vector with n -1 coordinates x i = (x i,1 , ..., x i,i-1 , x i,i+1 , ..., x i,n ) where x i,j ∈ [0, X] for each j ∈ N \ {i}, under the constraint that j =i x i,j ≤ X. Agent i forms a link with j if x i,j > 0. The set of strategies of agent i is denoted by X i . The set X = X 1 × ... × X n is the space of strategies of all agents. A strategy profile x = (x 1 , ..., x n ) can be represented as a directed and valued network g. We denote by G the set of (non negatively) weighted directed networks. The link x i,j > 0 is depicted as an edge between agents i and j with arrowhead pointing at agent j, and we denote by ij this directed and valued link for convenience when there is no confusion; thus ij ∈ g means that x i,j > 0, ij / ∈ g means that x i,j = 0. A path between agent k 0 and agent k p is a sequence of links k 0 k 1 , • • • , k p-1 k p without loops and such that q=p-1 q=0

x kq,k q+1 > 0.

The distance from agent i to agent j, and denoted d(i, j; g), represents the length of the shortest path from agent i to agent j in the network g. By convention if there is no finite path length the distance is infinite. We notice by d max (i; g) = max j =i,d(i,j;g)<+∞ d(i, j; g)

the largest among all finite distances from agent i to others in the network g.

Individual payoffs.

In agreement with the literature on communication networks, we assume that the benefit of an agent in a network is given by the sum of the benefits she receives from accessing every other agent in her component. The benefit b(i, j; g) that agent i captures from accessing agent j is the maximal value of the product of link strength over the set P (i, j; g) of all finite paths from agent i to agent j: b(i, j; g) = max p∈P (i,j;g)

x ia 1 • • • x a m-1 j
with the convention that b(i, j; g) = 0 if there is no finite path from agent i to agent j in the network g. Note that the value of any path is less than 1. Agent i's payoff in the network g is finally written π i (g) = j =i b(i, j; g). In this formulation individual payoffs are additive with respect to the benefits obtained from each accessed agent.

Efficiency and stability. The welfare W (g) of a network g is the sum of individual payoffs on that network, i.e. W (g) = i∈N π i (g). A network is said efficient if there is no other network such that the sum of individual payoffs exceeds that of the former, i.e.

W (g) ≥ W (g ) for all g ∈ G. Further, we apply the usual Nash criterion: a network is Nash if for every agent, her current strategy is a best-response to the current strategies of all other agents. Formally, a profile of individual strategies

x * = (x * 1 , • • • , x * n )
is a Nash equilibrium of the game if and only if, for every agent i ∈ N , for every strategy

x i ∈ X i , π i (x * i , x * -i ) ≥ π i (x i , x * -i ).

The three-player case

There are 16 architectures with three agents. Let us begin with efficiency issue. The welfare value of network architecture g is the maximum welfare that can be obtained on that architecture over all possible allocations. An efficient network basically satisfies that every agent invests her total resource X over her partners ('no waste' condition).

The following notation will be useful: sometimes, the best allocation is such that certain agents reduce the weight of some link to zero; that is, the best alternative consists in modifying the architecture. If z represents the value the architecture asymptotically tends to, we will notice W (g) → z (and not '='). This means that the value is not reached by this architecture although its value tends to that. This precludes de facto this architecture from being efficient.

in the table), whose value is W (g 1 3 ) = 3X + 3X 2 .
Let us turn to stability analysis. Given that there is no link formation cost, a Nash network basically satisfies that each agent forms at least one link; then we restrict attention to networks containing at least 3 links. Furthermore, for each agent in a Nash network, the sum of all direct benefits equals X (no waste). Hence, a Nash strategy maximizes the indirect benefits that the agent can capture.

• 3-link architectures: the wheel g 1 3 (in which each link's weight is X) is Nash: indeed, for each agent the payoff is X +X 2 , while any distribution of allocation (x, X -x), x > 0 (the amount x is invested on the link on which she currently invests X) would induce a payoff equal to either X or x(1 + X) (depending on X -x ≶ xX). The architectures g 2 3 and g 4 3 are not Nash, as one can identify one agent forming no link. Also g 3 3 is not Nash, as some agent (that with one inward connection) finds profitable to replace her current link with one entailing the formation of the wheel.

• 4-link architectures: none is Nash. In g 1 4 the agent with two links is better off by investing all on the link such that the deletion of the other link entails the formation of the wheel. In g 2 4 , note that the two symmetric agents (forming one link) invest X on their current connection, while the agent with two links shares her allocation into two parts. Therefore, the indirect benefit of a one-link agent is less than X 2 . By replacing her link with a connection (of weight X) toward the third agent provides her with an indirect benefit equal to X 2 . In g 3 4 the agent forming two links is better off by investing all in one link (benefiting therefore from indirect benefit). Last, in g 4 4 one agent does not form any link.

• the 5-link architecture: g 5 is not Nash; some agent (that forming two links and with one inward link) finds profitable to cut the link with the agent who do not form a link with her.

• the 6-link architecture: g 6 is not Nash; investing all on a single agent is strictly beneficial, since a positive benefit is in that case obtained from the agent at distance two.

To sum up, our environment exhibits no tension between stability and efficiency in the three-player case.

The general case

Our first result pertains with efficiency. We will use the following lemma: Lemma 1 For every agent i in a network g, π i (g) ≤ dmax(i;g) k=1 X k .

Considering any agent i in some network g, lemma 1 guarantees that her payoff does not exceed that obtained by some agent starting a chain of length d max (i; g) and composed of links with weights equal to X. The reason lies in the resource-constraint nature of our model. When the individual resource constraint is saturated, forming an additional link requires to decrease the investment put over some current links. This entails the following result: whatever a given set of paths of length k issued from some agent i, the sum of their values cannot exceed that of one unique path of length k and composed of maximal weights. The following proposition derives the set of efficient architectures:

Proposition 1 The complete wheel is the unique efficient network.

Our first conclusion is that a unique architecture is socially desirable. Indeed, this architecture is such that all agents access others by the means of a chain of length n -1 and composed of links of weight equal to X (when all individual resource constraints are saturated). Since no finite distance can be larger than n -1, we use lemma 1 to conclude that no individual payoff can exceed that obtained in the wheel architecture.

Uniqueness is derived by remarking that the wheel is the unique architecture such that all agents can obtain that maximal payoff (one can even see that whatever a distinct architecture, at most one agent can obtain such a payoff).

Our second result ensures that there is no tension between stable and efficient networks. Indeed, Proposition 2 The complete wheel is the unique Nash network.

The proof relies on two points: First, we consider one agent with maximal payoff on the network, say agent i, and one agent at maximal distance from her, say agent j; noticing that agent j is not critical for agent i, we derive that (i) agent j's payoff is not smaller than that of agent i, and (ii) for obtaining equality, agent i's payoff must be exactly equal to dmax(i;g) k=1

X k (we know from lemma 1 that it cannot be greater). We deduce that for each non terminal agent on the path from agent i to agent j, she invests amount X in a unique connection. Second, we notice that if some agent say p is not accessed by agent i, then a possible strategy entails for agent p a greater payoff than that of agent i, a contradiction. We deduce that agent i observes all agents. Combined with the first point, we see that to join agent j, agent i begins a chain of n -1 links with weight equal to X. We conclude that agent j's unique best response is to invest amount X on agent i. The complete wheel is therefore uniquely selected.

Hence, a unique architecture is selected 3 and there is a coincidence between efficient and stable networks. This result is in sharp contrast with the one-way flow model of [START_REF] Bala | A Noncooperative Model of Network Formation[END_REF]. The authors obtain in their benchmark case (without friction) that the complete wheel is both Nash and uniquely efficient, but many other architectures are also Nash (in their model Nash networks are minimally connected).

Let us go back to our proof and see why the results of the two models diverge. In both models a Nash network is connected. In BG, in a connected network all agents observe all others, so they obtain the same gross benefit. That is, payoff differences only stem from distinct number of costly connections. This is not the case in our context, in which links are valued. This generates the following major distinction: consider some agent i with maximal payoff and agent j at maximal distance from agent i (thus not critical to her). In both models, if agent j forms a unique link with agent i, she obtains the same payoff as agent i (hence, in both models at least two agents -i.e. agents i and j form a unique connection). But in BG, the fact that agent i's payoff is not smaller than that of agent j does not impose the existence of a chain from agent i to agent j: connectedness is sufficient. This originates the multiplicity of Nash networks.

In our resource-constraint setting, different connected architectures generate different 3 The result on stability is basically robust to stability criteria allowing for collective deviations, like strong Nash equilibrium [START_REF] Dutta | Stable networks[END_REF]) or Strong stability [START_REF] Jackson | Strongly stable networks[END_REF]). Indeed, we know that (i) the wheel architecture ensures to all agents the maximal individual payoff and (ii) for any distinct architecture at most one agent obtains such a payoff.

payoffs. The only compatible network must contain a chain of maximal length from agent i to agent j (and maximal weight X on each arc), otherwise the presence of multiple branches lowers too much agent i's payoff, contradicting that her payoff is maximal.

Extensions

A more general setting on the value of paths

We define the set Ψ l (i, j; g) of all paths of length l from agent i to agent j in the network g:

Ψ l (i, j; g) = {(x a 0 a 1 , • • • , x a l-1 a l ) ∈]0, X] l /a k = a k ∀k = k , a 0 = i, a l = j}
This formulation excludes paths containing some arcs of weight equal to zero. Then we define a function

   φ l : Ψ l (i, j; g) -→ R + (x ia 1 , • • • , x a l-1 j ) -→ φ l (x ia 1 , • • • , x a l-1 j )
with l arguments in ]0, X], and generating a positive real number:

φ(z 1 , • • • , z l ) > 0,
i.e. a path (composed with positive weights) has positive value. Second, we suppose that the function is increasing and (weakly) convex in each argument. By convexity, we mean that for every indexes l and p:

φ l (a 1 , • • • , a p , • • • , a l ) + φ l (a 1 , • • • , b p , • • • , a l ) ≤ φ l (a 1 , • • • , a p + b p , • • • , a l ).
The benefit obtained by agent i from agent j is written:

b(i, j; g) = max l=1,••• ,n-1 max p∈Ψ l (i,j;g) φ l (p)
with the convention that b(i, j; g) = 0 whenever there is no finite path length from agent i to agent j. For instance, the individual payoff in the complete wheel is written

n-1 k=1 φ k (X, • • • , X k times )
Indeed, there is a unique finite path from agent i to any agent j, and no two paths have equal length. Note that in this formulation we do not impose any ranking on the functions φ k , we only impose positivity.

Among all paths starting from agent i and issuing to agent j, we denote with P * ij (g) the subset of them with greatest value, i.e.

P * ij (g) = {p ij (g) ∈ argmax Ψ 1 (i,j;g),••• ,Ψ n-1 (i,j;g) φ l (p)}.
For convenience, we arbitrarily select a unique best path, say Θ(i, j; g) ∈ P * ij (g). We denote for each agent i in the network g the profile P * i (Θ, g) = {Θ(i, j; g)} j =i . This profile associates with each agent j = i one path from i to j of greatest value. We define the set Γ l i (Θ, g) composed of the paths in P * i (Θ, g) of length l. Then, convexity implies the following lemma (this lemma is a direct extension of lemma 1):

Lemma 2 For every index l ∈ {1, • • • , n -1}, and whatever selection process Θ, we have:

p∈Γ l i (Θ,g) φ l (p) ≤ φ l (X, • • • , X l times
) This lemma bounds above the sum of benefits that agent i captures from accessing agents placed at a given finite distance from her. To fix ideas, consider agent i's payoff in the following network (see figure 2):

Figure 2:

In this example P * ij (g) is reduced to a unique element for any j = i. We obtain:

π i = φ 1 (a) + φ 1 (X -a) +φ 2 (a, b) + φ 2 (a, X -b) + φ 2 (X -a, c) + φ 2 (X -a, c ) + φ 2 (X -a, X -c -c ) +φ 3 (a, b, X) + φ 3 (X -a, c, X)
As the function is increasing,

   φ 3 (a, b, X) ≤ φ 3 (a, X, X) φ 3 (X -a, c, X) ≤ φ 3 (X -a, X, X)
Further, the function is convex, so

         φ 1 (a) + φ 1 (X -a) ≤ φ 1 (X) φ 2 (a, b) + φ 2 (a, X -b) ≤ φ 2 (a, X) φ 2 (X -a, c) + φ 2 (X -a, c ) + φ 2 (X -a, X -c -c ) ≤ φ 2 (X -a, X)
That is,

π i ≤ φ 1 (X) + φ 2 (a, X) + φ 2 (X -a, X) + φ 3 (a, X, X) + φ 3 (X -a, X, X)
Using again the convexity of φ 2 and φ 3 , we obtain

π i ≤ φ 1 (X) + φ 2 (X, X) + φ 3 (X, X, X)
Hence, confirming lemma 3, we note that for each distance l = 1, 2, 3, the sum of benefits that agents i captures from agents placed at distance l from her is bounded above by the value φ l (X, • • • , X).

The following lemma extends lemma 2:

Lemma 3 Whatever network g ∈ G distinct from the complete wheel, there is at most one agent with payoff equal to n-1 k=1 φ k (X, • • • , X). Hence at least n -1 agents have a payoff strictly smaller than that they obtain in the wheel.

As a direct consequence of lemmata 3 and 4: Proposition 3 The complete wheel is both Nash and uniquely efficient.

Uniqueness (of efficiency set) comes from the fact that the complete wheel is the unique architecture providing φ k (X, • • • , X) from k = 1 up to k = n -1 as benefit to all agents. Hence, we observe that a crucial determinant of uniqueness is that all functions

φ k (X, • • • , X) are positive.
We obtain uniqueness of the stable set under the following more specific environment:

φ l (x ia 1 , • • • , x a l-1 j ) = θ(x ia 1 ) × • • • × θ(x a l-1 j )
with function θ(.) : R + -→ [0, 1[, increasing and convex, and such that θ(x) > 0 iff

x > 0. For instance, consider the function θ(x) = δ • x, δ ∈ [0, 1] and x ∈ [0, X], X < 1.
We obtain here a geometric decay function, in the sense that the value of a path

(x ia 1 , • • • , x a l-1 j ) is δ l • x ia 1 × • • • × x a l-1 j .
The additional result obtains:

Proposition 4 In such a specification, the complete wheel is the unique Nash network.

Indeed, when φ is a product of identical functions, the proof of proposition 2 is easily extended.

A dynamic process of network formation

We present a dynamic process close to that presented in [START_REF] Bala | A Noncooperative Model of Network Formation[END_REF] and pursued with slight modification by [START_REF] Kim | Network Formation and Stable Equilibrium, forthcoming in the[END_REF]. We consider some initial network g 0 ∈ G and a discrete time sequence 1, • • • , t, • • • At each time t, a strict and non empty subset A t of agents is stochastically drawn from the population; the selected agents i ∈ A play with probability 1 a weakly payoff-improving strategy; this includes possibility of playing the same strategy as the current one. The resulting network is labeled g t . In our process, the new strategies are payoff-improving, but not necessarily best-responses (in BG they are such). In terms of individual informational requirements, our hypothesis is less demanding than that of BG (the authors have also developed in a working paper some analysis under alternative informational assumptions -see footnote 22 pp. 1192 in the related article).

The following result obtains:

Proposition 5 This dynamic process converges with probability 1 toward the wheel architecture, regardless the initial network it starts from.

The result is close to that of [START_REF] Bala | A Noncooperative Model of Network Formation[END_REF] (or even [START_REF] Kim | Network Formation and Stable Equilibrium, forthcoming in the[END_REF]).

Yet, our proof contains a difference inherent to our context, although similar in spirit.

In BG, the proof relies on the following mechanism: starting from any initial network, choose some agent j 1 with maximum profit on it, consider some agent j 2 furthest away from her (at maximal distance), and let agent j 2 form a unique connection with agent j 1 . This strategy is a best-response: agent j 2 observes all the agents observed by agent j 1 while forming a unique connection. Then replicate the process by replacing agent j 1 by agent j 2 and by finding some agent at maximal distance from agent j 2 . The process goes on until achievement of the wheel. In our context, starting from any network, choose some agent j 1 with maximum profit on it, consider some agent j 2 furthest away from her, and make agent j 2 invest all resource X on agent j 1 . Then she obtains at least agent j 1 's payoff, and consequently she weakly improves her current payoff; now because links are valued, this new strategy may not be a best-response: indeed, she obtains exactly X + X(π j 1 (g) -b(j 1 , j 2 ; g)) from the new strategy. It is easy to find a network containing another agent j 1 with maximal payoff, and such that both agent j 2 is furthest away from her and b(j 1 , j 2 ; g) > b(j 1 , j 2 ; g).

Concluding remarks

We study a non-cooperative model of network formation in which agents invest a fixed and infinitely separable amount of resource into links. That is, the strength of links is endogenously determined by individual investment decisions. In this resourceconstraint setting, we find that the wheel architecture is both uniquely efficient and Proof of lemma 1. The proof is given by induction.

Inductive argument P (k), k ∈ {1, • • • , n -1}: if d max (i; g) = k, then π i (g) ≤ k
q=1 X q . We trivially have P (1). Suppose now P (1), • • • , P (k). We will derive P (k + 1). Indeed,

if d max (i; g) = k + 1, π i (g) = j =i x ij (g) + j =i x ij (g)π i j (g)
where π i j (g) represent the relevant part of agent j's payoff for agent i. That is, that derived from agents distant until at most order k of agent j; if some agent p is at distance k + 1 or more than agent j, then either the distance is infinite and it does not count for agent i's payoff, or the distance is finite which means by construction that agent p is at smaller distance from agent i (since d max (i; g) = k + 1).

Since the relevant network used by agent i to access others through agent j is a tree starting from agent j and with path length bounded above by k, we deduce as applying P (1), • • • , P (k) to agent j (more precisely to the relevant part of agent j's payoff for agent i) that π i j (g) ≤ k q=1 X q . Hence,

π i (g) ≤ j =i x ij (g) + j =i x ij (g) k q=1 X q
And since j =i x ij (g) ≤ X, we obtain that

π i (g) ≤ k+1 q=1 X q
which proves P (k + 1).

Proof of proposition 1. We notice that a link ij is not necessarily active in the sense that the link is used to extract benefit. For instance, consider the following three-player network such that x 12 = X -, x 13 = , x 23 = X. Then agent 1 may join agent 3 with two paths. And the two-arc path worths more than the one-arc path whenever (X -)X > . This is true for sufficiently small values of . But in such a case a more efficient network exists in which agent i reallocates the investment x ij into the connection x ik 1 where ik 1 is the arc used by agent i to join agent j (each agent using the path ik 1 , ..., k p j to extract value would be better off, and no agent was using the link ij to extract value by definition). Hence we can restrict attention to networks such that for every agent, the sum of benefits associated with agents at distance 1 from her is exactly equal to X.

From lemma 1 we easily derive that the complete wheel is efficient: in the wheel architecture all agents access others by the means of a chain of length n -1 and composed of links of weight equal to X (when all individual resource constraints are saturated). Since no finite distance can be larger than n -1, we use lemma 1 to conclude that no individual payoff can exceed that obtained in the wheel architecture.

We now derive uniqueness by showing that whatever network g ∈ G distinct from the complete wheel, there is at most one agent with payoff equal to X +

• • • + X n-1 .
Consider a network g distinct from the complete wheel. Suppose without loss that

π 1 (g) = X + • • • + X n-1 .
Then there is a chain, starting from agent 1 and containing all agents, and all agents except the terminal node, say agent n, invest X on her link.

Thus, these agents have a unique connection. EITHER agent n forms a set of links distinct from investing her total resource on agent 1; and clearly agents 2 to n obtain less than agent 1's payoff; OR agent n invests amount X on agent 1, in which case the complete wheel forms. We deduce that the complete wheel is the unique architecture such that all agents obtain a payoff equal to X + X 2 + ... + X n-1 . We conclude that the complete wheel is uniquely efficient.

Proof of proposition 2. We notice that the wheel architecture with all weights equal to X is Nash, since no other network can guarantee larger individual payoffs. We then turn to uniqueness. We first show that some agent with maximal payoff begins a chain composed of links of weight equal to X. Second we deduce that this agent accesses the whole society. Third we conclude that the wheel is uniquely selected.

Assume without loss that agent i ∈ argmax k∈N π k (g). Consider agent j such that d(i, j; g) = d max (i; g). Then agent j is non critical to agent i. Hence, if forming a unique link toward agent i, agent j obtains π j = X + X(π i (g) -b(i, j; g))

since agent i's payoff is not affected by agent j's strategy. Recalling that agent i's payoff is maximal, we know that π j (g) ≤ π i (g), inducing X + X(π i (g) -b(i, j; g)) ≤ π i (g).

But b(i, j; g) ≤ X dmax(i;g) . Hence the latter inequality implies that: dmax(i;g) or equivalently π i (g) ≥ X +X 2 +...+X dmax(i;g) . But by lemma 1 agent i's payoff cannot exceed that value. Thus it must be the case that π i (g) = X + X 2 + ... + X dmax(i;g) .

π i (g) ≥ X 1 -X 1 -X
We also derive two points. First π j (g) = π i (g). Second, for agent i's payoff to be such, agent i accesses all others through a chain of length d max (i; g) and composed of links with weights equal to X.

Suppose that some agent p is not accessed by agent i, i.e. d max (i; g) < n -1. Then by investing amount X on agent i, agent p obtains a payoff equal to X + Xπ i (g). As agent i's payoff is maximal, it must be the case that X + Xπ i (g) ≤ π i (g), which means that π i (g) ≥ X 1-X . But as X < 1, X + X 2 + ... + X n-1 ≤ X 1-X , a contradiction. Hence, d max (i; g) = n -1.

We know that there is a unique agent j such that d(i, j; g) = n -1 and that π j (g) = π i (g). Given that agent i begins a chain of length n -1 and composed of links of weights equal to X, the unique strategy for agent j to obtain agent i's payoff consists in forming a unique connection of weight X with agent i. The complete wheel obtains.

Proof of proposition 5. The proof is given by proposing from any network g 0 a sequence of individual beneficial deviations at each period, and leading to the wheel network in a finite delay. We do not impose individual deviations to be best-response strategies; rather we demand that the new strategy does not deteriorate the individual payoff.

Figure 1

 1 Figure 1 depicts the 16 architectures with three players and gives their values: The

uniquely Nash. Therefore, this environment is sufficient to generate a unique architecture making compatible private incentives and collective outcome.

Further advances may investigate some lines of research: introducing convex cost of link formation generates an immediate trade-off between distance and cost. Relaxing homogeneity in agents' types may also constitute an interesting extension, as well as introducing information asymmetry.

Consider some initial network g 0 ∈ G and some agent j 1 ∈ argmax k∈N π k (g 0 ), one agent with maximum profit on the network g 0 . Choose some agent j 2 such that d(j 1 , j 2 ; g 0 ) = d max (j 1 ; g 0 ), so agent j 2 is furthest away from agent j 1 in the network g 0 . This may include the possibility that d(j 1 , j 2 ; g 0 ) = +∞. Clearly, agent j 2 is not critical to agent j 1 (dropping all links formed by agent j 2 does not affect agent j 1 's payoff).

Then by investing X on agent j 1 in period t = 1, that is if agent j 2 's new strategy is

-out:j 2 the resulting new network (g 0 -out:j 2 represents the network g 0 minus all links formed by agent j 2 ), agent j 2 gets a payoff π j 2 (g 1 ) ≥ π j 1 (g 0 ), so that the new strategy weakly improves agent j 2 's payoff (i.e., π j 2 (g 1 ) ≥ π j 2 (g 0 )) (it strictly improves her payoff in the case d(j 1 , j 2 ; g 0 ) = +∞).

Consider now the network g 1 . By construction, agent j 2 ∈ argmax k∈N π k (g 1 ). Again, consider some agent j 3 such that d(j 2 , j 3 ; g 1 ) = d max (j 2 ; g 1 ); equivalently agent j 3 is such that d(j 1 , j 3 ; g 0 -in:j 2 ) = d max (j 1 ; g 0 -in:j 2 ), if g 0 -in:j 2 denotes the network g 0 minus all links pointing at agent j 2 (again, it can be the case that d(j 1 , j 3 ; g 0 -in:j 2 ) = +∞). By the same argument, assume that agent j 3 is selected at time t = 2 and that she invests all X on agent j 2 . Then she obtains at least the same profit, so her new profit is not less than the maximum profit in g 1 , so this strategy is payoff-improving.

Computing the process further, we go through until selecting agent j 1 in a finite time, who similarly improves her payoff by completing the wheel architecture (no individual payoff exceeds that obtained in the wheel).

The proof is completed by remarking that the wheel architecture is the unique absorbing state of the process: first whatever network distinct from the wheel, at least one agent can improve her strategy (since the wheel network is uniquely Nash). Second, the wheel architecture is Nash strict; hence no agent in the wheel possesses another strategy giving her an equal payoff.