Nicoleta Stroe
email: nicoleta-alexandra.stroe@renault.com

Sorin Olaru
email: sorin.olaru@centralesupelec.fr

Guillaume Colin
email: guillaume.colin@univ-orleans.fr

Karim Ben-Cherif
email: karim.ben-cherif@renault.com

Yann Chamaillard
email: yann.chamaillard@univ-orleans.fr

Time-varying MPC-based energy management for HEV including engine stop & start

In this paper, the energy management of hybrid electric vehicles is considered and two problems are addressed: torque split and engine stop-start policies. Model predictive control potential is proven especially for the decision to stop the engine at the proper moment and also to anticipate engine restart. A piecewise linear polynomial with respect to torque is used to approximate the fuel consumption and it will be shown that its partition can be exploited to implicitly consider engine stop decision. The strategy is applied to a dual-clutch transmission hybrid architecture which was simulated on a high-fidelity vehicle model.

I. INTRODUCTION

Hybrid electric vehicles (HEV) potential in minimizing fuel consumption makes them an attractive alternative to conventional vehicles. In the last decade, a considerable number of publications have enriched the research activity concerning the strategies to minimize fuel consumption and emissions or to improve drivability. From a control perspective, the problem of power split between the engine and electrical machine(s) and engine stop-start decisions are among the highest challenges for an HEV.

The literature contains already methods for calculating the power distribution and the most common are: dynamic programming, which gives the global optimum, provided the drive cycle is entirely known in advance [START_REF] Sundstrom | Optimal hybridization in two parallel hybrid electric vehicles using dynamic programming[END_REF], rule-based [START_REF] Goerke | Optimal control based calibration of rule-based energy management for parallel hybrid electric vehicles[END_REF] and ECMS -Equivalent Consumption Minimization Strategy [START_REF] Sciaretta | Optimal control of parallel hybrid electric vehicles[END_REF]. The latter can also be found under the form of Adaptive-ECMS [START_REF] Musardo | A-ECMS: an adaptive algorithm for hybrid electric vehicle energy management[END_REF] or Telemetric-ECMS [START_REF] Sciarretta | A real-time optimal control strategy for parallel hybrid vehicles with on-board estimation of the control parameters[END_REF]. In addition, the nowadays possibility to foresee traffic conditions and the ability to integrate them in real-time algorithms, as well as the ability to handle constraints have turned the attention to Model Predictive Control (MPC) method, in its deterministic formulation [START_REF] Cairano | Engine power smoothing energy management strategy for a series hybrid electric vehicle[END_REF], [START_REF] Lu | MPC based torque distribution strategy for energy management of power-split hybrid electric vehicles[END_REF] or stochastic [START_REF] Ripaccioli | A stochastic MPC approach for series hybrid electric vehicle power management[END_REF], [START_REF] Josevski | Energy management of a parallel hybrid electric vehicles based on stochastic model predictive control[END_REF].

Engine stop-start functionality did not receive the same attention in conjunction with power split problem. There are two main approaches: introduction of a binary decision variable in the optimization procedure (as in [START_REF] Beck | Comparison of two real-time predictive strategies for the optimal energy management of a hybrid electric vehicle[END_REF], where a mixedinteger quadratically constraint linear problem is formulated) or separation into a two-layer optimization. The latter is more often encountered: in [START_REF] Hahn | Extension of a linear optimal control strategy for HEV[END_REF] a constrained LP is used to compute the power distribution at the top level, whereas the engine operating point optimization is performed at a bottom level; in [START_REF] Nuesch | Convex optimization for the energy management of hybrid electric vehicles considering engine start and gearshift costs[END_REF] the upper level uses dynamic programming for engine ON/OFF decision and gear choice, whereas the lower level uses convex optimization to provide the torque split. The MPC benefit for stop-start strategy has been analysed in [START_REF] Dudek | Start -stop system for a city bus based on model predictive control[END_REF] for a city bus, where the energy management is also handled as a separated, low-level problem.

In this paper, the velocity profile and the gear engaged are assumed known in advance for a short time horizon. As a first contribution we show that this information will be exploited in a MPC formulation of the energy management problem, where a new optimization criterion will be introduced, with a generalized method for the calculation of the equivalence factor. A methodology to simultaneously include the torque split and engine stop-start objectives but without introducing an additional discrete variable is presented formally. The stop decision is implicitly handled by appropriate penalties on torque variations and it is taken a-posteriori, with respect to the values of computed vector of optimal commands. Finally, an additional contribution resides in the fact that a restart anticipation is introduced in order to improve the vehicle speed tracking.

The article is organized as follows: first, a powertrain control -oriented model is presented; second, the energy management problem is formulated. The simulation results obtained and the conclusions complete the paper.

NOTATIONS

• ICE -internal combustion engine • EM -electric machine • DCT -dual-clutch transmission • R i -gear ratio engaged on i th shaft (includes neutral definition), i ∈ {1 : odd, 2 : even} • C i -clutch status (0 -open, 1 -closed) • N i = min(R i , 1
) -used to define the case where one of the shafts is decoupled For an HEV, the torque at the wheel level includes complete information about the system configuration, i.e. EM position and transmission type. In this paper, a DCT hybrid with EM connected to the even primary shaft (Fig. 1) will be considered, although the approach is generic for any parallel hybrid architecture, with one EM and one battery, as shown in [START_REF] Stroe | Towards a generic control-oriented model for HEV predictive energy management[END_REF]. As mentioned in the introduction, the engaged gears are pre-imposed, the variables to be determined on-line are the component torques (T ice , T em ), which are considered as control signals. The wheel torque is calculated from the vehicle speed (v), slope (α) and vehicle parameters such as the mass (m), aerodynamic drag coefficient (c d ), frontal area (A f ), rolling friction coefficients (c r0 , c r1 ) and air density (ρ air ). The shaft inertias are neglected and only the static relations for wheel torque and rotational speeds are considered, which are expressed below. The ratios r w ice , r w em depend on the driveline states, they are therefore time-varying, which complexifies the prediction model. A description of the functional modes is given in Table I and more explanations about these expressions can be found in [START_REF] Stroe | Towards a generic control-oriented model for HEV predictive energy management[END_REF].

T w = 1 2 ρ air A f c d v 2 + (c r1 v + c r0 )mgcos(α) +mgsin(α) + m v R w (1)
T w = r w ice T ice + r w em T em (2) r w ice = F D (R1) R 1 C 1 + F D (R2) R 2 C 2 (3) r w em = F D (R1C1C2+R2) (R 1 C 1 C 2 + R 2 ) rat em (4)
ω ice = r w ice v R w + (1 -C 1 -N 2 C 2 ) ω ctrl ice ( 5
)

ω em = r w em v R w + rat em C 2 (1 -C 1 ) (1 -N 2 ) ω ctrl ice (6)
These static relations can be reduced to the compact form below, if the terms in ω ctrl ice are neglected:

T w = r w ice r w em T ice T em , ω ice ω em = r w ice r w em v R w (7)
ṠOC = - OCV (SOC) -OCV 2 (SOC) -4R (SOC) P b 2R (SOC) Q max ( 8
)
with the battery power given by:

P b = π 30 ω em T em + loss (ω em , T em ) .

A. Fuel consumption approximation

The considered system is equiped with an 1.2 L SI engine whose fuel mass flow ( ṁf [g/s]) with respect to torque is depicted in Fig. 2, for a family of rotational speeds. This dependance is extracted from a nonlinear map, defined for a torque grid with 20 values. As a contribution, in the present work we advocate the representation of the fuel consumption in terms of a piecewise first order polynomial with respect to torque, whose coefficients depend on rotational speed. For rotational speed values between two curves, an interpolation is performed:

ṁf = α i (ω ice , T mini,maxi ice )T ice + β i (ω ice , T mini,maxi ice ) for T mini ice ≤ T ice ≤ T maxi ice , i = 1 . . . N part (9)
For the particular case of the engine under study, a number of N part = 8 partitions was found to give accurate results.

III. ENERGY MANAGEMENT -PROBLEM FORMULATION

The main goal of this work is to minimize the fuel consumption for an HEV without an external recharge, by computing the optimal torque distribution between the two energy sources. It is assumed that the wheel torque demand and the engaged gear can be accurately predicted for several seconds (around 5 s). This prediction horizon value favors engine stop inclusion and thus, a higher consumption gain can be obtained, without the risk of affecting the vehicle speed tracking. First, an optimization criterion for the torque split problem will be introduced and then it will be shown how it can be modified to tackle the engine stop-start.

A. Optimization criterion

The optimization problem can be translated into a trade-off between the use of two power sources. The EM can provide alone the demanded torque or it can assist the engine, but in the absence of an external recharge, attention must be paid to battery consumption during driving. The criterion chosen in this paper incorporates this trade-off and its formulation is inspired from the ECMS design:

min u k P f k + λ k P e k P f k = H LV ṁf k , P e k = OCV k ṠOC (10)
where k is the current step, u k is the engine torque, H LV is the lower heating value of the fuel and λ k is an equivalence factor. As a difference with ECMS, the squared values of the powers are used instead, in order to obtain convexity structural advantages:

min u k+N -1 i=k P 2 f (i) + λ 2 k (P e k (i) -P emin k (i)) 2 ( 11
)
where N is the prediction horizon, u is a vector with ICE torques of length N and P emin k is the electrochemical power minimal value. By using the squared values of the powers, all sign information is however lost. Hence, the introduction of P emin aims to properly handle the sign of electrochemical power, that can be either positive or negative. The value of λ that assures a proper balance between the two sources is strongly dependent on the drive cycle characteristics, which cannot be entirely known in advance. Here, a generalized method calculates the weighting λ as the ratio between the variations of the two powers:

λ k,i = - P f (u i ) -P f (u i ) P e (u i ) -P e (u i ) (12)
where u i , u i are the minimum and the maximum limit, respectively of engine torque at the ith step as they are determined from constraints activation. As an alternative, to decrease the sensitivity with respect to the anticipatory information on the constraints activation, a constant λ k is set during the prediction, calculated as an average in a receding horizon manner. The goal is to find at each step an equivalence factor that approximates the average of future variations of the two powers:

λ k = 1 N pred k+N pred -1 i=k λ k,i (13)
where N pred ≥ N . Such an average-based formulation has been further retained for the present study. This second choice of the prediction horizon for λ calculation relies on the lookahead possibilities, that can be greater than those exploited in the MPC: for instance, a two-minute traffic prediction is a reasonable value to define the driving style variability and it is coherent with the actual possibilities of data acquisition. In principle, this choice leads to a variation of the cost function at a slower rate than the MPC optimal control sequence. However, for the MPC optimization itself a smaller horizon is chosen for two main reasons:

• computational requirements • to limit the impact of prediction model inaccuracies that arise as a consequence of linearization, as will be shown in the next subsection.

B. Problem formulation within time-varying MPC framework

The model-based predictive control strategy considered next involves the resolution of a finite-time horizon optimization problem, at each time instant k:

min U N i=1 ∆x T k+i Q∆x k+i + ∆u T k+i-1 R∆u k+i-1 (14a)
s.t.

x min ≤ x k+i ≤ x max , i = 1, ..., N u min ≤ u k+i-1 ≤ u max , i = 1, ..., N (14b)
where ∆x, ∆u denote the state and command variations, respectively and U is a compact vector integrating all the future commands along the prediction horizon. The problem is reformulated as a quadratic programming:

min U 1 2 U T HU + F T U (15a) s.t. A ineq U ≤ b ineq A eq U = b eq (15b)
where H is positive definite and all matrices are obtained using algebraic manipulations over [START_REF] Stroe | Towards a generic control-oriented model for HEV predictive energy management[END_REF]. For the current optimization problem, the only state variable is SOC whose model is nonlinear as given by ( 8). Here, a linearization at the operating point is performed, under the assumption that OCV and R are constant during the prediction. A Linear Time Varying (LTV) system is thus obtained:

x k+1 = A k x k + B k u k + D k (16) where x = SOC, A k = 1, u k = T ICE (k),
B k = π 30 Q max ω em k f k r w ice k r w em k ∆t opt (17a) D k = -OCV + f k 2RQ max ∆t opt -B k u k-1,0 (17b)
where ∆t opt is the MPC sampling time and u k-1,0 is the current engine torque. Let T k|k-1 em

= 1 r w em k T w (k) -r w ice k u k-1,0
. Then:

f k = OCV 2 -4R π 30 ω em k T k|k-1 em + loss(w em k , T k|k-1 em
)

After linear algebra manipulations, we have:

X k = Φ k x k + Ψ k U k + V k (18a) Φ k (i) = ones(N, 1) (18b)
Ψ k (i, j) = B k+j-1 if j ≤ i 0 otherwise (18c) V T k = D k D k + D k+1 . . . N i=1 D k+i-1 (18d)
where x k is the SOC value at instant k and X k , U k the vectors of future states and commands, respectively. Penalties on torque variations can also be introduced for drivability reasons. Let:

∆U k = D ∆ U k + U k0 (19a) U T k0 = u k-1,0 0 . . . 0 (19b) D ∆ (i, j) =      -1 i = j 1 i = j + 1 0 otherwise (19c)
With these notations, the criterion (11) can be reformulated as [START_REF] Serrao | A comparative analysis of energy management strategies for hybrid electric vehicles[END_REF], with:

H k = ᾱ2 k + q 2 k B2 k (20a) F k = ᾱk βk -q 2 k Bk U max k ( 20b
)
where

q k = λ k 1 H LV Q max OCV k , Bk = diag (B k+i-1 ), U max k is a vector with the input upper bounds, β(i) = β ω ice,k+i-1 , T min,max ice,k+i-1
and ᾱk = diag α ω ice,k+i-1 , T min,max ice,k+i-1

, as in [START_REF] Josevski | Energy management of a parallel hybrid electric vehicles based on stochastic model predictive control[END_REF]. If penalties on torque variations are also considered, the relations (20) are updated as below, with R ∆ k ≥ 0 the matrix of penalties:

H k = ᾱ2 k + q 2 k B2 k + D T ∆ R ∆ k D ∆ (21a) F k = ᾱk βk -q 2 k Bk U max k + D T ∆ R ∆ k U k0 (21b)
The ICE and the EM have physical limitations of torque and power, which are directly determined from the rotational speeds. For SOC distance-varying limits are considered, as defined by the equations below and depicted in Fig. 3:

SOC min k = SOC 0 -(SOC 0 -SOC min ) e 1- 1 1- d k dT ot SOC max k = SOC 0 + (SOC max -SOC 0 ) e 1- 1 1- d k dT ot (22)
where SOC min , SOC max are the physical SOC limitations (here set at 20% and 90%, respectively), d k is the current distance and dT ot is the total distance. For an HEV without an external recharge, battery depletion must be avoided and these constraints profile on SOC enforces convergence at the end of the drive cycle toward the initial value, if the total distance is known. If this information is not available a reset distance is used instead, whose value can be determined from the driver history data. In simulation, this distance can be adjusted to either 5 or 10 km. The constraints can therefore be expressed as:

T min ice (ω ice ) ≤ T ice ≤ T max ice (ω ice ) (23a) T min em (ω em ) ≤ T em ≤ T max em (ω em ) (23b)
P min em (ω em ) ≤ π 30 ω em T em ≤ P max em (ω em ) (23c) SOC min k -k ≤ X k ≤ SOC max k + k (23d)
where k are slack variables used to relax the SOC constraints and hence, avoiding infeasibility problems which may occur especially toward the reset distance, where the range of variation is narrow.

As mentioned in section II, the fuel consumption map is partioned in several torque-dependent regions to which a first order polynomial approximation is associated. A drawback of such piecewise approximation approach is that the physical constraints are parameterized according to the partitions and thus, upon the optimization the fuel consumption parameters will evolve. The first step is to verify if a unique interval can be found:

if ∃j s.t . U min k (i); U max k (i) ⊆ T minj ice ; T maxj ice then n pwl (i) = j, j = 1 . . . N part (24)
where n pwl (i) denotes the region for the ith command and T minj ice , T maxj ice represent the torque limits for the jth region, as in [START_REF] Josevski | Energy management of a parallel hybrid electric vehicles based on stochastic model predictive control[END_REF].

If this condition is not satisfied, the choice of the consumption region relies mostly on the current engine torque value. Let n 0 be the corresponding region. Then:

if T minn 0 ice ; T maxn 0 ice ∩ U min k (i); U max k (i) = ∅ then n pwl (i) = n 0 (25)
Finally, if no intersection is found between the torque range and the current region, a neighbour interval is selected.

n pwl (i) = j, j = 1 . . . N part min j |n 0 -j| s.t. T minj ice ; T maxj ice ∩ U min k (i); U max k (i) = ∅ (26)
New limits are thus imposed:

U min k (i) = max U min k (i), T min n pwl (i) ice U max k (i) = min U max k (i), T max n pwl (i) ice (27)
After each optimization step, a saturation check is performed for the commands whose physical limitations span several consumption regions. If the command lies within its defined limits (27), no action is taken. If saturation occurs, n pwl is shifted toward left or right, depending on whether the lower or the upper bound was activated, and a new optimization is launched with this update. The procedure is repeated until a chattering effect is encountered, as in (28), or a maximal number of iterations is reached.

U (iter) k (i) = U min,(iter) k (i), n (iter)
pwl (i) = n (iter-1) pwl (i) + 1 U max,(iter) k (i), n (iter) pwl (i) = n (iter-1) pwl (i) -1
(28) where the superscript iter denotes the iteration number.

C. ICE stop-start inclusion

It is assumed that the driveline states are calculated at a high level (supervisor) and based on this data, a torque split is computed at a lower level. A simple decision of ICE stop-start can be taken based only on the anticipation of idling periods, longer than the cost of a restart, which is usually expressed in seconds of idling. However, further improvement in fuel consumption can be achieved with MPC, which calculates a sequence of future commands.

Let t idle be the number of seconds of idling which translates the cost of an ICE restart, ct on the number of steps since an ICE restart and ct of f the number of steps since an ICE stop. Then at step k the engine is shut off for:

U k (1 : N stop ) ≤ T thr ice (29a) ct on > N start (29b)
where N stop = t idle ∆topt and T thr ice is a threshold value for ICE torque below which the stop of the engine is preferable over the considered length of the prediction horizon. The condition (29b) assures that the engine is not stopped shortly after a restart decision. If the engine is already stopped, a restart is demanded if the first calculated command is superior to T thr ice . A N on -step anticipation for ICE restart is also introduced: if the command from the position 1 + N on exceeds the threshold torque value and the engine has already been stopped for at least N stop steps, a restart command is activated. For the case considered, N on was set to 1.

In addition, all stop conditions are subject to powertrain inhibition functions, where safety requirements are formulated (SOC level, fuel temperature, altitude value, aftertreatement etc). This stop-start strategy is described by the diagram in Fig. 4.

The PWL approximation introduced in the previous section is particularly suitable for this problem because it allows the demarcation of the low torque region, where T thr ice is included:

T min1 ice ≤ T thr ice ≤ T max1 ice
. The first step is to identify the sequences of commands with a length of at least N stop for which the first region was selected for the fuel consumption approximation. Then, for these sequences, penalties on torque variations are introduced through the matrix R ∆ k , as it appears in formulation (21) of the optimization criterion:

R ∆ k (i + 1 : j, i + 1 : j) = 0, for j -i ≥ N stop n pwl (i : j) = 1 (30)
The purpose of this strategy is to force the entire sequence to either go toward zero (and the engine can therefore be stopped) or to reach the upper limit of the interval and then to commute toward higher values. The cost of an ICE restart is implicitly handled by the length of the sequence, N stop . If the engine has been stopped for ct of f < N stop , then penalties are introduced for the first N stop -ct of f commands, in order to maintain the calculated torque close to zero and hence, the ICE stopped:

R ∆ k (1 : N stop -ct of f , 1 : N stop -ct of f ) = 0 ct of f < N stop (31)

IV. SIMULATION AND RESULTS

The proposed strategy was validated in Matlab/Simulink environment and for the vehicle dynamics a high-fidelity model designed in AMEsim was used. Two representative scenarios for engine stop-start are considered: Artemis urban and traffic jam, whose velocity profiles are depicted in Fig. 5. The future wheel torque demand cannot be perfectly known in advance, but for a short horizon the velocity can be accurately predicted and therefore, it is possible to obtain an accurate estimation of the wheel torque, for a given driver. First, simulations were carried without the MPC-based ICE stop-start decision. For this scenario, the engine is stopped during regenerative braking and electrical traction phases (imposed for very low speeds). A comparison is made with the ECMS method, defined by [START_REF] Beck | Comparison of two real-time predictive strategies for the optimal energy management of a hybrid electric vehicle[END_REF]. In [START_REF] Serrao | A comparative analysis of energy management strategies for hybrid electric vehicles[END_REF] is shown that this Hamiltonian-based method, with a constant, drive cycle dependent adjoint state, gives close to global optimum results. Therefore, for each speed profile a constant equivalence factor was found such that the final SOC was the same as for the case with an MPC energy management, in order to make a fair comparison between the two strategies. With MPC, a degradation of only 6% from the ECMS is observed for Artemis urban and of 14% for Artemis jam, according to Table II (initial SOC fixed at SOC 0 = 50%). For the MPC-based stop-start, the threshold torque value was chosen as T thr ice = 7 Nm, the cost of a restart was set to 2s of idling, t idle = 2, ∆t opt = 0.5s and therefore, N stop = 4.

It can be seen that an increase in the prediction horizon is especially relevant for the engine stop inclusion, for N > t idle . This is also depicted in Fig. 6, where it can be observed that there are more frequent stops for a prediction of 5s than for 1s and a fuel consumption gain up to 5% can be obtained. V. CONCLUSIONS AND OUTLOOK An MPC strategy for the energy management of an HEV has been proposed. The engine stop-start has been equally addressed, without the introduction of a binary decision variable in the optimization problem. The assumption of an accurate prediction for the wheel torque demand and the gear engaged is not restrictive due to a short prediction horizon choice. Encouraging results can be obtained with a prediction of only 5s, notably when the engine stop is included.

An alternative solution to the engine stop-start problem, where the cost of a restart is explicitly introduced in the optimization criterion, as well as the inclusion of a terminal cost to guarantee stability can be further investigated.

Fig. 1 .

 1 Fig. 1. DCT hybrid configuration

Fig. 2 .

 2 Fig. 2. Fuel consumption [g/s] with respect to torque; curves for different rotational speed values

Fig. 3 .

 3 Fig.3. SOC limits for 2 cases: total distance known and reset distance fixed at 10 km, respectively

Fig. 4 .

 4 Fig. 4. ICE stop-start strategy

Fig. 5 .

 5 Fig. 5. Reference speed for Artemis urban and traffic jam, respectively

Fig. 6 .

 6 Fig. 6. MPC-based stop-start included, ω ICE for two values of the prediction horizon: 1s and 5s, respectively; Artemis urban and traffic jam

  rat em -ratio between the EM and the corresponding shaft where it is connected • ω ctrl ice -idle speed or 0 rpm, in case of engine stop • R w -wheel radiusII. POWERTRAIN CONTROL-ORIENTED MODEL

• F D Ri -axle ratio corresponding to ith shaft • r w ice/em -ratio between the ICE/EM torque at wheel level and the component (ICE/ EM) torque 978-1-5090-2720-0/16/$31.00 c 2016 European Union •

TABLE II FUEL

 II CONSUMPTION [L/100KM] AND FINAL SOC[%] FOR MPC (WITH AND WITHOUT S&S) AND ECMS For a constant equivalence factor λ, determined offline for each drive cycle

	````````M ethod	Drive cycle	Artemis urban	Artemis jam
		ECMS *	6.34 (53.5%)	6.8 (53%)
	MPC	1s 5s	6.8 (54%) 6.72 (53.5%)	7.85 (54%) 7.76 (53%)
	MPC	1s	6.8 (54%)	7.85 (54%)
	(S&S)	5s	6.63 (53.5%)	7.4 (53%)

*


Nahuel Lascano

Guillermo Gallardo

Rachid Deriche

Dorian Mazauric

Demian Wassermann

Extracting the Groupwise Core Structural Connectivity Network: Bridging Statistical and Graph-Theoretical Approaches

Keywords: Group-wise connectome, core graph problem, brain connectivity, diffusion MRI

Finding the common structural brain connectivity network for a given population is an open problem, crucial for current neuroscience. Recent evidence suggests there's a tightly connected network shared between humans. Obtaining this network will, among many advantages, allow us to focus cognitive and clinical analyses on common connections, thus increasing their statistical power. In turn, knowledge about the common network will facilitate novel analyses to understand the structure-function relationship in the brain. In this work, we present a new algorithm for computing the core structural connectivity network of a subject sample combining graph theory and statistics. Our algorithm works in accordance with novel evidence on brain topology. We analyze the problem theoretically and prove its complexity. Using 309 subjects, we show its advantages when used as a feature selection for connectivity analysis on populations, outperforming the current approaches.

Introduction

Isolating the common brain connectivity network from a population is a main problem in current neuroscience [START_REF] Bullmore | Complex brain networks: graph theoretical analysis of structural and functional systems[END_REF][START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF][START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF]. Recent evidence suggests that there's a common and densely connected brain connectome across humans [START_REF] Bassett | Robust detection of dynamic community structure in networks[END_REF]. In this work we present a new approach for selecting these common connections, combining recent topological hypotheses [START_REF] Bassett | Robust detection of dynamic community structure in networks[END_REF] and current methods [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF][START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF].

Finding the common brain connectome across subjects has the potential to increase our understanding of the relationship between function and structure in the brain. This relationship is one of the main open questions in neuroscience [START_REF] Bullmore | Complex brain networks: graph theoretical analysis of structural and functional systems[END_REF][START_REF] Donahue | Using Diffusion Tractography to Predict Cortical Connection Strength and Distance: A Quantitative Comparison with Tracers in the Monkey[END_REF]. Moreover, knowledge about the most common connections in a population will facilitate clinical and cognitive Diffusion MRI analyses by reducing the number of surveyed connections, increasing the statistical power of those analyses. Finding the common connectome will also allow us to increase our knowledge about the brain structure by comparing core networks across different populations.

We formalize the problem of selecting the common connections combining graph theory and statistics. Then, we prove that the problem is NP-Hard and propose a polynomial-time algorithm to find approximate solutions. To do this, we develop an exact polynomial-time algorithm for a relaxed version of the problem and prove the algorithm's correctness and complexity.

Currently, the most used algorithm to extract a population's core structural connectivity network (CSNC) [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF] uses an statistical approach: first, compute a connectivity matrix for each subject; then, analize each connection separately with a hypothesis test, using as null hypothesis that that edge is not present in the population; finally, construct a binary graph with the edges for which the null hypothesis was rejected. The main problem of Gong et al.'s [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF] algorithm is that the resulting graph can be a set of disconnected subgraphs. Moreover, recent studies have shown that the brain has a core network tightly connected and a sparsely connected outer one [START_REF] Bassett | Robust detection of dynamic community structure in networks[END_REF]. In other words, this approach ignores the resulting network's topology. Performing statistical analyses in a feature set chosen by hypothesis testing incurs in the double dipping problem [START_REF] Kriegeskorte | Circular analysis in systems neuroscience: the dangers of double dipping[END_REF].

A newer approach to solve the CSNC problem, designed by Wassermann et al. [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF], uses graph theory to get a connected CSCN: first, compute a binary connectivity graph for each subject using a threshold; for each possible connection compute the "cost" of including or excluding it from the common graph by evaluating in how many subjects that connection is present; finally, construct the binary graph with all the edges that is "cheaper" to include than to exclude and connect the resulting graph if it's disconnected, using the minimum possible cost. This algorithm guarantees that the resulting graph is connected, but the connection binarization discards significant information for the resulting common network. In other words, it discards information of the probability of each connection being in the brain. This is problematic because the resulting graph may include edges for which tractography assigned a very low existence probability across subjects. Also, the outer part of the brain, the connections which do not result in the core network, should also be sparsely connected [START_REF] Bassett | Robust detection of dynamic community structure in networks[END_REF], which this algorithm does not enforce.

In this work we propose, for the first time, a polynomial-time algorithm to obtain the CSCN of a population addressing the issues listed above. Our algorithm combines the recent graph-theoretical approach [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF] with the statistical awareness of the most popular one [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF]. We start by formalizing the problem, which allow us to prove that it's NP-Hard. Then, we propose a first algorithm that solves a relaxed version of the problem in an exact way, giving the best possible core graph for our formalization. Then, we adapt it to guarantee a connected result, agreeing with recent evidence on structural connectivity network topology [2, e.g.]. Finally, we validate our approach using 300 subjects from the HCP database and comparing the performance of the networks obtained by our new approach, Wassermann et al.'s [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF] and Gong et al.'s [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF] predicting connectivity values from handedness in the core network.

Definitions, Problems and Contributions

We want to develop a new algorithm to extract the core structural connectivity network, a problem that implies working with different brains. Thus, the first thing we need to do is unify them into a common connectivity model. This allows us to model all brains with graphs in which each node represents a cortical or subcortical region, and each edge represents a white matter connection between two regions. We choose the Desikan parcellation [START_REF] Desikan | An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest[END_REF] to uniformize the brain cortical and sub-cortical regions across subjects.

To compute the connectivity matrices we use a probabilistic tractography algorithm, which outputs one matrix per subject. The resulting matrices represent the existence probability of a connection across parcels in each subject [START_REF] Donahue | Using Diffusion Tractography to Predict Cortical Connection Strength and Distance: A Quantitative Comparison with Tracers in the Monkey[END_REF]. As these are symmetric, we interpret the matrices as weighted undirected graphs, sharing the node set across subjects.

Formally, we represent a sample of N brain structural networks by N complete weighted graphs G 1 = (V, E, w 1 ), . . . , G N = (V, E, w N ) with a common node set V . We call G 1 , . . . , G N the sample graphs. Each graph G i corresponds to a subject. Each vertex v ∈ V represents a cortical or sub-cortical region. Each edge e ∈ E = V × V represents a white matter bundle connecting two regions. Finally, the weight w i (e) is the connection probability for the edge e in the subject i obtained through tractography:

w 1 (e), w 2 (e), . . . , w N (e) ∈ [0, 1] ∀e ∈ E. (1)
Note that all graphs have the same ordered node set and all of them are complete: an edge weight, or connection probability, w i (e) of 0 represents an absent connection. Using this formalization we express the general core structural connectivity network problem as follows: find a core graph G * = (V * , E * ) densely connected such that G * keeps the more relevant connections E * ⊆ E in the sample and discards the less relevant ones, for some definition of relevance and density. For simplicity, once we select E * we can define V * as

V * = {v ∈ V : ∃u ∈ V, (u, v) ∈ E * } , (2)
the set of nodes that the edges in E * cover. Then, we can reduce the problem of finding G * to find E * alone. We want a formalization of relevance that represents the probability that a connection is present across subjects. Thus, we choose to model the groupwise relevance, w * (e) as the mean existence probability across subjects, factored by the standard deviation of these probabilities. In other words, w * (e) is the number of standard deviations that the mean existing probability of connection e is larger than 0. We use w * (e) as a statistical measure of edge presence across the population. Formally,

w * (e)
w(e) s(e) where w(e)

N i=1 w i (e) N , s(e) N i=1 w(e) -w i (e) 2 N . (3)
Note that w * (e) is the statistic of a hypothesis z-test which assumes a media of 0 for the population weight of e. We choose the z-statistic because of the normal distribution's properties, e.g. linearity, even if other distributions, such as Beta distribution, may be more appropriate for modeling the probability. In any case, note that for the purpose of our contribution w * can be any function E → R which grows with the relevance of the edges in the sample.

We also want a formalization that represents the density of the core subgraph. We use the relationship between the number of edges and the total statistical relevance w * that those edges sum:

α(w * , E * ) e∈E * w * (e) |E * | . ( 4
)
As we want also a sparse outer subgraph, we also define its density:

β(w * , E * ) e∈E\E * w * (e) |E * | . ( 5
)
Now we can express our objective informally as: choose E * such that α(w * , E * ) (Eq. 4) is large and β(w * , E * ) (Eq. 5) small. In accordance to recent evidence on the core network, we also want G * to be connected. Here, connected means that for every pair of vertices u, v in V * there is a path of edges in E * from u to v.

Let E c be the family of sets of edges that induce a connected graph. We now formalize the problem of finding this common graph G * in two different ways.

• The optimization version consists in computing:

max E * ∈E c f (w * , E * ) = λα(w * , E * ) -(1 -λ)β(w * , E * ) (6)
The parameter λ (between 0 and 1) can be adjusted to weight the density of the inner and the outer network. Note that if λ = 1, the solution to (6) only considers the density of the core network, and if λ = 0, it only considers the edges excluded of the core network. • Given A and B, the decision version consists in finding E * ⊆ E c such that:

α(w * , E * ) ≥ A β(w * , E * ) ≤ B (7)
Having formalized the Core Structural Connectivity Network into an optimization and a decision problem, we proceed with one of our main theoretical contributions: proving that the problem is NP-Complete.

CSCN Problem's NP-Completeness

We have formalized the problem of the Core Structural Connectivity Network taking into account the density and connectedness of the core subgraph and the sparsity of the outer one. We will now prove that, with this formalization, the problem is NP-Complete.

Definition 1 (Core Structural Connectivity Network problem).

Given G 1 = (V, E, w 1 ), G 2 = (V, E, w 2 ), . . . , G N = (V, E, w N ) weighted graphs (the sample graphs) with a common node set, a complete edges set (E = V × V ) and w 1 (e), w 2 (e), . . . , w N (e) ∈ R ≥0 ∀e ∈ E weights of their edges, and given A, B real numbers, find G * = (V * , E * ) connected graph (the core graph) such that

α(w * , E * ) ≥ A β(w * , E * ) ≤ B
for α and β as defined in Eq. ( 4) and Eq .( 5).

Here we prove that the Core Structural Connectivity Network problem, called CSCN problem, is NP-complete. In our reduction, we use the Steiner Tree problem [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF], called ST problem in the following. Given an edge-weighted graph G = (V , E , w), a subset S ⊆ V of nodes, and a real k ≥ 0, ST problem consists in determining if there exists a connected subgraph H such that S ⊆ V (H) and e∈E(H) w(e) ≤ k. The decision version of ST problem is NP-complete even if all weights are equal [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF].

Instance of ST problem. Consider any edge-weighted graph G = (V , E , w) such that w(e) = 1 2 for every e ∈ E . Given k ≥ 0, ST problem consists in determining if there exists a connected subgraph H such that S ⊆ V (H) and |E(S)| ≤ 2k. Without loss of generality, we assume that |E | ≥ 2k and that G is connected.

Reduction. We construct the instance of CSCN problem as follows. Let s = |S| and let t ≥ 1 be any positive integer. Let G = (V, E, w * ) defined as follows.

Let V = V ∪ {v i,j | 1 ≤ i ≤ s, 1 ≤ j ≤ t} and E = V × V . Let S = {u 1 , . . . , u s }. For every i, j, 1 ≤ i ≤ s, 1 ≤ j ≤ t, w *
vi,j ,ui = 1 and w * vi,j ,u = 0 for every u ∈ V \ {u i }. Furthermore, for every e ∈ E , set w * (e) = w(e) = 1 2 , and for every u, u ∈ V such that {u, u } / ∈ E , then set w * e = 0. Finally, we set A = s.t+k s.t+2k and B =

i,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t} if |E * | ≥ s.t.
(Otherwise we select a subset of this set of edges.) Indeed, by construction of G, we have w * vi,j ,ui = 1 for every i, j, 1 2 each. Recall that there are exactly s.t edges of weight 1, and the other edges have weight 0 or 1 2 . There are two cases. First, suppose that

1 ≤ i ≤ s, 1 ≤ j ≤ t. Then, if |E * | -s.t > 0, E * must contain |E * | -s.t edges of E , that is edges of E of weight
|E * | ≥ s.t. We get that e∈E\E * w * (e) = 1 2 (|E | -(|E * | -s.t)). Since |E * | -s.t < 2k, then we get that e∈E\E * w * (e) = 1 2 (|E | -(|E * | -s.t)) > 1 2 (|E | -2k). Furthermore, since |E * | < s.t + 2k, we get that 1 2 (|E |-(|E * |-s.t)) |E * | > 1 2 (|E |-2k)
s.t+2k . Thus, we proved that β(w We prove in Lemma 3 and in Lemma 4 that there is an admissible solution for CSCN problem if and only if there is an admissible solution for ST problem. Lemma 3. If there is an admissible solution for ST problem, then there is an admissible solution for CSCN problem.

Proof. Suppose there is an admissible solution for ST problem. We prove that there is an admissible solution for CSCN problem. Let H be a connected subgraph such that S ⊆ V (H) and e∈E(H) w(e) = 1 2

|E(|H)| ≤ k. If |E(H)| = 2k, then set E * = E(H) ∪ {{v i,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t}. If |E(H)| < 2k, then set E * = E(H) ∪ {{v i,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t} ∪ F , where F ⊆ E such that F = E(H) = ∅, w * (e) = 1
2 for every e ∈ F , and such that the graph induced by E(H) ∪ F is connected. The last condition comes from Corollary 1 in order to get the right number of edges in E * . This condition is always possible to satisfy because G is connected.

The graph induced by E * is connected. Indeed, H is an admissible solution for ST problem, E(H) ∪ F is connected by construction, and {{v i,j , u i } | 1 ≤ j ≤ t} is a set of edges all adjacent to u i ∈ S for every i, 1 ≤ i ≤ s.

Furthermore, we get

α(w * , E * ) = e∈E * w * (e) |E * | = s.t + k s.t + 2k = A and β(w * , E * ) = e∈E\E * w * (e) |E * | = 1 2 (|E | -2k) s.t + 2k = B.
Finally, we proved that there is an admissible solution for CSCN problem.

Lemma 4. If there is an admissible solution for CSCN problem, then there is an admissible solution for ST problem.

Proof. Suppose there is an admissible solution for CSCN problem. We prove that there is an admissible solution for ST problem. Let E * ⊆ E be such that the graph induced by E * is connected, and such that α(w * , E * ) ≥ s.t+k s.t+2k = A and β(w Finally, since E * is an admissible solution for CSCN problem, then it means that the graph induced by the set of edges E * ∩ E is connected and is such that for every u i , 1 ≤ i ≤ s, then there is an edge e ∈ E * ∩ E that is adjacent to u i . By the previous remark, every edge in E * ∩ E has weight 1 2 . Thus, it means that there is E * ∩ E is an admissible solution for ST problem considering the graph G . Indeed |E * ∩ E | = 2k and so e∈E * ∩E w(e) = k.

* , E * ) ≤= 1 2 (|E |-2k) s.t+2k = B. We first prove that {{v i,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t} ⊆ E * .
i,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t}, that is {e ∈ E | w * (e) > 1 2 } = {{v i,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t}. We get that {{v i,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t} ⊆ E * because,
We are now able to prove the NP-completeness of CSCN problem.

Theorem 1. CSCN problem is NP-complete.

Proof. The reduction is clearly polynomial. Furthermore, Lemma 3 and Lemma 4 prove the equivalence between CSCN problem and ST problem. Since the decision version of ST problem is NP-complete even if all weights are equal [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF], then we obtain the NP-completeness of the decision version of CSCN problem.

In Theorem 1 we have proved that CSCN problem is NP-complete. Hence, to be able to solve it in reasonable time we need a relaxation to make it tractable or an approximate algorithm for the complete version. In this article we will propose both.

Relaxation of the CSCN problem

We proved in the previous section that the connectivity constraint is the main reason of the difficulty of the problem. Without it, it becomes tractable. So we solve, in this section, a relaxed version of problem without the connectivity constraint. Then, we use this solution to approximate the full problem. Proof. Algorithm 1, in each step i, defines E * as the i maximum weighted edges and tries to use that to fulfill the constraints.

Theorem 2. The decision version of CSCN problem without the connectivity constraint is in P.

Algorithm 1 Maximum edges

Assume that there exists an E * that fulfills the constraints. There are two cases: 1) E * has the |E * | maximum weighted edges, 2) there are e j ∈ E * , e k ∈ E \ E * such that w * (e k ) ≥ w * (e j ). In 1), Algorithm 1 will find E * . In 2), let

E = (E * ∪ {e k }) \ {e j } another subset of E. Then α(w * , E ) = e∈E w * (e) |E | = e∈E w * (e) |E * | ≥ e∈E * w * (e) |E * | = α(w * , E * ) ≥ A
because the edges in E * are the same as the ones in E except from one that has a larger weight. For the same reason,

β(w * , E ) = e∈E\E w * (e) |E | = e∈E\E w * (e) |E * | ≤ e∈E\E * w * (e) |E * | = β(w * , E * ) ≤ B.
Thus, we found a new subset of E that stills fulfills the constraints. We can do the same process with E (replace an edge with another one of larger weight) iteratively, always getting subsets that fulfills the constraints, until we cannot do this anymore. At that point we will have a subset that has only the maximum |E * | edges and fulfills the constraints. Thus, algorithm 1 will find this subset. We now need to prove algorithm 1 runs in polynomial time in the size of |E|. The first operation, computing w * (e) for each e, implies computing the mean and standard deviation for each edge across the population, which can be done in O(N ) per edge (where N is the size of the population). This is O(N * |E|) for all the edges. The second step, sorting, can be done in O(|E| log |E|).

The main loop runs at most |E| times, and in each loop it adds an edge to E * , computes α and β and performs two comparisons. The comparisons can be done in constant time, as the addition to E * if we use a linked list of edges to represent it. To compute α and β it is needed to iterate once again E (the part in E * for α, the part in E \ E * for β) adding the weights together and then performing two divisions. This can be done in linear time in the size of E, and even quicker (constant time) if we optimize it by keeping the values of α and β across loops and updating them with the weight of the edge that changed sets.

Then, algorithm 1 solves the CSCN problem in O(max(|E| 2 , |E| * N ) or in O(|E| * N ) if a little optimization is used.

Heuristic approach

In Section 2.2 we developed Algorithm 1 to solve the problem of finding the Core Structural Connectivity Network in polynomial time. However, this algorithm does not guarantee a connected result. We solve the original problem, presented in Section 2, by first applying Algorithm 1 and then modifying the resulting core graph G * to guarantee its connectedness. This results in an approximate solution for the full problem computable in polynomial time.

To extend G * into a connected graph we add the necessary edges while decreasing the minimum possible the objective function f defined in Eq. 6. For this, we use the same approach that Wassermann et al. [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF]. Namely, we make a multigraph G cc with the connected components of G * as nodes, complete it with all the possible edges between those connected components, and run a Maximum Spanning Tree algorithm. This selects the edges needed to produce a connected subgraph with the maximum possible weight. For the full details, see Wassermann et al. [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF]. This way we get a connected subgraph close to the best possible subgraph, which we obtained using Algorithm 1.

Experiments and Results

L R L R

Fig. 1. Core structural connectivity network computed by our approach. On the left, we show the adjacency matrix for λ = 0.5, where 48.19% of the connections were included in the CSCN. In the central and right panels, we show the resulting CSCN for λ = 0.9 and 0.99 respectively. The percentage of included connections in the CSCN is 5.99% and 1.27% respectively.

We formalized the CSCN problem in section 2 and designed an algorithm to solve it in section 2.3. Now we will asses the performance of our method. For this, we compare it with the most used [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF] and with the recent one [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF] in the task of connectivity prediction performance.

We use a subset of the HCP500 dataset [START_REF] Sotiropoulos | Advances in diffusion MRI acquisition and processing in the Human Connectome Project[END_REF]: all subjects aged 21-40 with complete dMRI protocol, totaling 309. We compute the weighted connectivity matrices between the cortical regions defined by the Desikan atlas [START_REF] Desikan | An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest[END_REF] as done by Sotiropoulos et al. [START_REF] Sotiropoulos | Advances in diffusion MRI acquisition and processing in the Human Connectome Project[END_REF]. Examples of CSCN exctracted with our algorithm at different λ levels are shown in Fig. 1, which was generated using Nilearn [START_REF] Abraham | Machine learning for neuroimaging with scikit-learn[END_REF].

Consistency of the Extracted Graph

To compare the stability across different algorithms for CSCN, we use an analysis based on Wassermann et al. [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF]: we randomly take 500 subsets of 100 subjects each and computed the core graphs for all subsets. We then compute the number of unstable connections: connections that present in at least one core graph but not in all of them. Finally, in Table 1 we report each algorithm's stability:

stability of the algorithm 1 - #{unstable connections} #{total connections} .
This measure quantifies the CSCN consistency across subsamples. Due to the homogeneity of our sample, we expect the CSCNs obtained by an algorithm to be similar across subsamples. Hence, a stabler algorithm is preferable.

Predicting Handedness-Specific Connectivity

We evaluate performance of the methods by using the generated core graphs as a feature selection for handedness specific connectivity. We use a nested Leave-1 3 -Out procedure: the outer loop performs model selection on 1 3 of the subjects using the core graph algorithm and the inner loop performs model fitting and prediction using the selected features. Specifically, we first take 1 3 subjects randomly and compute the core graph for those subjects using the three different algorithms. Then we add the weights for the selected edges for each subject, and select the features F that are more determinant of handedness using a linear least-squares regression and the Bonferroni Our approach (λ = 0.50) Fig. 2. Performance of core network as feature selection for a linear model for handedness specific connectivity. We evaluate model prediction (left) and fit (right) for Gong et al. [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF] in green, Wassermann et al. [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF] in blue and ours, in red. We show the histograms from our nested Leave- correction for multiple hypotheses. This experiment is repeated 500 times. We quantify the amount of features that are selected after each procedure, which indicates how useful is the core graph algorithm for selecting the edges related to handedness. We show the results in Table 1.

To evaluate the prediction, we randomly take 1 2 of the remaining subjects and fit a linear model on the features F to predict connectivity weights using the handedness of each subject. Finally, we predict the values of the features F from the handedness in the subjects left out. We quantify the quality of the linear model fitting Akaike Information Criterion (AIC) and of the prediction performance with the mean squared error (MSE) of the prediction. For both measures a lower value indicates better performance. The outer loop is performed 500 times and the inner loop 100 times per outer loop, which totals 50,000 experiments. We show the results of this experiments in Fig. 2.

Discussion and Conclusion

We presented for the first time a polynomial algorithm to extract the core structural connectivity network of a population combining a graph-theoretical approach with statistic relevance of the connections, observing the recent evidence of the structural network topology.

Our results show that our algorithm outperforms, in the prediction experiment, the most used technique [START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF] as well as latest approaches [START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF]. In Table 1 we can see that our algorithm preserves, in average, more connections correlated with the handedness of the subjects. We can also see that despite being less stable than Wassermann et al.'s it is stabler than Gong et al.'s. Finally, Fig. 2 shows that, in the handedness prediction experiment, our method outperforms Gong et al.'s and Wassermann et al's: the number of cases with lower AIC and MSE is larger in our case. Hence, our CSCN is better as linear model relating connectivity with handedness in terms of model fitting and prediction.

In terms of theoretical contributions, we formalized the problem, proved its difficulty and gave a novel algorithm for dealing with it. We then validated our approach by showing its power as feature selector for getting connections related to handedness with 300 real subjects' data. The experiment shows our method performs better than the currently available. Moreover, our method avoids the double dipping problem by not choosing the feature set with hypothesis testing.

  Second, suppose that |E * | < s.t. We get that e∈E\E * w * (e) = s.t -|E * | + |E | 2 . Since s.t -|E * | + |E | 2 > 1 2 (|E | -(|E * | -s.t)), we obtain the result by the arguments described for the first case. Finally, if |E * | < s.t + 2k, then there is no admissible solution for CSCN problem. Lemma 2. If |E * | > s.t + 2k, then any solution for CSCN problem is not admissible because α(w * , E * ) < A. Proof. Suppose that |E * | > s.t + 2k. In order to maximize e∈E * w * (e), E * must contain {{vi,j , u i } | 1 ≤ i ≤ s, 1 ≤ j ≤ t} and |E * | -s.t edges of E .Indeed, by construction of G, we have w * vi,j ,ui = 1 for every i, j, 1 ≤ i ≤ s, 1 ≤ j ≤ t. Furthermore, w * (e) =1 2 for every e ∈ E . Recall that there are exactly s.t edges of weight 1, and the other edges have weight 0 or 1 2 . We get that α(w * , E * ) = Indeed, the average weight is lower when there are more edges of weight 1 2 (the number of edges of weight 1 is the same in both ratios). Finally, if |E * | > s.t + 2k, then there is no admissible solution for CSCN problem. By Lemma 1 and Lemma 2, we get the following corollary. Corollary 1. Any solution for CSCN problem is such that |E * | = s.t + 2k.

	s.t+ 1 2 (|E

* , E * ) = e∈E\E * w * (e) |E * | > B. * |-s.t) |E * | < s.t+k s.t+2k = A.

  By Corollary 1, we know that |E * | = s.t + 2k. Thus, it necessarily means that e∈E * w * (e) ≥ s.t + k. By construction of G, the set of edges of weight 1 is {{v

  otherwise, we would have e∈E * w

* (e) < s.t + k. Furthermore, every e ∈ E * ∩ E is such that w * (e) = 1 2 . Indeed, otherwise, we would have e∈E * w * (e) < s.t + k.

Table 1 .

 1 Stability of the algorithms and amount of features selected by linear regression in the core graph relating the weights with handedness. Our procedure gets more features selected than Gong et al.[START_REF] Gong | Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography[END_REF] and Wassermann et al.[START_REF] Wassermann | Extracting the Core Structural Connectivity Network: Guaranteeing Network Connectedness Through a Graph-Theoretical Approach[END_REF], showing better statistical power. It's also more stable than Gong et al., showing improved consistency.

	Algorithm	Features (mean) Features (std) Stability
	Gong et al. 2009	0.066	0.256	0.364
	Wassermann et al. 2016	0.415	0.723	0.644
	Our approach (λ = 0.50)	1.042	1.269	0.528

  1 3 -Out experiment. In both measures, our approach has more frequent lower values than Gong et al., showing a better performance.

Acknowledgements Authors acknowledge funding from ERC Advanced Grant agreement No 694665 : CoBCoM -Computational Brain Connectivity Mapping


Tamene Kenfack

Dominating set in circulant graphs

Keywords: elsarticle.cls, L A T E X, Elsevier, template

A Cayley graph on a group Γ with the generating set S ⊂ Γ is a graph costructed out of Γ. A circulant graph is a Cayley graph on a cyclic group denoted by Circ(n, S). In this paper, we address the problem domination in circulant graphs. We first give upper bounds for the domination number of certain circulant. The domination number of certain circulant graphs is obtained together with a corresponding dominating set. We also obtain the independent domination number and the total domination number for special cases of these circulant graphs.

Introduction

Cayley graphs are usefull in various domains of theoretical computer sciences such as computer architecture, routing problem, networking and coding theory. In the coding theory, the most interesting graphs are those having perfect e-codes, particulary regular graphs, vertex transitive graphs and distance transitive graphs. perfect 1-codes were introduced by Biggs [START_REF] Biggs | Perfect codes in graphs[END_REF] in the context of error correcting codes. A perfect 1-code is a single error correcting code, that is, it permits to correct one error. In the context of graph theory, a perfect 1-code is called an efficient dominating set. This is an independent dominating set in which every vertex not in the set is dominated by exactly one vertex of the set. Independent sets are also useful in the perspective of coding theory. In [START_REF] Chellali | Independent [1, k]-sets in graphs[END_REF] for example, Chellali et al. intoduce the independent [1,k]-sets and state that these sets are usefull in the context of error correcting codes as well. In fact, an independent [START_REF] Biggs | Perfect codes in graphs[END_REF][START_REF] Chellali | Independent [1, k]-sets in graphs[END_REF]-set will permit to correct most errors and those not corrected come from at most 2 sources. Cayley graphs are such regular graphs usable in coding theory. As example, hypercubes are used to represent Hamming codes.They are define out of groups. In fact, for a group(even infinite) Γ and a set S ⊂ Γ such that s ∈ S ≡ s -1 ∈ S and e(Γ) / ∈ S, the Cayley graph G on Γ with the generating set S ⊂ Γ is a graph with vertex set elements of Γ (V) and edge set E(G) = {(g, gs)|g ∈ Γ, s ∈ S}. Cayley graphs on finite cyclic groups are called circulant graphs and denoted by Circ(n, S).

Definitions. In a graph G = (V, E), a dominating is a set D of vertices such that every vertex of G is either in D or has a neighboor in D. The Dominating Set problem ask to find in a graph G = (V, E) a dominating set with minimum cardinality denoted γ(G). A dominating set is efficient if it is independent (i.e. no two vertices of D are adjacent) and each vertex v / ∈ D has exactly one neighboor in D. More generaly, an independent exact k-dominating set D of a graph G = (V, E) is a set of vertices such that every vertex not in D has exactly k neighboors in D. k-dominating set are usefull in the context of faulttolerence application. In this paper, we are investigating independent exacts 2-dominating sets of circulant graphs.

State of art. Minimum Dominating

Set problem ask to find a dominating set of minimum cardinality. This problem is an NP-complete for general graphs even when restricted to the class of planar graphs of maximum degree 3 [START_REF] Yannakakis | Edge dominating sets in graphs[END_REF]. When restricted to certain classes of graphs such as trees, we can get polinomial time algorithms even linear. Many autors address the problem in the class of circulant graphs [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF][START_REF] Dejter | Efficient dominating sets in cayley graphs[END_REF][START_REF] Obradovi | Efficient domination in circulant graphs with two chord lengths[END_REF][START_REF] Chelvam | Subgroups as efficient dominating sets in cayley graphs[END_REF][START_REF] Chelvam | Bounds for domination parameters in circulant graphs[END_REF]. In [START_REF] Dejter | Efficient dominating sets in cayley graphs[END_REF], Dejter and Serra give a method to construct countable family of circulant graphs each admiting an efficient dominating set. Obradovic et al. [START_REF] Obradovi | Efficient domination in circulant graphs with two chord lengths[END_REF] give conditions based on the relationship between cordes length for cayley graphs of degree at most 4, to admit efficient dominating set together with the structure of these sets. In [START_REF] Chelvam | Subgroups as efficient dominating sets in cayley graphs[END_REF] is gived a necessarily condition for a subgroup to be an efficient dominating set. In [START_REF] Kumar | Efficient domination in circulant graphs[END_REF], Kumar and MacGillivray investigate efficient dominating sets in circulant graphs of large degree. They prove that if such graphs admit efficient dominating sets, then either its elements are equally spaced, or the graph is wreath product of a smaller circulant graph with an efficient dominating set and a complete graph. Yung-Ping [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF] generalyse the condition gived by [START_REF] Obradovi | Efficient domination in circulant graphs with two chord lengths[END_REF] together with techniques in [START_REF] Kumar | Efficient domination in circulant graphs[END_REF] to completly describe efficient dominating set in circulant graphs with domination number prime. The problem of determining efficient dominating sets in circulant graphs is so largely study, but no general upper bounds are known for the domination number of circulant graphs. In this paper, we are investigating the problem for graphs with generating sets of the form S = {a, a + 1, ..., a + k -1, n -a, ..., n -(a + k -1)} where a, k, n ∈ N. We also get independent number in certain cases of these graphs.

Preliminaries

In the article by Deng [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF], the following theorem has been stated: Theorem 2.1 (Theorem 3.1, [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF]). Let n = |S 0 |d, where d is an integer. Then Circ(n, S) admits an efficient dominating set D which is a coset of some subgroup of Z n if and only is s i -s j = 0 mod (|S 0 |), ∀s i , s j ∈ S 0 = {0} ∪ S.

Remark 1. For any circulant graph G = circ(n, S) such that the connection set S is a symetric generating set of Z n and D is a subgroup generated by d(ussualy noted D =< d >), we have:

1. ∀s i ∈ S, s i ≡ 0 mod d is equivalent to the fact that vertex 0 ∈ D is adjacent to vertex s i which is also an element od D, hence, the set D is not independent.

2. ∀s i , s j ∈ S, if s i -s j ≡ 0 mod d then the vertices s i and s j have a common neighboor in D, hence the set D cannot be an exact 1-domination set (every vertex not in D has exactly one neigboor in D).

∀s

i , s j ∈ S, if s i -s j = 0 mod d then
2a k + d = (2k + 1)d then 2a k + d divides n implies that gcd(n, S) ≥ d = 1.
We can reduce such cases of circulant graphs to general cases by just treating the connected components since these connected components are circulant graphs of the same degree. The corresponding subgraphs are all isomorphic to circ(n/d, S )

with S = a i /d, a i ∈ S. More generaly, such reductions can allways be applied when gcd(n, S) ≥ 1.

Dominating sets of circulant graphs

Using the theorem 2.1, we obtain the following corollary.

Corollary 1. Let n, k ∈ N such that n = (2k + 1)d. For each a such that a ≡ 1 mod (2k + 1) or a ≡ k + 1 mod (2k + 1), G = circ(n, S), γ(G) = n 2k+1 Proof. Since n ≡ 0 mod (2k + 1), that is n = (2k + 1)d, d ∈ N.
In fact, we have for a ≡ 1( mod 2k + 1) or a ≡ k + 1( mod 2k + 1), and S 0 = {0} ∪ {a, a + 1, ..., a+k -1, n-a, ..., n-(k +a-1)} the evidence that ∀s i , s j ∈ S 0 , s i -s j = 0( mod 2k + 1). By applying theorem 3.1 of [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF], the conclusion follows.

It is a bit surprising that the domination of such circulant graph is not bounded by a constant integer (γ(G) ≤ n 2k+1 + c) when (2k + 1) n and generating set still those described in the previews corollary. The following theorem give an upper bounds for the domination number of circulant graphs where connection set contains consecutive numbers as describe early. To prove this theorem we will construct a dominating set of these graphs.

Proof. First of all, suppose that 2a-1 is the minimum of the two quantities that is d = min{2a -1, n -2a -2k + 1} = 2a -1. Without loss of generality, suppose that the vertice 0 ∈ D, the dominating set been in construction. By adding 0 as a dominator, vertices [a, a+k-1] and [n-a, n-(a+k-1)] are dominated together by 0. To dominate vertices lying into the set

I 0 = [1, a -1] ∪ [n -(a -1), n -1]
we use the following strategy: take vertices v i = v i-1 + k with v 0 = 0 until all vertices of I 0 are dominated. Let γ 0 = 1 + 2a-1 k . γ 0 is the number of vertices to use to dominate all vertices in I 0 . Morever, vertices v i also dominate (γ 0 +1)k vertices other than the one in I 0 . Finally, the number of vertices dominated by

the set V = {v i |0 ≤ i ≤ γ 0 -1} is nd = (2a -1) + (γ 0 + 1)k
To dominate the rest of the graph, let consider the sets I 1 to I d with d = n nd = n (2a-1)+(γ0+1)k . Vertices each I i , i = 1..d -2 can be dominated with the same process the one used for I 0 with the vertex 0 replaced by vertex .... of each I i . Doing so, all I i 0 ≤ i ≤ d -2 have the same number of dominators namely γ 0 .

For the sets I d-1 , the number of dominators is less or equal γ 0 . Since each set I i described bellow has at most γ 0 dominators, the whole graphs has at most γ 0 d dominators hence, γ(G) ≤ dγ 0 = γ 0 n (2a-1)+(γ0+1)k

As special cases of graphs describe in the preceding theorem, the next theorems gived most tight bounds for circulant graphs with connection set containing small consecutive numbers relatively to (a ≤ k + 1 ≤ n). In theorem 3.2 the dominating sets found have equally spaced vertices but are not independent.

Theorem 3.2. Let n, k, a ∈ N s.t k 2 + 1 ≤ a ≤ k + 1, a + k ≤ n+1
I 0 = [1, a + k -1], I 1 = [a+k+1, 2(a+k)-1], ..., I d-2 = [(d-2)(a+k)+1, (d-1)(a+k)-1], I d-1 = [(d -1)(a + k) + 1, n -1]. For each 0 ≤ i ≤ d -2, element of I i are dominated by i(k + a) and (i + 1)(k + a). More precisely, the set I i , 0 ≤ i ≤ d -2, contains two subparts. the upper side [i(a + k) + a, (i + 1)(a + k) -1] dominated by the vertice i(a + k) and the lower part [i(a + k), i(a + k) + k] dominated by the vertice (i + 1)(a + k). for the set I d-1 Corollary 2. Let n, k, a ∈ N s.t k 2 + 1 ≤ a ≤ k + 1, a + k ≤ n+1
I 0 = [1, a + k -1], I 1 = [a + k + 1, 2(a + k) -1], ..., I d-2 = [(d -2)(a + k) + 1, (d -1)(a + k) -1], I d-1 = [(d -1)(a + k) + 1, n -1] with n -1 = d(a + k) -1. For each 0 ≤ i ≤ d -2,
element of I i are dominated by i(k + a) and (i + 1)(k + a). Thus D is a dominating set. To see the independence of the set D, one can use the remark 1. In fact it is easy to see that between a and a + k -1 there is no divisor of k + a since these numbers are all less than k + a in the settings of the theorem.

We assume the minimality of domination sets found in the two preceding theorems even in those coming down. However, we think it is provable using remark 1 and techniques inspired by Yung [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF].

Theorem 3.3. Let n, k, a ∈ N s.t 2 ≤ a ≤ k 2 , n = 2(3k + 2a -1)d, d ∈ N and S = {a, a + 1, ...a + k -1, n -a, ..., n -(a + k -1)}. The domination number of Circ(n, S) is γ(G) = 2d and < 3k + 2a -1 > ∪ < 3k + 2a -1 > +k is a dominating set. Proof. Let d = n 3k+2a-1 and v = d-1 2
. Vertices of G can be partioned into sets I i as follows.

I 0 = [0, 3k + 2a -1], I 1 = [3k + 2a, 2(3k + 2a -1) -1], I v-2 = [(v-2)(3k+2a-1)+1, (v-1)(3k+2a-1)] and I v-1 = [(v-1)(3k+2a-1)+1, n-1] where n -1 = v(3k + 2a -1) -1. Each intervalle I i = [i(3k + 2a -1), (i + 1)(3k + 2a -1) -1], 0 ≤ i ≤ v -1 is
dominated completely by two vertices namely x i1 = i(3k + 2a -1) + k + a -1 and x i2 = i(3k + 2a -1) + 2k + a -1.

Since Cayley graphs are vertex-transitive, we remove (4k + 3a -2) from each dominator we will get the set describes in the theorem.

The following theorem generalise the situations of theorem 3.3 to situations . Theorem 3.4.

Subgroups as independent domination sets

Dominating sets of graphs described in the theorem 3.3 are not independent dominating sets. To obtain independent dominating sets in certain cases, it is possible to use the following theorem. The independent domination number of G is i(G) = n k+a . Proof. Using techniques of theorem 3.2, it is easy to show that the subgroup generated by k + a is a dominating sets. to see the indepence, recall that these sets have equally spaced vertices hence, the distance between two consecutive elements is allways the same. This distance has the value a + k which is greater than the maximum distance between two adjacent vertices (a + k -1). Then the subgroup < a + k > is also an independent sets for these circulant graphs.

The theorem bellow gives a constructive tool to get circulant graphs that admit subgroups as independent domination sets. We thing that it is possible to give a caracterization as those gives in [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF] for circulant graphs admiting subgroups as independent dominating set. The next theorem will give a caracterisation of those graphs with independent dominating sets that are subgroups and their cosets. Using notations of theorem 2.1, we can make the following remarks. Base on these remarks, we can formulate the following theorem. 1. ∀s i ∈ S, s i = 0 mod d;

2. the number of tuples s i , s j such that s i -s j = 0 mod d is at least 2d The prof of this theorem is based on the reamark 1. The first condition ensures the independence of the subgroup and the second ensures that the set D is a dominating set. It worth noticed that in certain cases, well choosed D will also be a minimum dominating set. Theorem 3.1 describes such cases for graph with generating sets that are consecutive integers. The following corollary complete theorem 3.1 and theorem 4.2 by giving a criterion like the one in corollary 1 for circulant graphs with generating that are consecutive integers to admit subgroups as minimum independent set. It is easy to see that this theorem is clearly an extension of corollary 1, in fact, by setting a = k + 1 we clearly get the corollary. In this corollary, if the order n of the graph is not a multiple of

Conclusion and future work

In this work we give some bounds for the domination number of circulant graphs where connection sets are consecutive integers. We got some subgroups that are minimum dominating sets and independent dominating sets for these circulant graphs. We are planning to use these results and those in [START_REF] Yung-Ping | Efficient dominating sets in circulant graphs with domination number prime[END_REF] in the perspective of coding theory. The problem of finding domination sets of circulant graphs remains not completely solve. Is this problem solvable or MDS is also NP-Complete for circulant graphs?

Theorem 3 . 1 . 1 .

 311 Let a, k, n ∈ N such that 2k + 1 ≤ n, S = {a, a + 1, ..., a + k -1, n -a, ..., n -(a + k -1)}. Let d = min{2a -1, n -2a -2k + 1} and γ 0 = d k +The domination number of the circulant graph circ(n, S) is upper bounded by γ ≤ γ 0 n (2a-1)+(γ0+1)k .

2 andS

 2 = {a, a + 1, ...a + k -1, n -a, n -(a + 1), ..., n -(a + k -1)}. The domination number of the circulant graph G = circ(n, S) is upper-bounded by γ(G) ≤ n k+a . Proof. Let d = n k+a and D = {0, k + a, 2(k + a), ..., (d -1)(k + a)}. Vertices of the graph G = Circ(n, S) can be partioned into sets

2 andS

 2 = {a, a + 1, ...a + k -1, n -a, n -(a + 1), ..., n -(a + k -1)}. If n divides k + a then γ(G) = nk+a and the subgroups < k + a > is an independent domination set for Circ(n, S). Proof. Let d = n k+a and D =< k + a >= {0, k + a, 2(k + a), ..., (d -1)(k + a)}. As we did in the proof of theorem 3.2, we partitioned the vertices of the graph into following sets

Theorem 4 . 1 .

 41 Let n, k, a ∈ N such that k+a divides n, a ≤ k+1 and n ≥ 2k+1. Let S = {a, a + 1, ..., a + k -1, n -a, ..., n -(a + k -1)} and G = Circ(n, S).

Theorem 4 . 2 .

 42 Let n ∈ Z, S with |S| = 2k a symetric generating set of Z n , and D be a subgroup of Z n generated by some integer d = max k+1 a=2 {k + a}. D is an independent dominating set if the following conditions are meet.

Corollary 3 . 2 .

 32 Let n, a, k ∈ N such that a + k ≤ n+1 2 , a ≤ k + 1, and (k + a) | n.Let a ∈ N such that a ≡ a mod (a + k) or a ≡ 1 mod (a + k), and a + k ≤ n+1 The circulant graph Circ(n, S) where S = {a , a + 1, ..., a + k -1, na , ..., n -(a + k -1)} admits the subgroup generated by a + k and its cosets as minimum independent dominating set.


la réécriture du fantasme à l'adolescence pourrait utiliser, dans l'après-coup du stade du miroir, une sorte de stade du jeu vidéo conçu comme répétition générale du rapport à l'autre. Jean-luc Godard dit du cinéma qu'il est le « musée du réel », le jeu vidéo en serait l'atelier. comme les pilotes d'avions s'entraînent au préalable sur des simulateurs de vol, l'adolescent s'entrainerait au rapport à l'autre, à l'autre sexe.

cependant, cette répétition générale ne saurait surprendre le jour J de la représentation que lors d'une rencontre (au sens de tuchê) avec l'extérieur. car si le jeu vidéo a valeur d'espace transitionnel, il ne saurait se substituer à la rencontre, à la relation véritable, physique, concrète.

« -Je peux en un clic devenir ami avec une personne qui habite à l'autre bout du monde sans sortir de chez moi.

-Je peux m'acheter une robe en allemagne et être livrée en 24h sans sortir de chez moi.

-Je peux acheter ma musique sur itunes et l'écouter immédiatement sans sortir de chez moi.

-Je peux voir des milliers de films sans sortir de chez moi.

-Je peux trouver un homme beau, intelligent et sincère sans sortir de chez moi.

-Je peux avoir à ma disposition un savoir infini et gratuit sans sortir de chez moi.

-Je peux être en contact avec tous mes amis sans sortir de chez moi.

-Je peux tout faire sans sortir de chez moi.

et maintenant je peux même sortir de chez moi sans sortir de chez moi.

ah bon ? oui comment tu fais ? et bien… je… sors A l'épreuve du sujet virtuel Michèle Benhaïm, Jamel Jouault, Karine Méolans-lafon

Adolescence, 2016, 34, 2, 309-318.

donc t'es dehors. donc tu ne peux pas sortir de chez toi sans sortir de chez toi, puisque tu sors de chez toi. donc t'es sortie de chez toi. donc tu peux pas.

si. avec la maison tortue. la maison tortue est, comme son nom l'indique, une maison que l'on porte sur son dos, comme les tortues. il suffit de se l'accrocher sur le dos et sur la tête. en prenant bien garde à ne pas se trancher le cou.

-Ben dis donc c'est pas très rassurant comme système.

et sortir de chez soi tu crois que c'est rassurant ? oui enfin il faut pas exagérer, qu'est-ce qui peut t'arriver ? -M'arriver ? M'arriver ? Mais me retrouver en contact avec des gens. des gens que je ne connais pas. Être obligé d'entrer en relation avec eux. communiquer réellement. Je veux dire, peut-être même les toucher.

et alors ? et alors ? et alors ? et les maladies ? les agressions ? les virus ? la gastro ? les odeurs ? le bruit ? les voix ? les corps ?

-Mais c'est la vie ça.

la vie ? la vie ? Mais enfin dans quel monde tu vis ma pauvre ? dans le monde réel.

-… … » 1 . la pratique clinique avec les adolescents permet d'évoquer la question du savoir, la question du transfert et de proposer quelques réflexions sur les enjeux de la cure avec les adolescents, enjeux que nous articulerons avec les concepts de division subjective et de castration.

lors de la présentation de « l'espace psy » aux adolescents, il est important de leur signifier qu'ils sont des sujets de droit et notamment qu'ils ont le droit de venir parler avec nous ; l'idée consiste à inverser d'emblée le schéma habituel consistant à dire aux ados qu'ils doivent parler, c'est-à-dire à leur signifier une injonction de parole. pacte nouveau, faisant passer de l'aliénation en tant qu'objet du droit au statut de sujet de droit 2 . il s'agit ensuite de calquer l'offre de parole sur la formulation du secret (et non l'inverse) : quelque chose comme la possibilité de pénétrer un espace privé à condition d'être obligé de se taire (aliénation-séparation du discours au secret, séparation d'avec le discours ambiant, aliénation au , 1. dollé J.-c. ( 2015). E-Génér@tion. paris : l'harmattan (citations disséminées dans l'intégralité de l'oeuvre).

2. discussion avec Xavier canonge lors du séminaire de recherche de master 2 et doctoral de psychanalyse, université aix-Marseille, mai 2014. secret garanti par une figure autre). il est en effet important de le signifier à l'adolescent : c'est une première présentification de la division subjective sur laquelle nous reviendrons.

Arthur est rencontré dans une institution de soins pour adolescents. il est addict aux jeux vidéo depuis l'âge de treize ans. il en a dix-sept. auparavant il a été diagnostiqué hyperactifàcinq ans et a pris un psychostimulant de cinq à treize ans. depuis, d'ailleurs, sont entrés dans ce lieu de soins trois adolescents addicts aux écrans tous auparavant diagnostiqués « hyperactifs » et tous auparavant traités par psychostimulant.

arthur profite de la coupure, de la scansion que représente l'admission pour soins pour, dit-il, « se poser et ne plus entendre parler de rien ». la formulation est intéressante lorsque l'on sait à quel point de nombreux adolescents errent aujourd'hui au coeur d'une invalidation du langage qui va de pair avec un champ de l'image exacerbée où les limites du visible et de l'invisible se forcent (le montrable/le non-montrable, le champ/le hors-champ, etc. souvenons-nous des films de lars von triers qui mettent en scène l'irreprésentable…). cette sorte de dévoilement ob-scène de la culture de l'intime adolescent serait à réfléchir… arthur est en errance affective et géographique car il a quitté le domicile familial depuis l'âge de quinze ans. il ne possède qu'un ordinateur dont il dira à plusieurs reprises : « c'est toute ma vie ». treize ansdébut de l'addiction aux jeux vidéo -, ce qui recouvre pour arthur cette effraction de réel qu'est la puberté avec son cortège de modifications des enjeux symboliques et imaginaires donnant lieu à la question adolescente par excellence : « Qui suis-je ? » la question du « qui suis-je ? » semble être précédée par celle du « qui suis-je pour le jeu vidéo ? ». le jeu me demande, et je lui réponds ; si je me trompe, la réponse est immédiate, mais ne me menace pas dans mon intégrité, je peux toujours me rattraper, j'ai toujours une chance de répondre correctement. la demande est claire, l'environnement est totalement compréhensible : compréhension de l'attente, des règles du jeu sans l'ambiguïté du sexuel, sans la menace de castration. la récompense est immédiate, accessible : retour de jouissance immédiat dans la demande, jeu ne menaçant pas car jeu non pas avec l'autre mais jeu avec l'objet, dans l'objet, jouant avec la séparation sans la faire apparaître. narcisse se baigne dans son reflet et ne s'y noie pas, dans le reflet de son reflet, ou plutôt dans l'écho de son reflet, lorsqu'il dit une action avec son doigt et qu'elle lui revient, identique, même. l'ordinateur, écho, condamné à répéter ce que lui, en tant que joueur, lui indique. si l'adolescence, le phénomène pubertaire, conduit généralement le jeune à deux découvertesune dans le champ des générations (le père est un fils) et une dans le champ de la sexuation (ma mère est une femme) -, arthur ne semble pas être encore ne serait-ce qu'au seuil de cette question. en effet, tout se passe comme si arthur n'était pas encore entré dans ce moment juvénile où l'on devient un homme, une femme, où l'on attend de l'autre le savoir qui donnerait le code de ce devenir. l'effraction pubertaire a fonctionné aux dires d'arthur comme un ravage, comme si cela avait produit un court-circuit collusif avec le réel. frôlant une destinée tragique oscillant entre la folie et la violence, avec pour horizon la mort, et plutôt que de se heurter au constat dépressiogène chez l'adolescent que le savoir ne peut être obtenu, simplement parce qu'il n'existe pas, arthur se barde, se « remparde » comme le dit J. lacan du petit hans, de codes informatiques grâce auxquels il fait ce qu'il veut, comme il veut, quand il veut dans ses jeux vidéo, c'est-à-dire grâce auxquels il défie toutes les règles et effectue des missions impossibles. on peut supposer qu'arthur, non psychotique, possède le signifiant mais ne sait pas bien s'en servir pour autant, il ne peut se présenter lui-même et peut seulement être représenté, héros de tous ses jeux auxquels il s'aliène. les jeux vidéo ne laissent pas de place au vide, ni au trou dans le savoir, ils sont une solution idéale pour combler ce trou. c'est cet aspect qui crée l'addiction : sorte de passion pour une croyance en un savoir qui existerait.

les enjeux cliniques à l'adolescence vont consister à tenter de philosopher avec cet adolescent ; de la même façon que nous jouons avec les enfants, nous philosophons avec les adolescents. cela peut nécessiter des détours visant à rencontrer l'être adolescent dans le lieu où, pour l'instant, il évolue. pendant plusieurs mois, confrontée au silence d'arthur, silence dû à une impossibilité de penser, d'associer, de dire, silence pesant pour arthur, nous le rompions très vite pour « discuter » (lorsque les adolescents nous disent « je veux pas vous voir », nous répondons souvent que nous non plus mais que nous pouvons discuter néanmoins. et discuter, ils le veulent bien), pour nous faire inviter, pour être digne de rencontrer son monde (c'est le travail avec les autistes qui nous l'enseigne). pendant des mois, il nous a expliqué les jeux, les stratégies, les personnages, les objectifs, les impossibles, les défis, l'héroïsme, etc. Bref, nous avons discuté. ce pasàsa rencontreaoeuvréàla construction d'une relation qui prenait une forme nouvelle, car là où auparavant nous extirpions arthur de son écran pour aller discuter, nous nous sommes mis à trouver un arthur debout et qui nous attendait, bousculant les autres pour passer le premier. peu à peu, puisque arthur pensait que l'on partageait tous les deux à présent un savoir, le rapport à l'écran et aux jeux vidéo s'est modifié et il a pu dire l'enfermement et l'extrême solitude. rentrer avec arthur dans sa passion (du vide) afin de le rencontrer a reposé sur l'hypothèse que, confronté seul à ce vide, arthur risquait une sorte de néantisation. parler beaucoup, face à son silence, reposait sur l'hypothèse que pour arthur, l'autre n'existait pas ou si peu, en tout cas cet autre semblait en panne. en parlant nous le faisions, de fait, exister. en effet, l'autre n'existe certes pas, mais il suffit quand même que quelqu'un parle pour le faire exister. « -" tante, parle-moi ; j'ai peur.

à quoi cela te servirait-il, puisque tu ne me vois pas ? " à quoi l'enfant répond : " il fait plus clair lorsque quelqu'un parle " » 3 . puis, une fois la rencontre établie sur le mode de l'humour relatif à notre bêtise non feinte quant aux jeux vidéo, nous n'apparaissions plus, l'autre n'apparaissait plus menaçant, inclue à notre tour dans son monde parallèle, ce monde dont il dit qu'il est le seul lieu où il se sent exister. nous pouvonsàprésent, à deux, envisager une sortie de l'enfermement. ainsi, nous passons à une autre étape : nous sentons arthur plus solide psychiquement, apte aujourd'hui à supporter que nous bousculions ses résistances et que nous attrapions la moindre formation de l'inconscient ; ce qui était inenvisageable au début et qui aurait même eu, pensons-nous, un effet traumatique. c'est là que nous concevons une distinction de taille entre division et castration. les premiers temps de la psychothérapie n'auraient pu voir arthur supporter la moindre perte. inutile de tenter de mener l'adolescent à devoir accepter la castration, vu que cela est justement ce dont il souffre et ce dont il se plaint. c'est bien parce que rien n'est d'abord envisageable du côté de la perte que le manque, à force d'être contourné, se réduit au vide. ce sont des moments délicats où l'on sent que forcer ce contournement, cet évitement, équivaudrait quasiment à un acte traumatisant. arthur est fragile comme tous les adolescents mais… un peu plus. donc parler, philosopher, présentifier, figurer de l'autre signe la division du sujet. le langage en tant qu'ensemble des signifiants, c'est bien ce que lacan appelle aussi le grand autre. et la présence du langage a pour effet de diviser le sujet ; mais cette opération a plusieurs dimensions : une coupure qui permet que des bords se dessinent et la castration proprement dite. Quant à la coupure, elle distingue elle-même une division subjective et une division temporelle. lorsque la division s'opère dans le sujet, par le signifiant, c'est également la division temporelle qui est introduite. si entrer dans le langage, c'est entrer dans le temps, la division espaçant s de a 1 , le jeu vidéo serait à la fois lieu de l'immédiat, de la suspension du temps, et lieu d'une réunification du sujet, re-trouvaille avec le phallus perdu. le vir, racine du virtuel et de virilité, pouvant se déployer à tout moment, immédiatement. Jouer aux jeux vidéo serait alors une façon de s'affranchir de la temporalité chronologique en pointant que, lorsque l'on donne comme critère à un jeu excessif ou abusif le temps passéàjouer, on ' 3. freud, 1923, p. 384. omet peut-être la dimension logique du temps de jouer (car il n'y a pas que le jeu pour contrer l'angoisse, on peut tout aussi bien travailler, et jouer dès que l'occasion se présente. ou repousser toutes les occasions pour jouer).

le trait commun de l'addiction au jeu pourrait être ainsi son utilisation comme fixation du temps pour voir. le jeu comme image actée, comme fixation d'un instant de sidération. l'adolescent refuserait de rester suspendu au temps de l'attente et de l'élaboration symbolique, n'acceptant pas que vienne d'un autre défaillant et menaçant ce qu'il en est de son identité de sujet désirant. fixé dans l'instant de voir, temps d'effraction du réel pubertaire, le sujet s'engage dans une économie dialectique, évitant ou plutôt retardant ainsi le risque d'une confrontation avec l'altérité, s'engageant dans la recherche d'un hors-temps, sorte de mise entre parenthèses (temporelle et sexuelle ?), son détachement à l'égard de l'autre se traduisant par un détachement à l'égard du langage et un recouvrement du vide par l'écran (couvrant le vide, d'une part, et l'image, d'autre part). l'acte télécommandé apparaît ici comme dépassement de l'image, répétition de la conclusion, tentative d'un mouvement duel voir-conclure écourtant, accélérant le temps pour comprendre. toutes les facultés cognitives, les ressources psychiques, l'attention, la mémoire, les stratégies, les tactiques étant tendues, unifiées vers ce but, mobilisées pour conclure, le plus hâtivement possible. si nous ne nous précipitons pas dans le même, alors il y a décalage, la frustration du décalage entraîne l'arrêt du jeu ou la poursuite, la répétition jusqu'à satisfaction. sans affronter l'énigme du sexuel, fixer l'écran pour ne plus voir l'image, regarder l'écran pour ne plus fermer les yeux, ne plus voir les images du dedans, ce réel in-montrable (ce réel qui fait effraction, pouvons-nous supposer).

une distinction se situerait là, entre addiction et divertissement. entre répétition mortifère, recouvrement du fantasme et travail du fantasme. entre rêvasserie et fantasme. le jeu vidéo vient, par la circulation des objets imaginaires à l'écran, attirer la libido pour ensuite la condenser dans un fantasme. comme si la vitesse du flux des objets, et des actions, importait plus que les objets eux-mêmes. l'addiction à la vitesse faisant disparaître le temps, lorsque la vitesse des commandes dépasse celle du langage, la diction devient a-diction. , dès lors, la division subjective est opérée lorsqu'arthur accepte de parler de son jeu à cet autre qui parle pour exister. car, le mot sépare de la chose4 , le discours sur le jeu, à propos du jeu, permet de développer les images à partir de soi, de s'en distancier. en racontant le jeu, les images, bien qu'actées, passionnent le joueur. en transmettant, le partage des images embarque dans le véhicule des signifiants, et le jeu qui le signifiait (lui qui était représenté par le signifiant virtuel) est réarticulé à partir de son je (division subjective), devenant signifié pour les signifiants qu'il développe. il redevient là maître du jeu en parlant de lui, en se faisant garant de sa parole. l'objet disparaît, à mesure qu'il parle de lui ; et s'effaçant, l'objet introduit le sujet au temps pour comprendre, incluant le déroulement de la chaîne des signifiants enfin apprivoisés, l'élaboration de ce qu'il en est de l'autre et de sa propre division. cette élaboration autorise le temps pour conclure, ce jour J de la représentation, conclusion identitaire incluant la hâte, précipitée par la rencontre (tuchê), solution qu'il faut saisir quand elle se présente. Moment de l'assertion sur soi et du jugement. acte, mouvement, par lequel du jeu enfin peut émerger le je d'un individu capable de s'affirmer comme sujet par un acte.

d'autre part, le jeu multiplie les signifiants du phallus afin de recouvrir le manque contournant la castration qui permet l'accès au sexuel. en effet, dans la relation à l'ordinateur, le joueur n'est pas contraint de se positionner en tant que sexué. il est indifférent à l'ordinateur, le jeu ne réagit qu'à la commande, le joueur n'est jugé que sur sa performance, sur ses réactions. le jeu vidéo est un objet qui ne déçoit pas, qui répond à la satisfaction hallucinatoire du besoin. c'est comme une mère-jeu-vidéo, apportant une réponse à l'enfantréponse qu'il a choisie en choisissant le jeu, en choisissant la question, en s'y confirmant et en s'y conformant. c'est une hallucination reproductible en presque tout point. si la béance s'ouvre dans la discordance entre satisfaction hallucinatoire du besoin et ce que la mère apporte à l'enfant, on pourrait dire que le jeu vidéo apporte une réponse concordante à la satisfaction recouvrant la béance. puis, en parlant, ces objets tombent devant lui comme d'une poche. arthur peut alors s'apercevoir de la quantité qu'elle contenait. enfant phallus de la mère, mère phallus de l'enfant ? entrer dans l'image c'est aussi sortir du monde, se « décastrer », en quelque sorte… ce n'est qu'une fois ce déplacement subjectif effectué (ici, le moment où arthur nous attend, parle aux éducateurs de l'entretien du soir avec nous, fait le caïd pour passer avant les autres) que l'on peut envisager l'interprétation qui va confronter l'adolescent aux limites, c'est-à-dire à la seconde dimension. lorsque le lien est toujours menacé d'être suivi d'une perte, il y a conflit d'altérité.

ce point de basculement ne se fait pas attendre, nous trouvons arthur en larmes, très énervé, une main bandée d'avoir cogné un mur, une cheville foulée d'avoir pris une porte à coups de pied. nous lui demandons ce qu'il se passe et arthur nous raconte que son ordi est tombé, que le disque dur est mort et qu'il a perdu tous ses jeux et tous ses mangas ; il ajoute : « Mon ordi c'est toute ma vie »… nous lui demandons comment cette chute a-t-elle été possible et là, il nous montre où il avait posé l'ordi, en équilibre sur un muret… un peu désemparée face à un tel acte manqué, la sonnerie de notre téléphone portable retentit alors. tandis que nous la coupons aussitôt, arthur nous dit : -« non, mais répondez si vous voulez ». nous répondons : -« non, plus tard, même si notre téléphone portable c'est toute notre vie et que nous ne risquons pas de le poser ici en équilibre. » il semble stupéfait, nous interrompons la séance. cette interprétation a fait point de bascule, elle a entraîné une connexion d'arthur avec son inconscient : arthur se montre « prêt » à faire le deuil… l'intérêt que nous avons manifesté pour comprendre les jeux vidéo semble avoir troué un espace clos et avoir permis que s'envisage pour arthur une construction altéritaire. à cette condition seulement, il peut laisser échapper quelque chose de son désir dans le fracas de l'ordinateur. la séance qui suit, arthur est gêné, timide, nous dit qu'il doit nous faire part de quelque chose mais que ce n'est pas facile. il ajoute : « Mais je crois que je peux vous parler en toute confiance. » il finit par dire qu'il est tombé amoureux, qu'il sait bien que cette jeune fille est un objet impossible mais que peu lui importe. ce qu'il souligne c'est son étonnement : « c'est la première fois que je ressens quelque chose…»n'empêche qu'arthur vient d'apprendre que ce qui fait la différence des sexes a trait à la jouissance, homme et femme ne font pas un et « il n'y a pas de rapport sexuel »… à partir de là, de cette vérification d'un fait de structure, pourrait-on dire, arthur s'apaise, lâche son ordinateur, raconte qu'il a regardé pour la énième fois sa série manga préférée mais que maintenant, « je ne suis pas dupe, c'est too much, c'est impossible, c'est du foutage de gueule ». de plus, d'avoir pu ressentir un sentiment amoureux l'a reconnecté avec le champ affectif, celui qui est traversé du manque et non plus du vide, et arthur a demandé à quitter le centre une semaine pour aller voir ses parents qui, exprime t-il pour la toute première fois, lui manquent… on pourrait dire que le voile levé (celui de l'écran vidéo ou de fumée pour d'autres jeunes) ne montre rien et que arthur a pu le supporter, faisant appel à l'amour pour suppléer la béance, opération bloquée auparavant. ce moment de la puberté, comme épreuve d'une vérité qui fait horreur au sujet, est difficile. ce savoir de la castration qu'on n'acquiert pas mais qui s'impose au sujet et le marque, ce savoir qu'à treize ou quatorze ans on évite mal… ou douloureusement. cependant, pour arthur, il semblerait que la castration 5 soit devenue envisageable… si l'adolescence ordinaire est un état-limite (rassial, 1999), l'adolescence en grande souffrance se révèle être un état-extrême. autrement dit, ces adolescents contemporains en dérive nous mettent bien au coeur de la question adolescente mais en excès. c'est pourquoi, dans un premier temps, nous ne pouvons nous satisfaire d'« accompagner », il nous faut véritablement « étayer » cet adolescent, tout-petit qui joue au grand, en errance de n'avoir jamais été « adopté », pour lequel tout peut faire signe et réduire l'autre, porteur de signes, à un mauvais objet absolu, à un autre à détruire. si la haine relève ici d'un collapsus entre réel et imaginaire et que, par conséquent, elle exclut le symbolique, alors ce n'est pas une affaire de limites, c'est une affaire d'existence.

Kateb, quatorze ans, sort du bureau, revient, met fin à l'entretien,«j'en ai marre de toi, tu veux toujours qu'on parle ! », va à l'ordinateur, met un clip de rap qui hurle « j'ai pas les mots », nous ironisons (interprétons), il change de clip, énervé, et met « crève en enfer »… affaire d'existence, les mots, le langage, crever, vivre, survivre, exister ? Être adolescent, c'est une difficulté en soi, c'est un passage qui mobilise toutes les dimensions de la structure. parfois, sans point d'ancrage, sans point d'appui, on peut errer comme un fantôme… BiBlioGraphie freud s. (1923( ). l'angoisse. in : Introduction à la psychanalyse. paris : payot, 1961, pp. 370-388. , pp. 370-388.

« la coupure qu'instaure dans la vie de l'homme la présence même du langage »(lacan, 1959-1960, p. 325).