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Keywords: phytoplankton, temperature, evolution, Adaptive Dynamics, microalgae, modelling

Phytoplankton are ectotherms and are thus directly influenced by temperature. They experience temporal variation in temperature which results in a selection pressure. Using the Adaptive Dynamics theory and an optimization method, we study phytoplankton thermal adaptation (more particulary the evolution of the optimal growth temperature) to temperature fluctuations. We use this method at the scale of global ocean and compare two existing models. We validate our approach by comparing model predictions with experimental data sets from 57 species. Finally, we show that temperature actually drives evolution and that the optimum temperature for phytoplankton growth is strongly linked to thermal amplitude variations.

Modelling the effect of temperature on phytoplankton growth across the global ocean

Ghjuvan Micaelu GRIMAUD * , Valérie LE GUENNEC * , * * Sakina-Dorothée AYATA * * , * * * , Francis MAIRET, * Antoine SCIANDRA * * , * * * , Olivier BERNARD *

INTRODUCTION

Phytoplankton gathers planktonic autotrophic organisms, mostly unicellulars, and forms the base of marine food web. Phytoplankton is the input point of inorganic carbon in the trophic net, thanks to photosynthesis. It plays therefore a key role in biogeochemical cycles at global scale [START_REF] Falkowski | Biogeochemical controls and feedbacks on ocean primary production[END_REF]. Its activity depends on many factors, primarily light, nutrient availability and temperature [START_REF] Falkowski | Aquatic photosynthesis[END_REF]. In the context of global warming, predicting how the ocean temperature increase will affect marine phytoplankton is a challenging issue.

Several models exist to take into account the direct temperature effect on phytoplankton growth [START_REF] Eppley | Temperature and phytoplankton growth in the sea[END_REF][START_REF] Geider | A dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature[END_REF][START_REF] Norberg | Biodiversity and ecosystem functioning: A complex adaptive systems approach[END_REF][START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF]. The models from [START_REF] Eppley | Temperature and phytoplankton growth in the sea[END_REF] and [START_REF] Norberg | Biodiversity and ecosystem functioning: A complex adaptive systems approach[END_REF] are mostly used in global scale studies [START_REF] Dutkiewicz | Modeling the coupling of ocean ecology and biogeochemistry[END_REF][START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF]. However, they assume that the maximum growth rate of a given species (µ opt ) increases exponentially with the optimal growth temperature (T opt ).

We focus here on temperature as an evolution driver in phytoplankton at global scale. Evolution of phytoplankton facing realistic temperature conditions has already been modeled by [START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF] using [START_REF] Norberg | Biodiversity and ecosystem functioning: A complex adaptive systems approach[END_REF] model and by [START_REF] Grimaud | Modelling thermal adaptation in microalgae: an adaptive dynamics point of view[END_REF] using [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] model. In line with [START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF] and [START_REF] Grimaud | Modelling thermal adaptation in microalgae: an adaptive dynamics point of view[END_REF], we use the Adaptive Dynamics theory to study a given temperature-dependent growth model in an evolutionary perspective. Nevertheless, we propose an original approach based on function optimization which allows to predict the evolutionary outcomes at the global ocean scale. We validate our approach on a data set of 194 observations (extracted from [START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF]) of the temperature response for different species for which isolation sites are known. We compare the different existing models, and we address the questions of how phytoplankton adapts to in situ temperature variations, and investigate the implications at global scale. We firstly present a model of the direct temperature effect on phytoplankton growth, we then use it in an evolutionary perspective and finally we compare the results to experimental data.

MODELLING THE DIRECT EFFECT OF TEMPERATURE ON PHYTOPLANKTON GROWTH

The direct temperature effect on phytoplankton growth is an asymmetric curve with three key parameters, the minimal, optimal, and maximal temperatures for growth (T min , T opt , T max respectively) called cardinal temperatures (Fig. 1). The maximal growth rate (µ opt ) is defined as the growth rate at T opt . As a first step, we use the model developped by [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] based on [START_REF] Rosso | An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model[END_REF] to represent the thermal growth curve (Fig. 1):

µ B (T ) =    0 if T ≤ T min µ opt φ(T ) if T min < T < T max 0 if T ≥ T max (1) with φ(T ) = λ(T ) β(T ) λ(T ) = (T -T max )(T -T min ) 2 β(T ) = (T opt -T min )[(T opt -T min )(T -T opt ) -(T opt -T max )(T opt + T min -2T )]
(2)

with

T opt > (T min + T max )/2
(3) and where T is the temperature in • C. This model does not make any assumption of structural link between the optimal growth temperature T opt and the optimal growth rate µ opt (an increase of T opt is not followed by an increase of µ opt ). Moreover, the model has only four parameters with biological meanings, which facilitates their estimation with experimental data.

All along this study, we compare [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] model with Eppley-Norberg temperature model [START_REF] Eppley | Temperature and phytoplankton growth in the sea[END_REF][START_REF] Norberg | Biodiversity and ecosystem functioning: A complex adaptive systems approach[END_REF] defined as follow:

µ E (T ) = ae bT 1 - T -z w/2 2 (4)
where a and b are parameters without biological meanings (a = 0.81, b = 0.0631), and w is the thermal niche width corresponding to T max -T min in Bernard and Rémond model. Instead of considering the T opt parameter, this model uses a parameter called z. It corresponds to the maximum of the quadratic expression of eq. ( 4) and it is determined by the "Eppley curve" T → ae bT [START_REF] Eppley | Temperature and phytoplankton growth in the sea[END_REF] (Fig. 1 B). However, T opt can be expressed as a function of z:

T opt = bz -1 + (w/2) 2 b 2 + 1 b (5)
Using parameter z, the model assumes that there is a structural link between T opt and µ opt : the higher T opt , the higher µ opt .

Using data sets for 26 species, [START_REF] Grimaud | Modelling thermal adaptation in microalgae: an adaptive dynamics point of view[END_REF] showed that there exists a linear link between cardinal temperatures:

T max = mT opt + p T min = rT opt -n (6)
and with m = 0.93, p = 9.83 and r = 0.97, n = 21.85. We thus replaced T max and T min in eq.( 1) by their function of T opt in eq.( 6). Given the parameter values, w is almost constant, in accordance with Thomas et al. ( 2012) approach (w = 30). We thus obtain two different temperature models depending only on one parameter T opt , µ B (T, T opt ) and µ E (T, T opt ).

EVOLUTIONARY MODEL FOR THERMAL ADAPTATION

Slow-fast dynamical system

In line with [START_REF] Grimaud | Modelling thermal adaptation in microalgae: an adaptive dynamics point of view[END_REF], we include eq.( 1) and eq.( 4) in a simple chemostat model of phytoplankton growth with varying temperature: 2012), in black circles. However, it is not possible to adjust the maximal growth rate in [START_REF] Eppley | Temperature and phytoplankton growth in the sea[END_REF] model due to its structural dependence to the "Eppley curve" (dashed line).

M :    Ṡ = f S (S, X, T (t)) = D(S in -S) -µ(T (t))ρ(S)X Ẋ = f X (S, X, T (t)) = [µ(T (t))ρ(S) -D]X Ṫ = f T (t) = cos(ωt) (7)
S is the nutrient concentration in the chemostat, S in is the inflow substrate concentration, X is the algal biomass

concentration, µ(T (t)) is either µ B (T (t)) or µ E (T (t)), f T (t) is a periodic function (reflecting seasonality), D is the dilution rate with D < µ(T (t))ρ(S in
) ∀t, and ρ(S) is the substrate uptake defined as:

ρ(S) = S K + S ( 8
)
where K is a half-saturation coefficient. We consider that growth, in term of carbon fixation, is fast compared to temperature fluctuations, i.e. is a small positive parameter. It is possible to analyze system (M ) using the Singular Perturbation Theory [START_REF] Tikhonov | Systems of differential equations containing a small parameter multiplying the derivative[END_REF]. The fast dynamics, where T (t) = T (0), corresponds to the classical chemostat model:

Ṡ = f S (S, X, T (t)) Ẋ = f X (S, X, T (t)) (9)
System (9) has a unique positive globally asymptotically stable equilibrium (S * (T (t)), X * (T (t))) ∈ R 2 + (see e.g. [START_REF] Grimaud | Modelling thermal adaptation in microalgae: an adaptive dynamics point of view[END_REF]), where:

S * (T (t)) = KD µ(T (t)) -D X * (T (t)) = (S in -S * (T (t))) (10)
The slow dynamics is given by:

M 0 :      S * (T (t)) = KD µ(T (t)) -D X * (T (t)) = (S in -S * (T (t))) Ṫ = f T (t), T (0) = T 0 (11)
The reduced system (M 0 ) admits a unique solution T * (t):

T * (t) = T 0 + ω sin(ωt) (12)
Tikhonov's theorem [START_REF] Tikhonov | Systems of differential equations containing a small parameter multiplying the derivative[END_REF] allows us to conclude: Proposition 1. For sufficiently small values of > 0, system (M ) admits a unique positive solution (X (t); S (t); T (t)) on [0; τ ], where 0 < τ < +∞. Moreover:

lim →0 S (t) = S * (t) lim →0 X (t) = X * (t) (13)
From a biological point of view, prop.1 assumes that phytoplankton populations are always at equilibrium because growth is faster than long-term temperature variations.

Assuming small annual temperature fluctuation of amplitude δ, we obtain = δω << 1. The long-term (i. e. annual) dynamics of the algal biomass can thus be approximated by:

S * (T (t)) = KD µ(T (t)) -D X * (T (t)) = (S in -S * (T (t))) T (t) = T 0 + δ sin(ωt) (14)
3.2 Evolutionary model using Adaptive Dynamics theory

We study system (M ) in an evolutionary perspective using the Adaptive Dynamics theory [START_REF] Dieckmann | The dynamical theory of coevolution: A derivation from stochastic ecological processes[END_REF]. To do so we allow one parameter to evolve, called the adaptive trait, here T opt . One mutant X mut appears in the resident population at equilibrium with a different value of T opt , T mut opt :

M mut :          S = S * (T (t), T opt ) X = X * (T (t), T opt ) Ẋmut = f Xmut (T (t), T opt , T mut opt )X mut = [µ mut (T (t))ρ(S) -D]X mut Ṫ = f T (t) (15)
Assuming that the mutant is initially rare, we compute the mutant growth rate in the resident population, f Xmut (T (t), T opt , T mut opt ). Depending on the sign of f Xmut (T (t), T opt , T mut opt ), the mutant can invade and replace the resident or not. Prop.1 insures that resident population is actually at equilibrium during mutant invasion.

Here, because T is a periodically time varying variable of period τ , we use the time average mutant growth rate [START_REF] Ripa | Mutant invasions and adaptive dynamics in variable environments[END_REF]:

< f X mut (Topt, T mut opt ) >= 1 τ τ 0 f X mut (T (t), Topt, T mut opt ) dt ( 16
)
We then compute the selection gradient g(T opt , T mut opt ) which gives the selection direction (e.g. growing or decreasing values of T opt are selected through evolution). The selection gradient is defined as the partial derivative of the time average mutant growth rate with respect to T mut opt evaluated in T mut opt = T opt :

g(T opt , T mut opt ) = ∂ < f Xmut (T opt , T mut opt ) > ∂T mut opt T mut opt =Topt (17)
At the evolutionary equilibrium, the selection gradient is equal to zero:

∂ < f Xmut (T opt , T mut opt ) > ∂T mut opt T mut opt =Topt=T * opt = 0 (18)
The evolutionary outcome of the model is thus given by the selection gradient. It is possible to simplify the way to find T * opt : Proposition 2. The evolutionary equilibrium T * opt is given by:

T * opt = arg max Topt < ln(µ(T(t), T opt )) > (19)
Proof. According to [START_REF] Grimaud | Modelling thermal adaptation in microalgae: an adaptive dynamics point of view[END_REF]:

∂ < f Xmut (T opt , T mut opt ) > ∂T mut opt T mut opt =Topt = D τ . τ 0 µ (T (t), T opt ) µ(T (t), T opt ) dt (20)
Moreover:

∂ < ln(µ(T(t), T opt )) > ∂T opt = 1 τ τ 0 µ (T (t), T opt µ(T (t), T opt ) dt (21)
If T opt = T * opt (evolutionary equilibrium), eq.( 20)=0. Thus:

∂ < ln(µ(T(t), T * opt )) > ∂T opt = 0
which is equivalent to say that:

T * opt = arg max Topt < ln(µ(T(t), T opt )) >

GLOBAL OCEAN SCALE SIMULATIONS

Evolutionary model with realistic temperature signal

We will now study phytoplankton thermal evolution at global ocean scale. Let us consider an ubiquitous phytoplankton species which has evolved locally at each sea surface location (i, j) in response to environmental pressure. In a first assumption, each point of latitude/longitude (i, j) can be viewed as a chemostat with growth equations given by eq. ( 7). Assuming that the sea surface temperature is a proxy of the temperature experienced by the phytoplankton cells, we use a realistic temperature signal T (t, i, j) from in situ observations. The sea surface temperature data for the global ocean have been downloaded from the European short term meteorological forecasting website (http://apps.ecmwf.int). The data cover the years 2010 to 2012 and the spatial resolution is 1 • in latitude and longitude with a temporal resolution of 3 hours.

We calculate for each time step (3 hours) at a given location, the value of the function φ(T (t, i, j), T opt ), depending on the perceived in situ temperature T (t, i, j) and the optimum growth temperature T opt . We then calculate the average of the integrated function over 3τ = 3 years (2010,2011,2012):

ψ(T opt ) = 1 3τ • 3τ 0 ln(µ(T(t, i, j), T opt ))dt (22)
Using prop.2, we search the maximum of eq.( 22) and the corresponding T opt to find the evolutionary optimum temperature T opt *.

Global scale simulations

Global scale simulations for Bernard and Rémond model (Fig. 2 A) show that for any range of temperature experienced by phytoplankton, the evolutionary temperature T * opt at a given place (i, j) is always higher or equal to the average temperature T (i, j). In the tropical zone, where the average temperature is high (near 26 • C), T * opt (i, j) T (i, j). In temperate and coastal zones, where the average temperature is between 10 and 20 • C, T * opt (i, j) > T (i, j). This corresponds to areas where the temperature range max(T(t)) -min(T(t)) is higher than 10 • C (Fig. 2 C). We suppose that due to the thermal growth curve asymetrical shape, it is more suitable to have higher T opt when temperature fluctuates. This assumption is in good aggreement with [START_REF] Kimura | Growth temperatures of archaeal communities can be estimated from the guanine-plus-cytosine contents of 16S rRNA gene fragments[END_REF] observations for Archae; these organisms live near their maximum temperature, with a T opt much higher than the environmental T .

Simulations for the Eppley-Norberg model (Fig. 2 B) show similar results, but the evolutionary temperature is always higher than the average temperature (for about 6 • C).

Comparison with experimental data

Using model (1), we determine the cardinal temperatures ( Tmin , Topt , Tmax ) for 194 phytoplankton strains thanks to growth rate versus temperature data sets from [START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF]. Because µ opt is a function of light and nutrient availability, we normalize the data sets. For each strain, an algorithm developped by [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] based on the Quasi-Newton Broyden-Fletcher-Goldfarb-Shanno minimizing error method is used to determine the cardinal temperatures. The calibration is coupled to a Jackkniffe statistical test evaluating the confidence interval of the parameters as in [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF]. We then only consider strains associated with data sets providing a confidence interval smaller than 5 • C for the estimated Topt , i. e. 57 strains.

Because the geographical coordinates of the isolation of the 194 stains are known, it is possible to compare Topt to in situ T (i, j) (Fig. 3 A). Results support the fact that T opt is much higher than T (here the maximum difference is 10 • C) in temperate areas and almost the same as T in tropical and polar areas. We simulate the two models at the isolation coordinates of the 57 selected strains (Fig. 3 B, green points). Bernard and Rémond model presents the same non-linear trend previously stated (Fig. 3 B, blue points) whereas Eppley-Norberg model mostly captures a more flattened relationship between T * opt and T (Fig. 3 B, red points). For Bernard and Rémond model, ∆ (defined as ∆ = T * opt -T ) is awlays higher or equal to zero due to the assumption that µ opt does not depend on T opt . Because it is observed that ∆ is nearly equal to zero in tropical zones, this assumption seems justified. Quite the opposite, ∆ is always higher than 6 • C for Eppley-Norberg model. The linear shape obtained with Eppley-Norbeg model plus ∆ which does not allow to match the points situated in tropical zones are due to the "Eppley curve" hypothesis, which should probably be reparametrised.

We then compare predicted T * opt to observed Topt (Fig. 4). For Bernard and Rémond model, mean error calculated as |T * opt -Topt |/ Topt is 21.7% whereas for Eppley-Norberg model, mean error is 25.5%. Error is thus quite similar for the two models, but the Eppley-Norberg model overestimates T * opt because of its constraint linked to the "Eppley curve". Model predictions are accurate despite some sharp simplifications. Thus, direct temperature effect must drive evolution at global scale, contrary to what is claimed by [START_REF] Marañón | Resource supply overrides temperature as a controlling factor of marine phytoplankton growth[END_REF].

There are some unavoidable biases in our approach. The first one is due to the age of the cultures which have been used to provide the data. In general, the measurements were not performed right after in situ isolation. It results that the strains may have evolved, due to the temperature where the strains are stored in the culture collection. Second, we only consider the effect of temperature. Effects of light and nutrients as a master of µ opt are not taken into account. Finally, we use sea surface temperature. Phytoplankton can migrate and are advected along the water column, and experience temperatures different from the surface. Despite these biases, we estimated accurately the T * opt , e.g. for the cyanobacteria group, which lives in areas where the average temperature is high. It is thus possible that the sea surface temperature, where light is available, is actually a good proxy to predict thermal adaptation.

CONCLUSION

We have presented a new method based on Adaptive Dynamics theory to study the outcome of phytoplank- ton adaptation at global ocean scale. We defined a standard ubiquitous phytoplantkon species and we compared two different thermal growth models describing it in the context of evolution. We found, in agreement with our theory, that the evolutionary optimal temperature T * opt is always equal or higher than the average temperature experienced by the phytoplankton T . Moreover, the area with high difference between T * opt and T characterizes large temperature fluctuations. When based on Bernard and Rémond, our model successfully fit the data, contrary to the Eppley-Norberg model which linearly link T * opt and T , and overstimates T * opt . We inferred that direct temperature effect actually drives evolution at the scale of the global ocean. This work is only a first approach to understand thermal adaptation at global scale. The next step will be to study the evolution of the maximum temperature T max in a context of global warming.

Fig. 1 .

 1 Fig. 1. Thermal growth rate curves using model from Bernard and Rémond (2012) (A) and from Eppley (1972) (B). Both models are calibrated on experimental data from Synechococcus strain WH8501 extracted from Thomas et al. (2012), in black circles. However, it is not possible to adjust the maximal growth rate in[START_REF] Eppley | Temperature and phytoplankton growth in the sea[END_REF] model due to its structural dependence to the "Eppley curve" (dashed line).

Fig. 2 .

 2 Fig. 2. World map of the difference between the optimal temperature for growth and the mean temperature ∆ = T * opt -T for Bernard and Rémond model (A) and Eppley-Norberg model (B) (red correspond to areas where ∆ ≥ 6 • C for (A) and ∆ ≥ 12 • C for (B)), and temperature range max(T (t))-min(T (t)) for the three years 2010, 2011, 2012 (C).

  Fig. 3. (A) Observed Topt for 194 phytoplankton strains as a function of T . The 57 selected strains are indicated in green points. (B) Predicted T * opt as a function of T for Bernard and Rémond model (blue points) and Eppley-Norbeg model (red points). The y = x curve is indicated in black. The green points correspond to the predicted T * opt for the 57 selected strains (light green for Bernard and Rémond model, dark green for Eppley-Norberg model).

Fig. 4 .

 4 Fig. 4. Predicted T * opt compared to experimental Topt for Eppley-Norberg model (A) and Bernard and Rémond model (B). Phytoplankton phylogenetics groups are indicated in color in (B): green, Dinoflagellates, pink, Diatoms, black, Cyanobacteria, blue, others.

ACKNOWLEDGEMENTS

This work was supported by the ANR Facteur 4 project. We are grateful to the anonymous reviewers for their helpful suggestions.


Christophe Betoule

Thomas Bonald

Remi Clavier

Dario Rossi

Giuseppe Rossini

Gilles Thouenon

Adaptive Probabilistic Flooding for Multipath Routing

In this work, we develop a distributed routing algorithm for topology discovery, suitable for ISP transport networks, that is however inspired by opportunistic algorithms used in ad hoc wireless networks. We propose a plug-and-play control plane, able to find multiple paths toward the same destination, and introduce a novel algorithm, called adaptive probabilistic flooding, to achieve this goal. By keeping a small amount of state in routers taking part in the discovery process, our technique significantly limits the amount of control messages exchanged with flooding -and, at the same time, it only minimally affects the quality of the discovered multiple path with respect to the optimal solution. Simple analytical bounds, confirmed by results gathered with extensive simulation on several topologies (up to 10,000 nodes), show our approach to be of high practical interest.

I. INTRODUCTION

Scaling layer-2 protocols to very large networks [START_REF] Myers | Rethinking the service model: Scaling ethernet to a million nodes[END_REF] let enterprises deal with simpler and less expensive hardware, which translates in lower capital expenditures and management costs. Every solution proposed so far tries to get, from one side, a scalable architecture in which packets are routed on the shortest path; on the other side, such architecture should be flat [START_REF] Kim | Revisiting route caching: The world should be flat[END_REF], self-configuring and possibly self-healing [START_REF] Mysore | Portland: a scalable faulttolerant layer 2 data center network fabric[END_REF]. Many works address this tradeoff, proposing different routing and forwarding schemes, trying to seamlessly fill the gap between local and wide area protocols: Viking [START_REF] Sharma | Viking: a multispanning-tree ethernet architecture for metropolitan area and cluster networks[END_REF], SEATTLE [START_REF] Kim | Floodless in SEATTLE: a scalable Ethernet architecture for large enterprises[END_REF], PortLand [START_REF] Mysore | Portland: a scalable faulttolerant layer 2 data center network fabric[END_REF] represent few recent examples of research efforts in this direction.

The routing process of such architectures is usually divided in two parts: the host routing process [START_REF] Kim | Floodless in SEATTLE: a scalable Ethernet architecture for large enterprises[END_REF] aims at resolving each host to a single switch; then, the switch routing process let each core machine learn the best path to the other switches [START_REF] Singla | Scalable routing on flat names[END_REF]. The latter process, to which we focus in the following, is commonly addressed [START_REF] Mysore | Portland: a scalable faulttolerant layer 2 data center network fabric[END_REF]- [START_REF] Kim | Floodless in SEATTLE: a scalable Ethernet architecture for large enterprises[END_REF] with a link-state algorithm (similar to OSPF or IS-IS) implemented at layer-2, and that generally yields a single, shortest, path to any other switch (with the exception of [START_REF] Sharma | Viking: a multispanning-tree ethernet architecture for metropolitan area and cluster networks[END_REF] that also keeps a backup tree).

We believe that using a link state algorithm at layer-2 is not the best choice. First, link state decouples topology distribution and routes computation -in the sense that all topological information needs to be received by all nodes prior that the routing table can be computed-which represents a useless waste of time. Then, notice that the core routing table computation algorithm is represented by Dijkstra algorithm, that results in a O(N logN ) complexity. However, Dijkstra only provides a single shortest path between any nodes pair, while additional paths computation produces even higher computational complexity (though still polynomial).

In this work, we propose Adaptive Probabilistic Flooding (APF) as a simpler solution to the aforementioned problem of switch routing, which takes an alternative approach for the computation of multiple disjoint paths. Aiming at simplicity, we devise a distributed greedy algorithm that drops computational complexity still remaining (almost) stateless and self-terminating. In our design, simplicity tradeoffs with the communication cost, as APF generates an higher number of messages with respect to link state algorithms. At the same time, we can tune (and analytically bound) the number of messages thanks to a simple parameter that drives the speed of the probabilistic procedure. Moreover, by means of simulation on several topologies (up to 10,000 nodes), we show that, with as few as 2-3 times more messages that a link state algorithm, APF is able to find the shortest path and optimal backup path in more than 90% of the cases. In the remaining 10% of the cases, non-optimality is due to a limited amount of overlap between primary and secondary paths (about 1 node on average).

The rest of the paper is organized as follows. Related work is presented in Sec. II. The Adaptive Probabilistic Flooding (APF), a path discovery algorithm, is described in Sec. III. Sec. IV is devoted to the performance analysis, where we provide analytical bounds for the number of messages, and simulation results for the APF path quality. Finally, Sec. V concludes the paper and outlines future work.

II. RELATED WORK

Our work focuses on two seemingly conflicting aspects: (i) abating routing algorithms complexity, to make them suitable for layer-2 devices (e.g., Ethernet-like switches), and (ii) enabling multi-path features. In this section, we explore the closest related work in both areas.

The first topic, i.e., the desing of light-weight routing protocol has been addressed mostly in wireless area [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF]- [START_REF] Caesar | Virtual ring routing: network routing inspired by dhts[END_REF]. DSR [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF] greedy determines routes by means of packet flooding, employing a TTL to limit the flooding depth. When a control message reaches the destination, it is sent back to the origin that thus discovers a path toward the target. In sensor-networks, SCOUT [START_REF] Kumar | Scalable object-tracking through unattended techniques (scout)[END_REF], routing in a hierarchy of sensors, follows a classical landmark routing [START_REF] Tsuchiya | The landmark hierarchy: a new hierarchy for routing in very large networks[END_REF] scheme. In VRR [START_REF] Caesar | Virtual ring routing: network routing inspired by dhts[END_REF] instead, the whole topology is arranged as a DHT, and the forwarding is done accordingly. However, solutions like [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF], [START_REF] Kumar | Scalable object-tracking through unattended techniques (scout)[END_REF], [START_REF] Caesar | Virtual ring routing: network routing inspired by dhts[END_REF] rely heavily on the notion of broadcast, so they are not directly applicable in a wired environments. Among them, the closest approach to ours is [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF], which does not offer multiple paths and is moreover critically affected by TTL tuning (unlike our proposed APF approach, as we shall see in Sec. III).

The second branch of work close to ours [START_REF] Kim | Enhanced non-disjoint multi-path source routing protocol for wireless ad-hoc networks[END_REF]- [START_REF] Lin | Improved ant colony algorithm for multipath routing algorithm research[END_REF] addresses the multipath problem. In wireless networks, a multipath extension to DSR is proposed in [START_REF] Kim | Enhanced non-disjoint multi-path source routing protocol for wireless ad-hoc networks[END_REF], where authors employ a reactive flooding algorithm, in which the TTL is gradually increased in order to find more than just one single response. However, while gradual TTL increase is able to provide multiple paths and is more robust to wrong TTL settings, it brings two shortcomings, namely a (i) slower convergence of the routing process, and a (ii) higher number of control messages. In wired network, the multipath problem is often treated jointly with traffic routing in the data plane, solving a multi-commodity flow problem [START_REF] Elwalid | Mate: Mpls adaptive traffic engineering[END_REF] where, given a traffic matrix and a topology, the objective is find the routing that minimizes network congestion. Another class of work closest to ours focuses instead on control-plane topology discovery [START_REF] Hopps | Analysis of an Equal-Cost Multi-Path Algorithm[END_REF]- [START_REF] Lin | Improved ant colony algorithm for multipath routing algorithm research[END_REF]. The simplest link state extension to multipath is Equal Cost Multiple Paths (ECMP) [START_REF] Hopps | Analysis of an Equal-Cost Multi-Path Algorithm[END_REF], that splits traffic on different shortest paths (if there exist) between each pair of nodes. Other works relax the hypothesis of equal cost, and develop algorithms to find k shortest paths on any given graph [START_REF] Eppstein | Finding the k shortest paths[END_REF], [START_REF] Hershberger | Finding the k shortest simple paths: A new algorithm and its implementation[END_REF]. Addressing multiple different paths (i.e., not necessarily the k shortest ones), [START_REF] Merindol | Low complexity link state multipath routing[END_REF] proposes a link state algorithm, modifying the downstream criterion to avoid loops. Notice however that, while path computation still faces the delay due to link state advertisement, the computation of alternative paths also adds further complexity beyond the O(N logN ) Dijkstra cost (respectively, O(E + N logN + k) in [START_REF] Eppstein | Finding the k shortest paths[END_REF], O(k(E +N logN )) in [START_REF] Hershberger | Finding the k shortest simple paths: A new algorithm and its implementation[END_REF] and O(E 2 ) in [START_REF] Merindol | Low complexity link state multipath routing[END_REF]). Finally, a different approach from link state algorithms is taken, e.g., in [START_REF] Blesa | Ant colony optimization for the maximum edge-disjoint paths problem[END_REF], [START_REF] Lin | Improved ant colony algorithm for multipath routing algorithm research[END_REF], that employs distributed Ant Colony Optimization (ACO) algorithms, based on swarm intelligence. In [START_REF] Blesa | Ant colony optimization for the maximum edge-disjoint paths problem[END_REF], [START_REF] Lin | Improved ant colony algorithm for multipath routing algorithm research[END_REF] a set of ants is spread through the network in order to discover disjoint multiple paths, and the pheromone left by the ants is employed in order to avoid already crossed paths. In this case however, the shortcoming is that the routing process is not guaranteed to converge to the shortest path (unlike APF).

III. PATH DISCOVERY ALGORITHM

In this work, we propose Adaptive Probabilistic Flooding (APF), a simple algorithm for multi-path topology discovery. Summarizing the main differences with related literature, APF sits at a different operational point in the tradeoff between algorithmic complexity vs. control message overhead. Unlike in link state algorithms [START_REF] Hopps | Analysis of an Equal-Cost Multi-Path Algorithm[END_REF]- [START_REF] Merindol | Low complexity link state multipath routing[END_REF], since in our approach the whole traveled path accumulates in the control message as in [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF], [START_REF] Kim | Enhanced non-disjoint multi-path source routing protocol for wireless ad-hoc networks[END_REF], every message brings useful information for path discovery, that can thus be computed online. At the same time, unlike in [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF], [START_REF] Kim | Enhanced non-disjoint multi-path source routing protocol for wireless ad-hoc networks[END_REF], our algorithm does not rely on critical parameters such as TTL. Furthermore, the actions that need to be taken to handle each message are simple operations, making thus the algorithm viable on simple hardware. Finally, unlike [START_REF] Blesa | Ant colony optimization for the maximum edge-disjoint paths problem[END_REF], [START_REF] Lin | Improved ant colony algorithm for multipath routing algorithm research[END_REF], the algorithm is guaranteed to find the shortest path, and as our simulation results confirm, the secondary path is very often the optimal one found by [START_REF] Eppstein | Finding the k shortest paths[END_REF], [START_REF] Hershberger | Finding the k shortest simple paths: A new algorithm and its implementation[END_REF] link state algorithms.

A. Overview

We aim at designing a distributed algorithm for path discovery, capable of finding multiple, possibly disjoint paths between any pairs of nodes. To do so, each node periodically advertises its presence by means of some flooding procedure described below. Specifically, each node sends an advertisement message every τ a seconds; typical values range from a few seconds to minutes [START_REF] Nelakuditi | On selection of paths for multipath routing[END_REF].

During the flooding procedure, each relay node adds its identifier to the advertisement messages it receives, so that these messages carry information concerning the whole traveled path. Upon the reception of an advertisement message, a node learns a path from the source of this message, as well as from any intermediate node on this path, as in [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF]. Flooding decisions are taken independently by each node, and constitute the core of the algorithm. The main idea is that nodes need to flood a received message at least once, so that shortest paths are discovered. Nodes actually need to flood the message multiple times, in order to discover further paths beyond the shortest one. The number of flooding decisions is critical with respect to both the quality of the path discovery and the overhead of the algorithm.

A simple option [START_REF] Johnson | The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF], [START_REF] Kim | Enhanced non-disjoint multi-path source routing protocol for wireless ad-hoc networks[END_REF] could consist in including a Time To Leave (TTL) field in the packet, so as to interrupt the flooding process when some pre-configured maximum path length is reached. The selection of a proper TTL value is critical in this case: if the TTL is shorter than the graph diameter D for instance, then connectivity cannot be guaranteed; if the TTL is too large, the overhead of the algorithm becomes prohibitive (as the number of relayed messages is exponential in the TTL).

We propose an alternative approach based on adaptive probabilistic flooding. Any node receiving some advertisement message from source s floods this message the first time, and floods it with some decreasing probability the following times. Specifically, node i floods an advertisement message generated by source node s over all its links (except the one from which it has received the message) with probability:

P = β ni,s (1)
where β is some fixed parameter and n i,s is a counter, stored at node i, of the number of times node i has already received an advertisement originated by node s. The flooding decisions are taken independently on each link, and the counter is reset periodically, as explained later. Note that node i floods the first advertisement message it receives for source node s since n i,s = 0 in this case. As further messages are received, flooding will become exponentially less likely, according to the backoff parameter β. Note also that, if we set β = 0 APF produces the same number of messages of a link state algorithm for propagating the network topology. The quality of the path discovery is expected to increase with β, at the expense of larger overhead. However, we shall see that performance is not very sensitive to this parameter, and rather degrades gracefully with parameter misguidance, which makes the algorithm robust and practically interesting.

B. Primary and secondary paths

Consider a network, modeled as an undirected graph G = (E, V ), composed of |V | = N routers, in which any pair of adjacent routers are connected by a single link for simplicity (the algorithm can be easily extended to the general case of multiple links between any pair of nodes). Between any two routers i, j ∈ V , we are interested in finding a pair of paths, i.e., sequences of edges connecting node i to j. We denote by P and S the primary and secondary paths, respectively, returned by the adaptive probabilistic algorithm on graph G 1 . We denote by L p and L s the respective lengths of these paths.

To gauge the quality of the primary and secondary paths found by our algorithm, we need to define target path properties. The primary path is expected to be the shortest path in number of hops; in other words, we say that P is optimal if it belongs to the set of shortest paths from i to j in G (as there may be several such paths). The secondary path is expected to minimize the similarity with the primary path, P ∩S. Note that this choice reduces the share of faith between these paths, improving network resilience against failures and traffic surges.

To find the optimal secondary path S, we consider a modified graph G in which the cost of links along the primary path P are increased by the network diameter [START_REF] Ogier | Distributed algorithms for computing shortest pairs of disjoint paths[END_REF], and other link costs are unitary. As links belonging to P are now discarded due to higher cost, running Dijkstra on G we retrieve a path S minimizing the similarity function P ∩S (notice that since nodes along the primary path are not removed from G , they can be included in S only if strictly necessary as the path would otherwise be disconnected). We say that the secondary path found by the algorithm S is optimal if |P ∩S| = |P ∩S | and L s = L s , i.e., the length L s of the secondary path is equal to the length L s of the optimal S (as there may be multiple disjoint paths minimizing the similarity with the shortest path).

C. Pseudocode

A pseudocode description of the algorithm is given in Fig. 1. A source node s initiates the advertisement process by flooding an advertisement packet ADV to all its neighbors. The flooded packet contains a list of node identifiers ID, initially set to ID[0]=s by the source, to which each node appends its own identifier. Upon reception of an advertisement packet ADV, 1 The aggregation of primary and secondary paths at a given node, forms two distinct trees of the network, as in [START_REF] Sharma | Viking: a multispanning-tree ethernet architecture for metropolitan area and cluster networks[END_REF]. ns++ // Update counter associated with source s 25: end while Fig. 1. Algorithm pseudocode for a generic node j of the network a node learns a (backward) path to the source s and to any intermediate node d = ADV.ID[i] along the path. In case the receiver j detects a loop (finding its identifier within the ID list), it discards the message and aborts the flooding procedure. Otherwise, it analyzes, and possibly stores, the newly learned path O j,d . Specifically, the primary (and secondary) path is first set if not existent yet. Also, if the newly overheard path is shorter than the primary path length(L j,d ) < length(P j,d ), then the primary path is updated with the overheard one. Similarly, if the overheard path has lower similarity than the current secondary path

if { S j,d ∨ |P j,d ∩ L j,d | < |P j,d ∩ S j,d | ∨ (|P j,d ∩ L j,d | = |P j,d ∩ S j,d | ∧ length(L j,d ) < length(S j,d ))} then 13: S j,d ← L j,d //
|P j,d ∩ L j,d | < |P j,d ∩ S j,d |, or if it has equal similarity but is shorter than the secondary |P j,d ∩ L j,d | = |P j,d ∩ S j,d | ∧ length(L j,d ) < length(S j,d ),
then the secondary path is updated.

Finally, after having added its own identifier to the ADV.ID, the node probabilistically floods the ADV message, with independent decisions per each neighbor (except the node ADV.ID[ -1] from which the message came), and updates the per-source counter n s . Not shown in the pseudocode for the sake of simplicity, ties in the secondary path selection are broken at random.

Notice that, we expect (i) messages on the shortest path to reach a node before messages that take longer paths. This definitively holds in case of homogeneous delay; otherwise, it may happen that (ii) messages traveling along the quickest path arrive first, which would be then stored as primary path; analogously, (iii) if control messages queue with data-plane messages without priority, the first message could have as well traveled along the less congested path. Notice that, by simple priority queuing, case (iii) can be ruled out. Then, notice that (ii) may only happen in networks having links with very long delays: in this case, the message traveling along the shortest path is not necessarily the first to be received. However, one of the subsequent messages will surely have traveled along the shortest path (since due to n s = 0 the messages are flooded at least once), so that the primary path is guaranteed to converge to the shortest (unlike [START_REF] Blesa | Ant colony optimization for the maximum edge-disjoint paths problem[END_REF], [START_REF] Lin | Improved ant colony algorithm for multipath routing algorithm research[END_REF]).

IV. PERFORMANCE EVALUATION

In this section, we precisely quantify the APF overhead vs path quality tradeoff, and that can be tuned through the β backoff parameter earlier introduced, by proceeding as follows. We first evaluate the overhead of the algorithm through a simple analytical model, then, we employ discrete event simulation to validate the analysis and evaluate performance in terms of the quality of discovered paths.

Simulations are carried on with Omnet++ [START_REF] Varga | Omnet++ website[END_REF] over four different network topologies gathered by mean of Rocketfuel [START_REF] Spring | Measuring isp topologies with rocketfuel[END_REF], whose most significant properties are summarized in Tab. I. Specifically, Tab. I reports the number of nodes N , the average and standard deviation of the node degree, δ and σ, the diameter D of the original graph G and the largest diameter D over the modified G graphs.

Note that we consider both real ISP topologies, corresponding to different network segments, as well as a set of 50 synthetic random graphs with N ≤ 10000 and δ = 4. For the time being, we use homogeneous settings (i.e., constant and equal delay on every link), and no failures happen within the network. The evaluation of more complex (heterogeneous delay, failures, etc.) scenarios is part of our ongoing work.

A. Overhead

We now evaluate the cost of APF in terms of communication overhead, space and computational complexity. The average amount of messages handled by any given node during the advertisement procedure is denoted with M a and represents the primary cost sustained by the algorithm. For comparison purposes, we take any link state algorithms like IS-IS or OSPF as a reference. We recall that such protocols are composed by two phases: (i) flooding of topological information, and (ii) path computation, usually fulfilled by the mean of Dijkstra algorithm on the graph information gathered on the first phase.

Fig. 2. Adaptive Probabilistic Flooding: Communication Overhead

Communication overhead: Consider a single advertisement from some source node s, and consider some relay node j with degree δ. The first time node j receives an ADV message originated by s, it sends a copy on each output link, except the one where the ADV message has been received. This generates δ -1 messages. The second time j receives an ADV message from the same source, it will forward the message over each of the δ -1 links with probability β < 1; so, at second reception, node j generates (δ -1)β messages on average. Iterating and taking into account N advertisement processes (one per each node), we bound the total number of control messages M a that are seen by the average node:

M a ≈ N [(δ -1) + (δ -1)β + (δ -1)β 2 + • • • ] = N (δ -1) ∞ n=0 β n = N (δ -1) 1 1 -β (2)
It turns out that this simple and conservative bound matches very well the empirical results found by simulation. It's worth to note as the bound in (2), when β = 0, represents the number of messages a link state algorithm would flood in order to distribute the whole topology (assuming that 0 0 = 1 so that messages are flooded exactly once). In this sense we can regard APF as an extension of link state algorithm, β being a parameter which tradeoffs between paths quality and number of additional messages.

Note also that, as the first flood is always performed, convergence of the primary P path to the shortest path is always guaranteed. Hence, the backoff parameter β affects only the quality of the secondary path S: by tuning β, we can upper bound the algorithm overhead M a while matching the required level of path quality.

Fig. 2(a) depicts, as a function of β, the upper bound (2) normalized over the number of messages produced by the link state flooding, along with M a gathered by simulations carried on random and real networks. Notice that, for β ≤ 0.7, APF sends at most 2-3 times more messages than link-state algorithms. Notice also that the bound in ( 2) is generally a close estimation of the number of messages. Finally, notice also that APF can be applied to fairly large networks, as our simulations up to N =10,000 nodes suggest. Fig. 2(b) plots the load of a single node for a single advertisement in function of the (slotted) time for a random topology with N =1000 nodes: we observe that, after some initial exponential growth due to the flooding process, the backoff factor kicks in and slows down the growth, which then dies out very fast, due to an increasing number of flooding paths being probabilistically cut out. This auto-termination feature is a very desirable property of the algorithm, and further suggests that advertisement periods do not need to overlap (as in case of parallel advertisements), but can rather be serialized. In this way, we would tradeoff also between nodes peak load (which increases with the degree of parallelization) and speed of routing convergence (that decreases with the degree of serialization). We remark that serialization is not feasible for link state solutions, as paths are computed just when the whole topology is fully known.

Space complexity: Space needed by APF to run is O(2N ) in order to store primary and secondary/backup tree of the network (as in [START_REF] Sharma | Viking: a multispanning-tree ethernet architecture for metropolitan area and cluster networks[END_REF]). Additionally, we need also O(N ) counters n s to safely count the number of messages generated by each source. This could be problematic in case the size of the network is not known a priory (though N could be set to a fairly large number for safe operation). At the same time, a staggered advertisement solutions as proposed above, implies that instead of keeping O(N ) counters (one for each source in case of advertisements in parallel), the system could perform advertisement in series and keep a small number of O(1) counters. This could be achieved by desynchronizing the start of each advertisement either with a simple policy (e.g., periodically at random within [0, 2τ a ]) or with more sophisticated schemes (e.g., CSMA-like solutions). This is an interesting direction for future research, that we aim at pursuing in the following.

Computational complexity: If Dijkstra and the other graph algorithms need to perform O(N logN ) and O(N logN + E 2 ) for the computation of the shortest and backup path respectively once the full graph is known, APF on the other hand needs to perform simpler operations on a packet-bypacket basis. More precisely, on the reception of each control message, switches need to perform: (i) a comparison for the shortest path (Fig. 1, line 9); (ii) an intersection for the best alternate path (Fig. 1 line 12). Since the overall number of APF control messages is bounded, we can bound APF computational complexity as well -which grows with N when all advertisements start at the same time. Instead, it's worth to note that in link state algorithms, a single link state advertisement for a topology change causes to start, for each node, another run of a O(N logN ) Dijkstra algorithm.

B. Path quality

Let us now focus on the quality of the paths that the adaptive probabilistic flooding algorithm is able to find. For simplicity, we let each node advertise itself once at time t = 0 and evaluate the connectivity and optimality of the primary and secondary paths. Since evaluating path quality of random networks is unrealistic, we now only consider the ISP topologies, reporting results over 20 simulations per topology.

We express path quality in terms of connectivity along the primary and secondary path (i.e., whether paths P and S joining any two nodes i, j ∈ V exist) and optimality (i.e., whether P and S are optimal according to the above definitions). We express connectivity in terms of the probability that, ∀i, j ∈ V , nodes i and j are connected by some primary/secondary path. We express optimality in terms of the probability that the primary path is also the shortest, and that the secondary path is the shortest most diverse path from the primary.

Fig. 3(a) depicts the connectivity probability of the primary and secondary paths as a function of β: as expected, primary connectivity does not depend on β and is always guaranteed. Since a primary path is always found, the connectivity index is relevant for the secondary path only: we see that all secondary paths are connected in all networks when β ≥ 0.7 (which correspond to limited overhead in Fig. 2(a)).

Fig. 3(b) reports the optimality probability of the primary and secondary paths as a function of β: again, since the shortest path is always eventually found, the optimality of the primary path is guaranteed. Thus, the optimality index is relevant only for the secondary path: we see that a significant percentage (from 60% to 85%, depending on the topology) of secondary paths are optimal even for a very low value of β = 0.1, and that at least 90% of secondary paths are optimal for all considered topologies when β ≥ 0.8. Moreover, we observe that optimality gracefully degrades β, and furthermore with similar (roughly linear) slope across all topologies. This is a desirable behavior: as no phase transition nor knee appear in the path quality slopes, tuning β between low overhead (low β) vs high path quality (high β) is not critical.

Finally, we dissect the reason behind the sub-optimality of some secondary paths. Recall that a secondary path is optimal if it is the shortest and most diverse path compared to the primary. Hence, sub-optimality of the secondary path may be due to either (i) a non-zero overlap between primary and secondary paths, |P ∩ S| > 0, or (ii) a path with a stretch over the optimal secondary path larger than one L s /L s > 1. Fig. 3(c) depicts the overlap, i.e., the number of nodes that primary and secondary paths have in common, conditioning over the sub-optimal paths (i.e., the overlap of optimal secondary paths is not accounted for in the picture). As shown by the figure, sub-optimality seems to be tied to slightly more than one node in common as |P ∩ S| ∈ [1, 1.5]. Furthermore, as the average overlap is always |P ∩ S| ≥ 1 for any β, we can conclude that overlapping paths are significantly more common that long-stretching paths.

V. CONCLUSIONS

We have presented a novel flooding based algorithm for multiple-path discovery: the algorithm trades a small amount of state in routers, i.e., O(N ) counters, in order to significantly limit the number of messages generated by flooding through an adaptive probabilistic algorithm.

Simple analytical bounds, confirmed by simulation results, show the overhead entailed by the advertisement procedure to be low (with respect to the amount of messages needed by classical link state algorithms) and auto-terminating (due to the multiplicative decrease of the flooding probability).

Simulation results also testify excellent performance in terms of path quality: connectivity and optimality of the primary path are achieved by design, while 90% of secondary paths are also optimal when β ≥ 0.8 (or otherwise decrease linearly for lower β). Interestingly, the low percentage of low path is due to a very limited amount of share of faith between paths. As part of our future work, we want to carry on more realistic experiments on a wider set of topologies and further reduce the amount of state to O(1), by exploring policies for the serialization of node advertisement.

1: while {receiving message ADV} do 2 : 3 :

 23 ← length(ADV.ID) for all { i ∈ [0, ] } do 4: if {ADV.ID[i] = j} then .ID[i] // Destination 8: L j,d ←(ADV.ID[ ], . . . , ADV.ID[i]) // Overhearing advertised paths from ADV; 9: if { P j,d ∨ length(L j,d ) < length(P j,d )} then 10: P j,d ← L j,d // Update primary path 11:

  Communication overhead w.r.t. link state algorithms, and comparison with analytical bound Time evolution of the number of advertisement messages seen by a single node and generated by a single advertisement process (N = 1000, δ = 4)).

Fig. 3 .

 3 Fig. 3. Adaptive Probabilistic Flooding: Path Quality.

  Update secondary path

	14:	end if
	15:	end if
	16:	end for
	17:	append j to ADV.ID
	18:	s ← ADV.ID[0] // Source
	19:	for all {next ∈ neighbors(j) } do
	20:
	21:	send ADV to next w.p. β ns
		// Adaptive probabilistic flooding
	22:	end if
	23:	end for
	24:

if { next = ADV.ID[ -1] } then

ACKNOWLEDGEMENT

The work presented in this paper has been carried out at LINCS (http://www.lincs.fr) and was funded by an Orange Labs grant.


Valentin Frédéric
email: frederic.valentin@telecom-paristech.fr

Keywords: Interdisciplinarity, Indiscipline, Mood-Board, Design, Design Education, Discretization

This paper analyses the use of mood boards in design education not only as a means of conception and communication but also as a method that brings interdiciplinarity into play. It presents a longitudinal study of two use cases in two different educational contexts: a design school and an engineering school. After analyzing mood boards, their production and presentation, the authors suggest that mood boards actually help practitioners to draw pertinent questions and responses because they organize three conceptual activities: discretization, coherence, and relevance. The discretization of fields, objects, facts, deconstructs the disciplines as such, but also organizes the interplay of disciplines. This interplay could be dismissed as undisciplined. But we suggest that the format-i.e. the formal meaning of the media-shapes the process and organizes a rationally and efficiently constructed space of conception. After Rancière, we suggest that the concept of "indiscipline" covers not only the deconstruction and the stepping out of disciplines, necessary to the process of conception, but also that it reflects the rigorous framing of this process through formal and practical meaning making.

level to demand a specific space for a multidisciplinary practice and at the epistemological level to establish the rationale of a discipline while it borrows and modifies concepts and methods from other disciplines as well as develops its own. More importantly in our perspective, it structures the way design education is defined and prescribed in France. After studying multiple texts formalizing design education in France (Arrêté du 17 juillet 1984; Arrêté du 30 juillet 2012; Arrêté du 5 avril 2012; Extraits du Bulletin officiel spécial du 29 avril 2010), we noted that one of the recurrent characteristics associated with the designer's practice is her "capacity to interact and work with other disciplines", her capacity to interact with various actors of different disciplines while understanding and integrating them into her own practice (as an example, a transversal capacity expected from HND students in graphical, product and environment design is to "communicate at all levels in a pluridisciplinary manner"(Arrêté du 22 août 2006 and Arrêté du 8 avril 2009).

Without repeating the entire debate on interdisciplinarity [START_REF] Dubreuil | Défauts de savoirs[END_REF][START_REF] Findeli | Rethinking design education for the 21st century: Theoretical, methodological, and ethical discussion[END_REF], Bremner & Rodger, 2013), we wish to identify the relations between design educational tools and scientific disciplines. We find that tools are often considered either in their creative or managerial capacity but here we want to question their role as interdisciplinary research tools. Following the principle of "design oriented research" as defined by Daniel Fallman (2008), we would argue that mood boards are not only design tools but also a means of production of knowledge and concepts that deconstructs disciplines as such, but also formally and pragmatically frame the research findings. Saying so does not mean a consideration of design and its tools as "disciplined"-that is following a set of well defined steps-nor "undisciplined"-in the sense that it would not follow any rigorous rationale-but rather that the pragmatic and semiotic traits of these documents and their circulation organize not only the deconstruction of disciplines but also the re-organization of data to produce a new object, an X.

In the first section of this article, we present a review on design education and the interplay of disciplines. We then draw on the literature on mood boards to point out the way they elicit "emotions" and their creative and managerial qualities. We also take into consideration the criticisms that they raise. After presenting our research setting, we move on to analyze a series of mood boards created in two contexts of education to better understand how they structure the design practice in terms of formal properties and cognitive operations. In the next section, we identify their epistemological properties. Then, we discuss how these properties operate a synthesis and the expansion of the design project. We ask the following questions: How are they related to the rationale of a scientific process? How do they organize a design space? We finally conclude on the "in-discipline" of design.

Design Education and the Interplay of Disciplines

In our review of design education literature, we identified three major topics of inquiry: the acquisition of a theoretical and visual culture (Brookes,1992;[START_REF] Dutton | The Hidden Curriculum and the Design Studio: Toward a Critical Pedagogy[END_REF][START_REF] Gall | Navigating a way through plurality and social responsibility[END_REF][START_REF] Chin | Critiquing commonly available multicultural art education resources[END_REF][START_REF] Hadjiyanni | Beginning with concept: Deconstructing the complexity of "culture" through art in design education[END_REF], the development of methodological and analytical techniques (Schon & Wiggins,1992;[START_REF] Goldschmidt | Variances in the impact of visual stimuli on design problem solving performance[END_REF][START_REF] Adams | Educating effective engineering designers: the role of reflective practice[END_REF][START_REF] Ozkan | Cognitive strategies of analogical reasoning in design: Differences between expert and novice designers[END_REF][START_REF] Daalhuizen | A personal matter? An investigation of students' design process experiences when using a heuristic or a systematic method[END_REF] and the fostering of students creative skills (Cross, Christiaans, and Dorst, 1994;[START_REF] Folkmann | Enabling creativity. Imagination in design processes[END_REF][START_REF] Van Dooren | Making explicit in design education: generic elements in the design process[END_REF][START_REF] Lu | The relationship between student design cognition types and creative design outcomes[END_REF]. While all these acquisitions are legitimate to become designers, what is also at stake is the interplay of these different competences in a design project [START_REF] Hatchuel | Teaching innovative design reasoning: How conceptknowledge theory can help overcome fixation effects[END_REF]. In particular: how do creative skills intersect with theoretical knowledge? [START_REF] Findeli | Research Through Design and Transdisciplinarity: A Tentative Contribution to the Methodology of Design Research[END_REF] explores the interplay between design practice and scientific practice. While he criticizes an approach of design education that focuses only on practice, he also points out the limits of design considered as an applied science. In the latter case, he emphasizes first the impossibility of listing the infinite number of sciences that could be applied by design. Second, he contends that design cannot be an applied science. Just as engineering sciences are not applied sciences for fundamental sciences but constitute a specific field with their own complexity of knowledge [START_REF] Schmid | Une nouvelle logique de l'interdisciplinarité[END_REF], design sciences constitute a field of concepts, methods, and theories that are autonomous from "fundamental" sciences. Yet design also builds a relation to other sciences. This relation is to be understood through the practice of design. Following Findeli's analysis of design education (2001), we want to understand the practice of design through the artefacts that are produced during design classes and projects and to see how these practices can be understood from an epistemological point of view. We are interested in the implied theories of design supporting these artefacts and their production. In this article, we focus on the data that is collected from contextual research, and how they are organized and presented. Among these artefacts, mood boards are tools that support designers' search of materials, and synthetic approach.

Mood boards in the literature

Mood boards are regularly cited as design tools [START_REF] Takano | Proposal of A Multispace Design Method Corresponding to Diverse Usage[END_REF][START_REF] Michura | Seeing the forest and its trees. A hybrid visual research tool for exploring look an feel interface design[END_REF]. First, mood boards elicit emotions and therefore tend to structure the user's experience (Eckeert & Stacey, 2000;[START_REF] Mcdonagh | Exploring the degree to which individual students share a common perception of specific mood boards: observations relating to teaching, learning and team-based design[END_REF]. Baxter contends that "the mood of a product is the sentiment feeling or emotion which the product engenders when first seen. […] A good mood board captures these feelings in images but without referring to specific product features which might limit the range of styling options considered" (Baxter, 1995 p. 222). Mood boards also collect styling features that have been successful in the past. This second characteristic of mood boards helps gather aesthetic cues for future designs (Tan & Melles, 2010). Mood boards therefore serve a double purpose in the design practice. First, they serve as a "reservoir" of qualities, aesthetic properties, for the design work (See also [START_REF] Gonçalves | What inspires designers? Preferences on inspirational approaches during idea generation[END_REF]. Second, they serve a communication purpose. They play a managerial role [START_REF] Mcdonagh | Mood boards as a design catalyst and resource: Researching an underresearched area[END_REF] as they convey styling objectives to members of the design team and customers. But, as pointed out by [START_REF] Garner | Problem interpretation and resolution via visual stimuli: the use of "mood boards" in design education[END_REF], mood boards also raise mixed feelings for practitioners, students and professors alike. First, there is no guarantee that the communicational purpose is fully conveyed as participants can have a very different interpretation of what the images mean. The inspirational function of mood boards is not obvious either. First, mood boards can be too abstract. They do not help to make design choices and are therefore difficult to relate to tangible positioning and to actually embed in a real project. Second, mood boards may induce fixation in the design process [START_REF] Cheng | A new strategy to reduce design fixation: Presenting partial photographs to designers[END_REF][START_REF] Crilly | Fixation and creativity in concept development: The attitudes and practices of expert designers[END_REF][START_REF] Purcell | Design and other types of fixation[END_REF]. The choice of images does not represent a starting point but rather serves to legitimate design choices already made. In between fixation and defixation, mood boards have therefore a delicate balance to strike. The consequence is that mood boards, like other reflective and messy synthesis processes, "are considered a 'waste of time,' as they aren't positioned as actionable or immediately predictive" [START_REF] Kolko | Abductive thinking and sensemaking: The drivers of design synthesis[END_REF]. As we observed in both use cases, mood boards can be qualified by students as a simple stage to reach the desired goal, and may be easily discarded (when asked, our students revealed that they did not keep their mood boards). Mood boards are not seen as "proper work" that should be archived, because they seem too specific in time and space to be considered as relevant for further design enquiries. These criticisms show that the art of doing good "mood boards" is not formally defined and that the method could finally be detrimental to the design process. Our first hypothesis is that by understanding the semio-pragmatic elements of mood boards, we should be able to better design them and show their functions and limits. Mood boards are specific documents that organize representations in a special way and that are meaningful within particular situations of design and design education. By analyzing both the documents and their reception, we want to understand their role in the design process and the type of knowledge they produce. Our second hypothesis is that mood boards, as part of a rationale of design, are related to fields of knowledge and disciplines. We want to better address this connection between practice and scientific disciplines.

Research Setting

To address the question of the semiotic and pragmatic properties of mood boards in the context of education, and how this particular practice is related to scientific disciplines, we selected two use cases of design education that involved mood boards in the design process.

The cases reported in this paper were selected for three main reasons. First, the observations could take place at the very beginning of the education projects, which was a condition for the success of our longitudinal approach, as we could follow the various steps from brief to brief refinement through moodboards, to the development of a project. From the first discussions on the overall purpose and format of the class in June 2014, to the more specific meetings preparing the class from September 2014 to march 2015, implying a continuous discussion with the professor in charges (through regular inquiries about the class and interviews) and participation in the definition of the different exercises, the authors could follow the different stages that framed the workshop of Use Case 1. For Use Case 2, the authors also participated in meetings that prepared, then evaluated the class that took place from March to June 2014. Second, the groups that participated in the project under study accepted that the authors could carry out an in-depth empirical study, which was another necessary condition to sustain our comprehensive approach to design practices at work. Third, the studied cases can be compared as they took place in different organizations-one design school and one engineering school-which makes it possible to see commonalities and differences.

Use Case 1 presents an observation done from January to February 2015 in a class of 16 master degree students enrolled in the "global design" program of a private design school in Paris. The design brief was presented as a reflection on "Design and Misbehaving" or "how to misbehave in design". The students had to propose uncanny design practices. In this context, the students were organized in four groups of four and presented four mood boards.

Use Case 2 took place in an engineering school in Paris from March to June 2014. The program is called "Design and Innovation" and brings together competences in design, management and economics. Students have to develop a detailed scenario of use, interaction specifications, and aesthetics recommendations. In this use case, the students had to develop scenarios and interfaces after the following brief : "what would be an intelligent car that cares for me?" The brief was given by a car company that wished to tap into the technical culture of students. Each of the 20 students had to present a mood board.

We will compare these two use cases to extract the distinctive principles of the use of mood boards.

Description of mood boards

In both use cases, mood boards had at least two explicit goals: creating a minimal culture for students who had never had a chance to work on these subjects and creating a resource for design cues.

Context and Pragmatic aspects

The students used digital tools to make their mood boards (Power point, Adobe Illustrator, etc.). There were no other media. The presentations were made of various numbers of slides with collected images on each. Some were annotated. Mood boards were shared amongst students, whether during a formal public presentation or informally. In the design school (Case 1), the term "mood board" was not explained. The professors relied on the tacit understanding that it was part of the students' culture of design practice, a common tool. In the second use case, examples were shown to help the engineering students to produce their own.

In both use cases, the students were evaluated on their capacity to be coherent but also "expansive" (Use Case 1's evaluation chart uses the criteria "richness of the universe"), that is, to develop an original approach to the project. The other explicit goal was that the students had to build a "minimum culture" on the subject. In Use Case 2, the professor also added that the process was part of the acquisition of a visual culture quite foreign to engineering students. She posited that they had to go "from ideas to concretization". While it was assumed that design students were comfortable with the use of visual references and their manipulation, it was assumed that engineering students were not.

Both curriculums have strong relations with the professional and industrial world. The mood board exercise also aimed at developing their professional skills. But at the same time, the purpose of the programs is also to disorient the students whether by the use of uncommon tools or methods, and/or by encouraging a deeply personal approach to an undefined problem.

Analysis of mood boards

In Use Case 1, mood boards presented visual elements from different origins: other design projects, art photographs or actual products. For instance, one of the mood boards presented a photomontage of a design project (the portrait of a man with a vegetal beard) on the same level as artistic photographs (see Figure 1). A number of images came from critical design [START_REF] Dunne | Hertzian Tales: Electronic Products, Aesthetic Experience, and Critical Design[END_REF][START_REF] Mollon | Critical design[END_REF] which emphasizes uncanny associations reinforcing the overall feeling of design as a source of tensions and questions. Pictures in the mood boards were also decontextualized and did not present obvious hierarchies between elements of various nature and origins.

But the documents also demonstrated a form of coherence. In the mood board already mentioned, coherence was achieved around the theme of vegetal hybridization. The two visuals on the left, while they lost the reference to the vegetal, kept the idea of a graft, here producing energy. Through this mood board, the students efficiently communicated an ecological discourse based on an aesthetical research around the hybridization of plants and human body. The visual of "TomTato", if it is not directly depicting the adjunction of vegetal to the body, denotes (as it is described in the caption) hybridization. Because this reference is a real commercialized product, it is possible that the students saw it as a proof of concept, as a way to make their fictional project more plausible.

When presenting their mood boards, students focused more on the visuals of the projects rather than an actual description or analysis of the quoted projects. When present, the captions referring to the artists and projects were hardly visible. Students admitted later that they used these elements with two distinct purposes. The first was to "reassure" the professor by showing references related to the exercise, demonstrating the assimilation of the brief. But the focus on the visual aspect of references was a deliberate extraction from the project to display a personal interest engaging a novel interpretation and perspective that could lead to an original production. Also, during the presentation, student commented less on the mood boards, and far more about their future projects. According to their professor, it is because the mood board is supposed to speak by itself, to express emerging aspects of the project that revolve around a set of ideas only revealed through the project. The mood board finds its resolution in the future project and is otherwise very vague.

In Use Case 2, students were primarily discovering an industrial field without previous knowledge. Their mood boards therefore reflected their exploration of the field. They did not come up with "ordinary" cars but with innovative -concept or prototype -cars that were getting them nearer to the design brief. Their mood boards showed different research fields related to cars. As we can see in Figure 2, they picked up innovation in terms of safety, new driving panels, new forms of energy. They also looked into cars that fly or swim. In terms of materials, they were interested in changing pigments, in pliable material, in eco-friendly cars. They also looked into infrastructure: connected cars in smart cities, evolving like a swarm. In terms of aesthetics, they chose very unusual vehicles that helped them reframe the vision of what a car is. Mostly their mood boards showed technological surveys of car innovations (see Figure 3). Nevertheless the students also looked for "things" that would be inspiring first in terms of physical sensations: for instance wet crabs, fish, water backgrounds that conveyed smooth, sliding, to the point of slimy sensations; second in terms of imaginary references: cartoons that showed a partnership between the person and her vehicle, whether "fusional" or friendly (Figure 4). The images also built an ambiguity between the hyper adapted, hyper connected, hyper controlled, and the chummy, comfortable, fun vehicle. One student who was thinking of going into marketing, actually came up with the most extreme choice of images as he picked up cars related to nature in all kind of ways: with tree growing out of the boot, garden cars, compost, etc.

Interestingly enough, the main difference between novice and expert practices of moodboards lies in their formal organization: design students used more rigorous ways of organizing materials and have a stronger inclination to pick up materials based on aesthetic properties.

Epistemological properties of mood boards

We suggest that the semiotic and pragmatic characteristics of mood boards can be analyzed as four cognitive and aesthetics operations: decontextualization, discretization, coherence, and relevance.

Decontextualization

The analysis of mood boards presents several striking features for the design process. First, most of these images were not situated-the original websites were not mentioned. Rather than functioning as part of complex situations of use or part of a historical tradition, they were presented as decontextualized items. There were no examples taken out of the "history" of design. All the examples were contemporary. The design projects were therefore less thought in terms of family of objects, tradition, or inscription in social uses, but more like radical innovations that needed to capture the latest aesthetic and technical traits.

Discretization

Students and professors consider what can be obtained from these images, not the images as such. Only certain aspects of the represented objects will be kept for the future design. Mood boards constitute a reservoir of traits that can be extracted from the representations. The aspects that are discretized can be formal: a shape, a color, a structure, a technology. But designers do not necessarily retain the tangible features of the objects. The discretization can be symbolical: representations of emotions or sensations, for example a fish that stands for a weightless nautical environment, or political and social, for example a graft of grass that questions our relation to nature. Either as symbols for something else, or as parts for a design production, the students discretize the representation of the objects. They display a design competence that sees a cue for something else through a particular object or representation, or situation. This competence is based on a discretization of the elements so that they can be moved, translated, or recomposed in a new object.

Coherence

The mood boards are not only a tool to discretize objects and representations. They strive towards a certain form of coherence. While gathered images can be considered as very heterogeneous at first sight, they are displayed on the same document. This collection is claimed to be meaningful for the design project. In other words, what is claimed behind the diversity is a convergence and a coherence of at least two kinds.

In the first type of coherence, images are presented as correlated in some way because they function as index of what can be done. For instance, in use case 2, students pointed out technical features that they would use in the design project. They therefore posited the articulation of different materials and/or technologies to build the design project. Coherence here is indexical because it focuses on the articulation of causes and effects.

Second, coherence can also be conveyed by a claim of similarity. Images function as iconic elements of an atmosphere, an aesthetic space. For instance, in case 1, colors and materials were evocative of the same palette. Coherence is therefore built in the mood boards either because the technical elements behind the images will be articulated, or because the aesthetic qualities of the design project are already staged.

The first type of coherence is generally associated with feasibility. Certain students stress more than others the need to make sure that the whole projected system will work (for instance in Use Case 2, a group of students supported a pragmatic approach of infrastructures versus services to counterbalance and support the more original propositions).

The second type of coherence is seen more in terms of originality. Another way to interpret the coherence of the document mood board is therefore to see that it tries to maintain a balance between feasibility and originality, something that is also at stake in brainstorming methods [START_REF] Kazakci | Brainstorming versus creative design reasoning: a the-ory-driven experimental investigation of novelty, feasi-bility and value of ideas[END_REF].

Relevance

In our analysis, relevance is the last expected property of mood boards. Students claim that the elements that they choose and display contribute to define the value of the design project. Mood boards are not media for their own sake. They support the design of future projects. "The mood board has to help showing 'where we are going'." "One of the objectives of the mood board is to spot a visual universe, materials, a cosmology of objects and productions in which to find 'something' of their project." Here the discourse that goes with mood boards has several functions. Not only does it describes for the audience what items should be seen "behind" the apparent heterogeneity and how to discretize the images for future design projects, it also anticipates the values of the design project. Students talk about future scenarios of use and the ethics of their projects. Every discretized item is a cue for a design that foregrounds the values that it wants to support.

Relevance in this context, means the social role and ethics of the designer "for a better world." For designers, there is a "real world" that needs to be changed. Relevance is a claim and property of projects that can produce better situations [START_REF] Harfield | On the roots of undiscipline[END_REF]. Second, relevance defines how the project will be integrated in society. When cars were concerned, the students were optimistic that they could get rid of the car as we know it, so that it could become a real, highly connected, partner in mobility. The design project has to display its relevance in terms of contemporary values. For instance, none of the students recommended cars that would use more energy. And all the students envisioned a highly connected and social car. There is a rationale of norms that connects the projects to society [START_REF] Habermas | The Theory of Communicative Action: Reason and the Rationalization of Society[END_REF][START_REF] Harfield | On the roots of undiscipline[END_REF].

Discussion: synthetic and expansive values of mood boards

The rational of mood boards is to create new knowledge based on observation of gathered images and items. But contrary to a traditional corpus in human sciences that helps to understand already existing objects [START_REF] Rastier | Arts et sciences du texte[END_REF], corpuses in design find their resolution in objects to be. In design projects, the rational of the corpus is present, in the sense that items are collected so as to address their commonality. But the commonality is not a given, it will only happen with the new design project. One strong sign that mood boards are not analytical tools is that the images are decontextualized. This decontextualization has at least one effect. It helps consider the object represented not as an example sustaining a theory demonstrated somewhere else but as a reservoir of traits that can be cut from the objects and recomposed or translated somewhere else. This first decontextualization helps define the represented object not as a whole within a specific situation but an ensemble of traits to be considered for its future potential in different situations.

In that sense mood boards are not primarily analytical but synthetic and expansive. First, we consider how mood boards are synthetic. Kolko in her article on design synthesis points out a number of "messy" properties of the design process: "Synthesis methods have been continually referenced as critical in sensemaking, organization, and in drawing the important connections between apparently unrelated elements. These are the keys for relating research to design-synthesis methods are the ways in which ethnographic insights lead to new, innovative, appropriate, or compelling ideas" (Kolko, 2010, p.17). Synthesis is an abductive sensemaking process. Through efforts of data manipulation, organization, pruning, and filtering, designers produce information and knowledge for a future project. The coherence that is achieved by mood boards, relies on strong synthetic activities that extract common points and organize them around a theme or a quality: for example, the "buddy car", the "sleek car", the "witch car" in Case 2. Students have to demonstrate the commonalities of the represented artefacts or situations from a certain angle. One of the professors describes the mood board as an "exercise to bring homogeneity out of heterogeneity". She emphasizes the need to "generate homogeneity".

At the same time, the collection of items organizes a tension. The representations are really incompatible at first sight. The work of interpretation is highly regarded if the capacity to organize tensions is generative of a variety of unpredictable combinations. For the professors, this is a key element to judge of the originality of the students' work. In other words, the corpus does not close on a type, a genre, but has to be expansive of a new object that will solve the "apparent" contradictions: as stated in an interview by the professor supervising Case 1, "for the professor, ideally, soliciting heterogeneous sources allows students to generate more fertile ideas out of analogies, metaphors, etc." Mood boards are therefore documents that structure expansion through a number of semiotic properties that we analyzed in the previous sections of this article.

Conclusion: Sciences of conception and in-discipline

Findeli (2001) following Moholy Nagy (1947) suggests that it is more important to focus on the design project and its formal operations -such as those that we describe in the practice of mood boards -rather than focusing on inter-disciplinarity per se. The attention is therefore drawn to relations within documents whose goal is precisely to be conceptive through formal meaning making as well as the pragmatics of these documents. Following Joyce S. R. Yee and Craig Bremner and on the basis of the analysis of the properties of these mood boards, we propose that a mood board "consists of combining methods from the social sciences, humanities, and hard sciences to derive a suitable model of inquiry." (Yee, Bremner, 2011, p.1). The following section discusses how these practices are related to disciplines and / or scientific methods.

At first sight, practices like mood boards seem un-disciplinary as suggested by Bremner and Rodger (2013). They might be described as un-disciplined because:

• Those documents need the author's and the audience's capacity of interpretation. These means of interpretation, the ways to make sense of the elements, are not specified in the exercise, hence an accusation of subjectivity; • They seem to pick indiscriminately between elements of different natures;

• Mood boards channel ways of doing work without disciplinary restrictions. They maintain a goal (an issue that needs to be solved between the elements) but remain vague on the ways to achieve it.

As we have seen in the previous sections of this article, this perception is in fact disproved by the analysis of the semio-pragmatic characteristics of mood boards. In our conclusion, we want to suggest that rather than un-disciplinary, they are part of an "in-discipline" as Rancière defines it [START_REF] Dubreuil | Défauts de savoirs[END_REF].

Following Rancière [START_REF] Dubreuil | Défauts de savoirs[END_REF], we suggest that a mood board is part of the indiscipline of design. Rancière, who is a philosopher of art and design, qualifies a certain number of forms that deliberately reframe what can be seen and what can be thought. As he studies contemporary art forms, he concludes that conception is based on formal constraints that help reframe the set of solutions. These "dispositifs", or arrangements, deliberately create a plane of conception independent of any set discipline. This autonomous plane of conception makes it possible to select items from disciplines-either their facts, methods, or theories-to compose new relations. In this respect, design entertains a special relation to other disciplines by its capacity to organize a rigorous method of interplay with the disciplines [START_REF] Schmid | Une nouvelle logique de l'interdisciplinarité[END_REF]) which, we suggest, could help define design as an indiscipline: that is a discipline that plays with disciplines in a rigorous formal way.

But the question is how do these autonomous conceptive spaces produce rigorous results. The tools of conceptive sciences (including engineering sciences) cannot be the same as sciences of nature that observe the real. Conceptive sciences build new artifacts and therefore need to address the question of a rigorous conception. What needs to be guaranteed from an epistemological point of view is the conceptive validity of the processes and tools.

In this respect, mood boards can be considered as a design rationale that organizes an indiscipline with formal meaning making. By creating a purposefully oriented set of heterogeneous objects within a document, mood boards create a cluster of possible relations.

In other words, the formal qualities of the documents frame the different operations that we suggest are the rationale of design: decontextualization, discretization, coherence, and relevance.

To summarize our findings:

• Moodboards have four epistemological properties: Decontextualization, Discretization, Coherence, Relevance;

• In Design Education, moodboards present synthetic and expansive values as they organize tensions while generating homogeneity, which helps to reframe what can be seen and what can be thought; • They suggest that design is an "in-discipline," as it picks elements of different natures to constitute a formal assemblage that plays with disciplines in a rigorous formal way.

He seeks understand the properties of an "open work of design" by analyzing current and previous work in art, design, etc.

Brulé Émeline

Émeline Brulé is a PHD Student in Design and Media Studies at Telecom ParisTech. Her research focuses on wearable technologies dedicated to health. She studies "smart" objects to develop a comprehensive framework of this family of objects. Using critical design, she explores the way we define our bodies via this new kind of prosthetics.

Figure 1 :

 1 Figure 1: Mood board from Case 1: Visual elements come from different origins, design projects, art photographs or actual products. Here a photomontage (the portrait of a man with a vegetal beard) is on the same level as artistic photographs and products.

Figure 2 :

 2 Figure 2: Mood board from Case 2 displaying various innovations innovation in terms of safety, new driving panels, infrastructure or new forms of energy.

Figure 3 :

 3 Figure 3: Mood board from Case 2 displaying unusual cars, such as an eco-friendly car, one made of lego, one crowd-funded and the last displaying an external airbag.

Figure 4 :

 4 Figure 4: Mood board from Case 2 displaying inspirations in terms of physical sensations such as wet crabs, fish, that conveyed smooth, sliding, to the point of slimy sensations and imaginary references: cartoons showing a partnership between the person and her vehicle, whether "fusional" or friendly. It induces an ambiguity between the hyper adapted / connected / controlled car and the chummy, comfortable, fun vehicle.

Author Biographies

Gentès Annie

Annie Gentes is associate professor in Information and Communication Sciences at Telecom ParisTech and Head of the Codesign Lab. Her research focuses on "extreme design"design in research and design in society -and the invention of ICT. She works with artists, designers and engineering scientists to understand their specific conceptive activities that turn a technology into meaningful artifacts.

Valentin Frédéric

Frédéric Valentin is a PHD Student in Design and Media Studies at Telecom ParisTech. His research takes interest in "openness" in design practices through an historical perspective.