Maxime Lucas

Iacopo Iacopini

Thomas Robiglio

Alain Barrat

Giovanni Petri

Simplicially driven simple contagion

come

I. INTRODUCTION

Contagion processes have been widely studied using complex networks as the underlying structure supporting the propagation of diseases, innovation, and opinions [START_REF] Daley | Epidemics and rumours[END_REF][START_REF] Pastor-Satorras | Epidemic spreading in scale-free networks[END_REF][START_REF] Rogers | Diffusion of Innovations[END_REF][START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF]. The most studied examples include simple contagion models (where a contagion event can be caused by a single contact), such as the paradigmatic Susceptible-Infectious-Susceptible (SIS), widely used to describe the diffusion of a single pathogen in a population [START_REF] Vespignani | Modelling dynamical processes in complex sociotechnical systems[END_REF][START_REF] Pastor-Satorras | Epidemic processes in complex networks[END_REF].

In reality, however, contagion processes often co-exist and affect each other [START_REF] Sanz | Dynamics of interacting diseases[END_REF]. Infectious diseases can indeed display complex comorbidity interactions, in which the presence of a pathogen impacts the individual susceptibility towards another [START_REF] Wang | Coevolution spreading in complex networks[END_REF], like HIV increasing susceptibility to other sexually transmitted diseases [START_REF] Røttingen | A systematic review of the epidemiologie interactions between classic sexu-ally transmitted diseases and hiv: How much really is known?[END_REF]. Modeling efforts in this direction include both cooperation [START_REF] Cai | Avalanche outbreaks emerging in cooperative contagions[END_REF][START_REF] Chen | Fundamental properties of cooperative contagion processes[END_REF][START_REF] Cui | Mutually cooperative epidemics on power-law networks[END_REF] and competition [START_REF] Karrer | Competing epidemics on complex networks[END_REF][START_REF] Poletto | Characterising two-pathogen competition in spatially structured environments[END_REF][START_REF] Li | Competing spreading dynamics in simplicial complex[END_REF] between diseases. However, to date, models of interacting contagion processes have been developed under two main assumptions: (i) the processes are simple contagions, and (ii) their interaction is symmetric, that is, bi-directional and of equal strength. Within these restrictions, cooperative models can display a discontinuous transition to the epidemic state [START_REF] Chen | Fundamental properties of cooperative contagion processes[END_REF], and become indistinguishable at the mean-field level from complex contagion models describing social reinforcement [START_REF] Hébert-Dufresne | Macroscopic patterns of interacting contagions are indistinguishable from social reinforcement[END_REF] (where exposure to multiple sources presenting the same stimulus is needed for the contagion to occur [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF]).

Interactions between spreading processes are naturally not restricted to infectious diseases: a social behavior can also dramatically impact the spread of a disease [START_REF] Funk | Modelling the influence of human behaviour on the spread of infectious diseases: a review[END_REF][START_REF] Perra | Towards a characterization of behavior-disease models[END_REF][START_REF] Granell | Dynamical interplay between awareness and epidemic spreading in multiplex networks[END_REF][START_REF] Scarpino | The effect of a prudent adaptive behaviour on disease transmission[END_REF][START_REF] Perra | Non-pharmaceutical interventions during the covid-19 pandemic: A review[END_REF]. A current and cogent example is the impact of the adoption of risky behaviors (no hand washing, no masks, no self-isolation or reduction of face-to-face contacts) during the COVID pandemic [START_REF] Perra | Non-pharmaceutical interventions during the covid-19 pandemic: A review[END_REF]. Motivated by this example, we challenge both restrictions described above. First, it is known that reinforcement mechanisms influence social behavior so that models of simple contagion-that assume independent pairwise exposures-do not offer the most adequate description [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF]. Simplicial contagion has been proposed as an alternative approach to account for simultaneous exposures via group-contagion events [START_REF] Iacopini | Simplicial models of social contagion[END_REF][START_REF] Battiston | Networks beyond pairwise interactions: Structure and dynamics[END_REF]. Such group ("higher-order") contributions induce discontinuous transitions, bi-stability and critical mass phenomena even for single processes [START_REF] Barrat | Social contagion on higher-order structures[END_REF][START_REF] Landry | The effect of heterogeneity on hypergraph contagion models[END_REF][START_REF] Ferraz De Arruda | Phase transitions and stability of dynamical processes on hypergraphs[END_REF][START_REF] Matamalas | Abrupt phase transition of epidemic spreading in simplicial complexes[END_REF][START_REF] St-Onge | Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs[END_REF]. Second, most contagion processes do not interact in a symmetric way. This can happen, e.g., for diseases with very different time scales [START_REF] Ventura | Role of time scale in the spreading of asymmetrically interacting diseases[END_REF], or when considering interactions between a disease and the adoption of prudent behaviors [START_REF] Scarpino | The effect of a prudent adaptive behaviour on disease transmission[END_REF], which is instead driven by a phenomenologically and analytically different social contagion process.

Here, we show that a simple contagion (describing infectious disease spreading) can exhibit the characteristics of a simplicial contagion when it is cooperatively driven by a simplicial contagion (describing the spread of a risky social behavior). Namely, a simple contagion in the epidemic-free regime can exhibit an abrupt transition to the epidemic regime, as well as bi-stability, if the cooperative driving by the social process is stronger than a critical value. In particular, in the asymmetrically driven case, discontinuous transitions can only take place when the driving process is simplicial, contrary to the case of symmetric interactions. We describe the phase diagram of the system through a mean-field (MF) approach, complemented by the numerical integration of coupled Markov-chain equations, and provide an analytical expression for the critical value of the cooperation. Finally, we identify effective infectivities as markers of the abrupt driven transition by rewriting the MF equations as a simple contagion with effective parameters.

II. RESULTS

A. Model for interacting simplicial contagion processes

We consider a model for two interacting spreading processes, denoted as A and B, which also include simplicial contagions [START_REF] Iacopini | Simplicial models of social contagion[END_REF][START_REF] Barrat | Social contagion on higher-order structures[END_REF]. Individuals are represented by a set of N nodes that can each be in one of four compartments, following the standard SIS framework [START_REF] Pastor-Satorras | Epidemic processes in complex networks[END_REF]: those susceptible to both diseases (S), infected exclusively by one of the two diseases (either A or B), or by both (AB) [see Fig. 1(a)]. The compartment membership of each node i is encoded in three binary variables x γ i ∈ {0, 1}, where γ ∈ {A, B, AB}. If node i is in state γ then x γ i = 1, otherwise it is zero: each node has either one non-zero or all zero variables. The density of nodes in state γ is given, at each time t, by ρ γ (t) = 1 N N i=1 x γ i (t). The densities ρ(t) = {ρ A (t), ρ B (t), ρ AB (t)} serve as macroscopic order parameters (with ρ S (t) = 1 -ρ A (t) -ρ B (t) -ρ AB (t) the density of susceptible individuals).

Nodes can interact in pairs or larger groups, so that contagion events, which cause nodes to change compartment, take place on top of a contact structure that allows for higherorder (non-pairwise) interactions [START_REF] Battiston | Networks beyond pairwise interactions: Structure and dynamics[END_REF][START_REF] Lambiotte | From networks to optimal higher-order models of complex systems[END_REF][START_REF] Bick | What are higher-order networks?[END_REF][START_REF] Battiston | The physics of higher-order interactions in complex systems[END_REF]. We mathematically represent a group encounter as a k-hyperedge, a set of k + 1 interacting nodes [START_REF] Torres | The why, how, and when of representations for complex systems[END_REF]. For simplicity, we allow for interactions up to dimension k = 2, i.e., on 1-hyperedges (links) and 2-hyperedges (triangles). Six parameters-three for each disease-yield contagion and recovery probabilities (Fig. 1). The infectivity of disease x ∈ {A, B} at order k = 1, β x,1 ≡ β x , is the probability per unit time for a node i susceptible to pathogen x to acquire x from an "infectious" 1hyperedge it is part of [Fig. 1(b)-(d)]. Similarly, β x,2 ≡ β x control infections coming from 2-hyperedges [Fig. 1(d)]. Note that all other nodes in the hyperedge need to be infectious for the hyperedge to be considered so. Finally, µ x ∈ [0, 1] denotes the standard spontaneous recovery probability (from x) per unit time.

The interaction between the two contagion processes is controlled via two additional non-negative parameters, the coupling factors AB and BA that multiply the transition probabilities to a double infection (AB) from a single infection (A or B). For example, the transition B → AB occurs with probability BA β A from a pairwise contact with A [see Fig. 1(e)-(g)]. The two processes cooperate if xx > 1 and compete if xx < 1, while they are independent if xx = 1. Note that the symmetry AB = BA does not need to hold. Furthermore, although the model is defined on a generic higher-order structure, we focus here on simplicial complexes, a particular class of hypergraphs [START_REF] Battiston | Networks beyond pairwise interactions: Structure and dynamics[END_REF]. In a simplicial complex K, by definition, groups of nodes are called simplices and respect downward closure: each sub-simplex ν ⊂ σ built from subsets of a simplex σ ⊂ K is also part of the complex K [in an infectious 2-simplex thus, contagion can occur both through the 1-hyperedges contained and through the 2-hyperedge itself, see Fig. 1(d),(g)]. We make this choice for coherence with previous work [START_REF] Iacopini | Simplicial models of social contagion[END_REF], but it can be relaxed to more general hypergraphs [START_REF] Barrat | Social contagion on higher-order structures[END_REF][START_REF] Landry | The effect of heterogeneity on hypergraph contagion models[END_REF][START_REF] Ferraz De Arruda | Phase transitions and stability of dynamical processes on hypergraphs[END_REF][START_REF] St-Onge | Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs[END_REF] without affecting the MF results.

B. Mean-field description

We consider the MF description of the model, obtained under a homogeneous mixing hypothesis [START_REF] Kiss | Mathematics of Epidemics on Networks: From Exact to Approximate Models[END_REF]. For simplicity, we assume identical recovery rates for the two processes, that is µ A = µ B = µ. In fact, for µ A = µ B the equations can be simply refactored in term of a new parameter δ = µ A /µ B leaving the asymptotic dynamics unchanged (see Sup. Mat. I). We also introduce the rescaled infectivity parameters λ x = β x k /µ and λ x = β x k /µ, for x ∈ {A, B}, where k and k respectively denote the average numbers of 1-and 2-hyperedges incident on a node. After rescaling time by µ, the general mean-field equations describing the evolution of the densities are:

ρA = -ρ A + λ A ρ S (ρ A + ρ AB ) + λ A ρ S (ρ A + ρ AB ) 2 + ρ AB -AB λ B ρ A (ρ B + ρ AB ) -AB λ B ρ A (ρ B + ρ AB ) 2 (1a) ρB = -ρ B + λ B ρ S (ρ B + ρ AB ) + λ B ρ S (ρ B + ρ AB ) 2 + ρ AB -BA λ A ρ B (ρ A + ρ AB ) -BA λ A ρ B (ρ A + ρ AB ) 2 (1b) ρAB = -2ρ AB + AB λ B ρ A (ρ B + ρ AB ) + AB λ B ρ A (ρ B + ρ AB ) 2 + BA λ A ρ B (ρ A + ρ AB ) + BA λ A ρ B (ρ A + ρ AB ) 2 (1c)
with the additional condition that

ρ S = 1 -ρ A -ρ B -ρ AB . (2)
In the following, we focus on a simplicial contagion A (representing a risky social behavior) that cooperatively and uni-directionally drives a simple contagion B (representing a disease). We thus set λ A > 0, λ B = 0, AB > 1 and BA = 1.

In this scenario, it is convenient to consider the total density of infectious individuals for each contagion, regardless of whether they are also infected by the other one. Formally, we introduce two new variables ρ Atot = ρ A + ρ AB and ρ Btot = ρ B + ρ AB . In other words, ρ Atot is the total density of people with a risky behaviour, having been infected by B (ρ AB ), or not (ρ A ). Similarly, ρ Btot is the total density of people infected by disease B, having a risky behavior (ρ AB ), or not (ρ B ). After introducing these two variables, we end up with the system of coupled equations (see Appendix A):

ρAtot = ρ Atot [-1 + λ A (1 -ρ Atot ) + λ A ρ Atot (1 -ρ Atot )], (3a) ρBtot = ρ Btot [-1 + λ B (1 -ρ Btot ) +λ B ( AB -1)(ρ Atot -ρ AB )] , (3b) ρAB = -2ρ AB + AB λ B (ρ Atot -ρ AB )ρ Btot + λ A (ρ Btot -ρ AB )ρ Atot + λ A (ρ Btot -ρ AB )ρ 2 Atot . (3c)
Equations ( 3) include two known specific cases. First, without any interaction between the processes ( AB = BA = 1), ρ Btot and ρ Atot evolve independently as a simple and simplicial contagion [START_REF] Iacopini | Simplicial models of social contagion[END_REF], respectively. Second, by considering only pairwise interactions, λ A = 0 = λ B , A and B evolve as interacting simple contagions [START_REF] Wang | Coevolution spreading in complex networks[END_REF]. In the general case we consider ( BA = 1), the dynamics of ρ Atot is decoupled from the other two variables and drives them. We first study the non-equilibrium stationary state (NESS) reached by the system of Eqs. (3) at large times ρ * Btot = lim t→∞ ρ Btot (t), by numerical integration. Figure 2 shows the resulting ρ * Btot values and their transitions. The most interesting case is given by λ B < 1 as, without a driving process A, the simple contagion process B would be in the epidemicfree absorbing state (ρ * Btot = 0). We thus illustrate how the B NESS ρ * Btot depends on the parameters of the driver A, on the coupling AB , and how it can transition to the epidemic active state, despite λ B < 1.

For λ A ≤ 1, we always obtain a continuous transition for ρ * Btot [see Fig. 2(a,b)]. On the other hand, if λ A > 1, the driven process can exhibit a discontinuous transition [see Fig. 2(c,d)]. More precisely, the transition changes from continuous to discontinuous when the coupling parameter AB becomes larger than a critical value c AB . Above this threshold [ AB > c AB , black circles in Fig. 2(c)], there is a discontinuous transition at a critical value λ c A that does not depend on AB . For weaker cooperation [ AB < c AB , white and gray symbols in Fig. 2 It is important to note that the epidemic-free absorbing state ρ * Btot = 0 remains stable as long as λ A < 1. As a consequence, there is a region of bi-stability λ c A < λ A < 1 (region shaded with yellow background) for AB > c AB . Bi-stability can also be observed when AB < c AB in the region λ c A < λ A < 1 (gray symbols) as long as λ c A < 1. For λ c A ≥ 1 (white symbols), the continuous transition occurs for values larger than one and there is no region of bi-stability. Finally, when λ c A < 1, the stability of the absorbing state implies the existence of a forward discontinuous transition at λ c A = 1 (upward arrows). In conclusion, the simple contagion B exhibits characteristics of a simplicial contagion-an abrupt transition and bi-stabilitydue to the driving of the simplicial contagion A.

To analytically explain this behavior, we need to find the NESS by setting ρx = 0 (see Appendix B for additional details). Solving ρAtot = 0, as Eq. (3a) exactly maps back to the single simplicial contagion analyzed in Ref. [START_REF] Iacopini | Simplicial models of social contagion[END_REF], leads to a trivial solution ρ * Atot = 0 and two other NESS ρ * ,± Atot . Similarly, solving the full two-dimensional system (ρ Btot , ρ AB ) leads to the absorbing state (0, 0) and the implicit solutions for ρ * Btot :

ρ * ,± Btot = 1 - 1 λ B + (ρ * ,± Atot -ρ * ,± AB )( AB -1). (4)
Equation ( 4) implicitly contains two solutions ± from ρ * ,± AB . The value of c AB is represented by shades of green. For visual clarity, the green scale is truncated at a maximum value of 5, so that larger values are represented by the same color as 5. The red cross corresponds to the case shown in Figs. 2(c,d). In region II (λB > 1, λ A > 1), c AB = 1 and the transition is discontinuous for all AB > 1. For λ A ≤ 1, that is regions III and IV, the transition is always continuous. The blue cross indicates the case shown in Figs. 2(a,b).

(see [START_REF] Iacopini | Simplicial models of social contagion[END_REF]). This implies that ρ AB also goes to the absorbing state if λ A < 1: if nobody is infected by A, nobody can be infected by both A and B. As a consequence, the term coming from the cooperation vanishes: ρ * ,± Atot -ρ * ,± AB = 0. Hence, from Eq. ( 4), we have lim λ A →1 -ρ * ,± Btot = 1 -1 λ B ≤ 0 (recall λ B < 1). If negative, it is not a valid solution and only the absorbing state is. As λ A increases above 1, ρ * ,+ Atot increases continuously [START_REF] Iacopini | Simplicial models of social contagion[END_REF], leading ρ * ,± Btot to also cross continuously 0 at a certain λ A ≥ 1. For λ A ≥ 1 instead, ρ * Atot has a discontinuous transition at λ A = λ c A which implies that ρ * AB has one too. Consequently, from Eq. ( 4), since λ B < 1, we have

lim λ A →λ c,- A ρ * Btot = 0, but lim λ A →λ c,+ A ρ * ,± Btot > 0 above a certain value AB > c
AB . Hence, this critical value c AB can be derived analytically by solving ρ * ,+ Btot = 0 at λ c A . In other words, we find the critical driving strength c AB by finding the curve ρ * ,+ Btot , between the gray and black curves in Fig. 2(c), such that it reaches zero at λ c A . This corresponds to the case λ B ≤ 1, λ A > 1 which we denoted region I in Fig. 3. Similarly, if λ B > 1 instead (with λ A > 1, region II), it suffices solving ρ * ,+ Btot = 1 -1/λ B at λ c A because 1 -1/λ B is now the pre-transition NESS. In summary, the discontinuity is controlled by the term coming from the cooperation, (ρ * ,± Atot -ρ * ,± AB )( AB -1) which will be discontinuous if A has a discontinuous transition.

In short, the nature of the transition depends on the new second term induced by the driving in Eq. ( 4) and, in particular, on the λ A value at which it becomes positive. This fact, allow us to obtain the critical c AB at which the discontinuous transition in B becomes possible (Fig. 3):

c AB =    √ λ A -λ B √ λ A -1 λ B in region I (λ B ≤ 1, λ A > 1), 1 in region II (λ B > 1, λ A > 1).
(5) In region I, increasing λ A or λ B makes c AB decrease, so that discontinuous transitions are obtained for smaller values of the driving strength AB . In fact, c AB → +∞ as λ A → 1 or λ B → 0. In region II, all values of cooperation AB > 1 yield a discontinuous transition. Finally, no critical value of cooperation can be defined in regions III and IV (λ A ≤ 1), where transitions are always continuous.

C. Effective formalism

In this section, we devise an effective contagion theory to highlight the origins of the observed transitions. In particular, we saw that, for AB > c AB -that is, when simplicial behavior is possible for B-the driven process B will exhibit discontinuous transitions as a function of λ A . We illustrate how this phenomenology emerges by rewriting the dynamics of B as an effective simple contagion following Ref. [START_REF] Hébert-Dufresne | Macroscopic patterns of interacting contagions are indistinguishable from social reinforcement[END_REF].

First, note that we can rewrite the MF equation of the single simplicial ρ Atot from Eq. (3a) as a simple contagion

ρAtot = -ρ Atot + λA ρ Atot [1 -ρ Atot ], (6)
with effective infectivity λA = λ A +λ A ρ Atot . As expected, the effective infectivity depends on both the simple and simplicial infectivities. Since Eq. ( 6) is written as a simple contagion, its well known stationary solutions given by 1 -1/ λA and the effective infectivity also has a critical value of 1.

Similarly, we can also rewrite Eq. (3b) of the driven ρ Btot as a simple contagion

ρBtot = -ρ Btot + λB ρ Btot [1 -ρ Btot ], (7)
with effective infectivity

λB = λ B + λ B ( AB -1) 1 1 -ρ Btot ρ A . (8)
Since we observed characteristics of the driver A in the driven contagion B, we may want to further cast its effective infectivity into a form similar to that of A: λB = λ B + λ B ρ Btot . This is achieved by defining an effective simplicial infectivity

λ B = λ B ( AB -1) ρ A ρ Btot (1 -ρ Btot ) , (9)
which implicitly depends on λ A and λ A through ρ A .

If there is no interaction ( AB = 1), we recover λB = λ B because the effective simplicial infectivity vanishes, λ B = 0, as expected. More importantly, since ρ Btot evolves according to the effectively simple contagion of Eq. ( 7), its stationary solution is given by 1 -1/ λB which yields a critical value λB = 1. This can help distinguish between the transitions observed in Fig. 2. Indeed, below the critical value, λB < 1, the only stable NESS of ρ Btot is 0, and above it there is also a positive solution. Thus, the driven contagion B has a transition to an epidemic state if and only if λB crosses 1 as λ A increases. Finally and most importantly, this transition is discontinuous if and only if the transition of λB across the values one is discontinuous. This can be seen by comparing the three curves in Fig. 4(a)-crossing the value one (white background) discontinuously (black) or continuously (gray and white)-with the corresponding curves for ρ * Btot in Fig. 2(c).

D. Temporal properties

So far, we lack information about the temporal trajectories, which, in practical settings, are often the only data available. Consider observing the spread of B via ρ Btot (t), while the driving social contagion process A remains unobservable. Interestingly, the observed B evolves differently depending on the initial conditions of the hidden process A.

We show the phenomenology described beyond the homogeneous mixing hypothesis, by shifting to a Markov-chain formalism [START_REF] Gómez | Discrete-time markov chain approach to contactbased disease spreading in complex networks[END_REF][START_REF] Soriano-Paños | Markovian approach to tackle the interaction of simultaneous diseases[END_REF]. With this microscopic approach we can encode any interaction structure between nodes-contrary to MF approaches that assume homogeneous mixing of the population-while keeping the computational cost lower than the one required for Monte Carlo simulations. The complete Markov-chain description of our model can be found in Appendix C. We build a synthetic random simplicial complex up to dimension 2 by means of the generative model introduced in Ref. [START_REF] Iacopini | Simplicial models of social contagion[END_REF]. This model, a direct extension of Erdös-Rényi-like models for graphs, allows to generate a simplicial complex starting from a number of nodes that get randomly connected to form simplices. We here generate a simplicial complex with N = 2000 nodes having k = 20 and k = 6 and integrate the associated Markov equations to follow the temporal evolution of the system.

We consider the scenario of the black curve ( AB = 1.75) of Fig. 2(b), that is, with a simplicial driver A (λ A = 2.5). We fix all other parameters, including the initial condition ρ Btot (0), but vary the initial condition of the driver, ρ Atot (0). As shown in Fig. 5, if the driver contagion A is in the endemic regime but not in the bi-stability region (e.g., λ A = 1.2), ρ Btot reaches the same NESS for all values of ρ Atot (0), but with different transient dynamics and even non-monotonic evolutions [Fig. 5(a)]. Moreover, if the simplicial driver is in the bi-stability region (λ A = 0.7), it induces bi-stability in B: ρ Btot (t) can reach two different states, depending on the driving initial condition, even though all "visible" B parameters are fixed [Fig. 5(b)]. Note that this bi-stability emerges only if the driving is simplicial with λ A > 1 (see also Sup. Mat. Fig. S1).

III. DISCUSSION

In conclusion, our results highlight that an abrupt transition in the observed process can occur as a function of the control parameter of a second-potentially hidden-driver process. Consider an observer of an epidemic process of unknown nature. A natural intervention would try to reduce the intrinsic infectivity of the spreading pathogen, e.g. through pharmaceutical interventions or reduction of social contacts (sanitary lockdowns): this would however lead only to a continuous change in the incidence. However, if the spread is driven by an underlying complex contagion, then acting on the hidden driver process (e.g. trying to reduce the social adoption of risky behaviors) could more effectively lead to an abrupt transition to the epidemic-free state (if the interaction is strong enough AB > c AB ). Finally, different populations could be characterised by different properties of the hidden behavioral contagion process (different values of λ A and λ ∆ A values), thus leading to a large diversity of temporal evolutions, and-potentially-of final outcomes of the pathogen's spread, without the need for different intrinsic infectivity properties of the pathogen across these populations.

Results also suggest that other driving spreading processes could yield a similar phenomenology if they exhibit a discontinuous transition (e.g. [START_REF] Gómez-Gardenes | Explosive contagion in networks[END_REF]), inducing a change from a continuous to a discontinuous transition in the driven process.

We note in this context that the framework of Ref. [START_REF] Kuehn | A universal route to explosive phenomena[END_REF] suggests a universal route to abrupt transitions, achieved through the addition of a control parameter to a process that displays a continuous phase transition. However, the situation that we have explored here broadens the picture. Indeed, if both spreading processes are simple contagions, it appears that a bi-directional interaction (leading to a feedback loop) is an additional necessary condition for a discontinuous transition to emerge. In the case of a uni-directional coupling, instead, the driving process needs to be itself simplicial with bi-stability. Our results thus provide a different route to the emergence of abrupt transitions in epidemic-like processes due to the asymmetric coupling of the contagion dynamics, as opposed to the addition of a control parameter [START_REF] Kuehn | A universal route to explosive phenomena[END_REF]. This resonates with recent results in synchronization phenomena [START_REF] Khanra | Explosive synchronization in multilayer networks through partial adaptation[END_REF]. The exact conditions under which these routes apply to coupled systems in general would be an interesting direction for future work. Another interesting perspective would consist in the analysis of real-world data and the development of tools to detect the footprints of simple, complex, or coupled processes from observed time series [START_REF] Brett | Detecting critical slowing down in high-dimensional epidemiological systems[END_REF] in order to discriminate them or, potentially, perform full reconstruction [START_REF] Peixoto | Network reconstruction and community detection from dynamics[END_REF].

setting the left-hand side to zero:

0 = -2ρ AB + AB λ B (ρ Atot -ρ AB )ρ Btot + λ A (ρ Btot -ρ AB )ρ Atot + λ A (ρ Btot -ρ AB )ρ 2 Atot , (B2) =ρ AB -2 -AB λ B ρ Btot -λ A ρ Atot -λ A ρ 2 Atot (B3)
+ρ Atot ρ Btot AB λ B + λ A + λ A ρ Atot , from which we already see that ρ * AB = 0 if ρ Atot = 0 or ρ Btot = 0. Now, we inject the expression of ρ * Btot from Eq. ( 4) and cast the equation into quadratic form in ρ AB :

0 = Aρ 2 AB + Bρ AB + C, (B4)
where

A = + AB λ B E - AB , (B5)
B = -2 -AB λ B (Λ - B + E - AB ρ * Atot ) (B6) -(λ A + E - AB K)ρ * Atot -λ A ρ * 2 Atot , C = ρ * Atot K(Λ - B + E - AB ρ * Atot ). (B7)
To shorten the notation, we have also defined

E - AB = AB -1, (B8)
Λ - i = 1 -1/λ i , (B9)
K = AB λ B + λ A + λ A ρ * Atot . (B10)
The non-zero solutions for ρ AB is the standard quadratic solution

ρ * ,± AB = -B ± √ B 2 -4AC 2A , (B11)
which, unfolded, is an expression in terms of the parameters of the system only. These together with ρ * Atot can be reinjected into Eq. ( 4) for ρ * Btot to close the system.

FIG. 1 .

 1 FIG.1. The model of interacting simplicial contagions. (a) Transition probabilities between the compartments: susceptible (S, gray), infected exclusively by one disease (A or B, respectively blue/red) or by both (AB, black). (b)-(d) A susceptible node i can acquire A after a contact with an infectious k-hyperedge (this also includes AB individuals). In (d), since i is part of a 2-simplex composed by two other infectious nodes, the infection can come both from each of the two 1-hyperedges (links) with probability βA and from the 2-hyperedge with probability β A . (e)-(g) If i is already infected with B, the probability of getting A for each contact is affected by the coupling factor BA. The same rules symmetrically apply to B instead of A.

FIG. 2 .

 2 FIG. 2. Abrupt transition induced by a simplicial driver. A simplicial driver process for A, with λ A = 2.5, can induce a discontinuous transition (c,d), contrary to a simple driver (a,b), with λ A = 0 [λB = 0.8, λ B = 0]. Note the different scales on the horizontal axes. (a,c) Stationary solutions ρ * Btot of the MF Eqs. (3) plotted as a function of the rescaled pairwise infectivity λA for three values of the driving strength AB . In (c), the transition of the simple contagion B becomes discontinuous above a critical value of cooperation c AB . (b,d) Heatmaps of ρ * Btot as a function of λA and AB . Dashed horizontal lines correspond to the selected AB values shown in (a) and (c) respectively. The blue dot in (d) highlights the critical point (λ c A , c AB ). The blue and red crosses represent a visual hint to locate the results within the full phase diagram of Fig. 3.

Atot 1 FIG. 3 .

 13 FIG. 3. Phase diagram of the system. The (λB, λ A ) parameter space exhibits four regions. In region I (λB ≤ 1, λ A > 1), ρ * Btot undergoes an abrupt transition if the driving cooperation is strong enough, AB > cAB . The value of c AB is represented by shades of green. For visual clarity, the green scale is truncated at a maximum value of 5, so that larger values are represented by the same color as 5. The red cross corresponds to the case shown in Figs.2(c,d). In region II (λB > 1, λ A > 1), c AB = 1 and the transition is discontinuous for all AB > 1. For λ A ≤ 1, that is regions III and IV, the transition is always continuous. The blue cross indicates the case shown in Figs.2(a,b).

1 . 40 FIG. 4 .

 1404 FIG. 4. The discontinuous nature of the driven contagion B can be determined from its effective infectivity λB. (a) We show λB against the infectivity λA for several values of cooperation AB , corresponding to the curves in Fig. 2(c). The full phase diagram as a function of both λA and AB is shown as a heatmap in panel (b), where the dot corresponds to the critical point (λ c A , c AB ). The background color in (a) corresponds to the colorbar in (b). It corresponds to the values of the vertical axis and highlight visually the critical value λB = 1

FIG. 5 .

 5 FIG. 5. The temporal evolution of the simple contagion B is affected by the initial conditions of the (hidden) driver A. We show ρB tot over time, resulting from the numerical integration of the Markov-chain equations for a simplicial complex with N = 2000 nodes, k = 20 and k = 6, for A in (a) the endemic region, λA = 1.2, and (b) the bi-stable region, λA = 0.7. Shades of red from dark to light represent a range of initial conditions of the driver ρA(0) from 0.001 to 0.35 [see Sup. Mat. Fig. S1 for the temporal evolution of ρA(t)]. In (b), the simple contagion process B can reach one of two stationary states, depending on the initial conditions of the driver A. Other parameters are set to λB = 0.8, λ B = 0, AB = 1.75, and λ A = 2.5.

ACKNOWLEDGEMENTS I.I. acknowledges support from the James S. McDonnell Foundation 21 st Century Science Initiative Understanding Dynamic and Multi-scale Systems -Postdoctoral Fellowship Award. A.B. acknowledges support from the Agence Nationale de la Recherche (ANR) project DATAREDUX (ANR-19-CE46-0008). M.L. and G.P. acknowledge partial support from the Intesa Sanpaolo Innovation Center during the preparation of this work. Code availability. The code used in this study is available at https://github.com/iaciac/ interacting-simplagions.

Appendix A: Derivation of MF description

As explained in the main text, we focus on the case BA = 1, λ B = 0, so that Eqs. [START_REF] Daley | Epidemics and rumours[END_REF] becomes

Then, we apply the following change of variables:

We further rewrite this by replacing all remaining ρ A and ρ B , and using the identity

which can be refactored to obtain Eqs. (3) from the main text.

Appendix B: Derivation of the MF fixed points

Equation (3a) is the same as the simplicial contagion from Ref. [START_REF] Iacopini | Simplicial models of social contagion[END_REF], and its non-trivial solutions are

(B1) For this isolated case, we know that λ A controls the type of transition to the epidemic state [START_REF] Iacopini | Simplicial models of social contagion[END_REF]. That is, for λ A ≤ 1, the bifurcation diagram has a continuous transition at λ = 1 from ρ * Atot = 0 to the epidemic state ρ * ,+ Atot . When instead λ A > 1, a discontinuous transition to ρ * ,+ Atot occurs at λ c A = -λ A +2 λ A ≤ 1. The epidemic-free state remains stable for λ A ≤ 1, but becomes unstable above: This leads to bi-stability in the parameter region {λ A > 1, λ c A ≤ λ A ≤ 1}. The discontinuous transition is therefore the direct consequence of a sufficiently strong three-body (higher-order) interaction in A (λ A > 1).

The remaining two-dimensional system (ρ Btot , ρ AB ) can be solved analytically by hand or with the help of software such as Mathematica [43]. As discussed, the implicit solution for Eq. (3b) is given by Eq. (4).

To solve for ρ AB , we rewrite Eq. (3c) by factorizing and Appendix C: Markov-chain approach Here, we write a system of coupled Markov-chain equations which govern the microscopic evolution of our model [START_REF] Gómez | Discrete-time markov chain approach to contactbased disease spreading in complex networks[END_REF][START_REF] Soriano-Paños | Markovian approach to tackle the interaction of simultaneous diseases[END_REF].

More precisely, we can write down the conditional probability P (x γ i (t + 1) = 1|x(t), θ, A) ≡ p i γ (t) of finding each node i in state γ = {S, A, B, AB} at time t + 1 given the probability vector representing the status of all nodes at time t x(t) = x γ i (t), the model parameters θ = {β A , β A , β B , β B , µ A , µ B , AB , BA }, and the structure A. Using the simplified notation p i γ (t), we impose that, at each time,

The Markov-chain equations for the three states are the following:

The different q i x (t) denote the probability of node i not being infected by disease x by any of the simplices it participates in. Considering again only contributions up to D = 2, we have:

where the first product of each equation accounts for the contagion through the links of the simplicial complex K. These links are fully specified by means of the standard adjacency matrix {a ij }, whose elements a ij = 0, 1 denote the absence or presence of a link (i, j). Similarly, the second product accounts for the contagion of i through the 2-simplices of K (triangles), which are analogously specified by the elements of the adjacency tensor {a ijl }. This tensor is the 3-dimensional version of the adjacency matrix, in which a non-zero element denotes the presence of a 2-simplex (i, j, l).

Finally, the factors f i A (t) and f i B (t) in Eq. (C2) denote the probability of transitioning from state S to one of the states A or B when exposed simultaneously to both pathogens. As-suming an equal probability for both diseases [START_REF] Soriano-Paños | Markovian approach to tackle the interaction of simultaneous diseases[END_REF], we can write:

where qi A (t) and qi B (t) correspond to 1 -q i A (t) and 1 -q i B (t), as given by Eqs. (C3), after setting AB = BA = 1.

Supplementary Material: Simplicially driven simple contagion I. CASE OF DIFFERENT RECOVERY RATES: µA = µB

In the main text we assumed identical recovery rates. Here, we remove this constraint and allow them to be potentially different, so that µ A = µ B . By rescaling all equations by µ A (instead of µ), we have the following-instead of Eqs. (A1):

which, introducing the total densities, becomes

and then

which, compared to the case of identical recovery rates, contain the additional µ B µ A factors. We denote that dimensionless ratio δ = µ B µ A and the equations become, after refactoring:

So, the equation for ρ Atot (simplagion) is unchanged, as expected. For ρ Btot , we notice a temporal rescaling by a factor δ, but the implicit solution is unchanged,

We can consider two limits where the timescales for A and B are of different orders. First, in the limit δ 1, which means that B heals much slower than A, ρBtot ≈ 0, that is process B is quasi-static compared to the timescale of process A. Thus, ρ Atot converges fast to its NESS and ρ Btot is driven by that NESS. Second, in the limit δ 1, B heals much faster than A, it is the opposite. It is possible then to rescale time by δ to see that process A now appears quasi-static compared to the timescale of B. So, ρ Btot converges fast to its NESS which is in fact adiabatically moving towards its asymptotic NESS, driven by ρ Atot that slowly converges to its own NESS. S1. The temporal evolution of the simple contagion B is affected by the initial conditions of the hidden driver process A. As for Fig. 5 of the main text, we show ρB tot over time (a-c), but together with the temporal dynamics of the driver process, as given by ρA tot (d-f). In (a,d) a simple driver process is used (λ A = 0.8), while in (b,e) and (c,f) the driver process A is truly simplicial (λ A = 2.5). The process A is placed either in the endemic region, λA = 1.2 [(a,d) and (b,e)] or in the bi-stable region (λA = 0.7). Different curves correspond to different initial conditions of the driver process, ρA(0). The other parameters are set to λB = 0.8, λ B = 0, and AB = 2.


Dr Yorck Mohr
email: yorck.mohr@evonik.com

Dr Alisa Ranscht

Dr Marcelo Alves-Favaro

Alessandra Elsje

Quadrelli

Dr Florian M Wisser

Jérôme Canivet
email: jerome.canivet@ircelyon.univ-lyon1.fr

Dr E A Quadrelli

Nickel-Catalyzed Direct Arylation Polymerization for the Synthesis of Thiophene-based Cross-linked Polymers

Keywords: nickel, catalysis, C-H arylation, thiophene, conjugated polymer

An earth-abundant nickel(II) bipyridine catalyst, combined with lithium hexamethyldisilazide as base, demonstrates its wide applicability in the direct arylation polymerization of di-and trithiophene heteroaryls with poly(hetero)aryl halides. With a nickel catalyst loading of 2.5 mol%, a series of twenty highly cross-linked organic polymers are obtained with 34 to 99% yields. Using mixed polytopic coupling partners allows obtaining alternating and optically active thiophene-based solids with intrinsic porosity.

Introduction

Conjugated polymers have shown to be key components in (opto)electronic devices with applications ranging from light capture to sensing and photo-/electrocatalysis. [START_REF] Reynolds | Conjugated Polymers: Perspective, Theory, and New Materials[END_REF][START_REF] Jayakumar | [END_REF][3][4][5][6][7][8] Among synthetic methods, the direct arylation polymerization (DArP) relies on a C-H/C-X biaryl coupling strategy and allows for reduced synthetic steps and waste compared to traditional biaryl coupling, like Suzuki-Miyaura and Stille-Migita reactions. [9- 16] The last decade shows the expansion of the site-selective DArP methodology to prepare a variety of mostly linear conjugated polymers. [13][14][15][16] However, despite crucial advances to increase DArP overall sustainability, [10,17] the developed processes rely on transition metal catalysis, most often on palladium catalysts. There is thus a need for earth-abundant catalytic systems able to promote DArP.

Inspired by molecular C-H arylation catalysis, [18] Thompson and coworkers developed copper-catalyzed DArP for the synthesis of conjugated linear polymers. [19] An optimized protocol described a Cu-phenanthroline catalyst loading of 5 mol% to get 20 to 80% yield at 140°C in the presence of potassium carbonate as base in a mixture of amide and aromatic solvents. [20,21] Mori and coworkers developed nickel-based systems using phosphine/carbene ligands for polythiophene synthesis from halogeno-thiophenes via Grignard-type intermediates. [22,23] Similarly, we recently reported the nickel 2,2'bipyridine (bpy) catalyzed direct C-H arylation of thiophenes and indole (Scheme 1a). [24,25] A variety of arylthiophenes has been obtained with 100% selectivity towards thiophene's C2 position in ethereal solvents in the presence of lithium hexamethyldisililylazide (LiHMDS) as base. Advantageously avoiding the homocoupling reaction, this Nicatalyzed C-H arylation would allow for alternating units though orthogonal synthesis strategy to prepare periodic, sequential conjugated copolymers. Such perfectly alternating conjugated polymers, without homocoupling pattern affecting light absorption and charge transport, are of a great interest for semiconductors and optoelectronics. [26][27][28][29][30] Furthermore, to introduce porosity within such polymeric networks, we envisioned alternating di-/tritopic heteroarenes and tri-/tetratopic (hetero)aryl electrophiles to get three dimensional polymeric frameworks with intrinsic porosity (Scheme 1b). The porosity of semiconductors is crucial for active site isolation and accessibility for photocatalysis purpose and has also been demonstrated to positively affect the light harvesting efficiency. [31,32] Here we report the first example of Ni-catalyzed DArP for the synthesis of porous conjugated organic polymers made by copolymerization of non-activated polytopic thiophenes and aryl halide subunits (Scheme 2). Analogous sulfur-rich organic polymers, synthesized mainly through imine condensation [33][34][35][36][37] and Pd-catalyzed cross-coupling, [38][39][40] already found applications in light harvesting and electron transport, especially for solar fuels production. [41,42]

Results and Discussion

The Ni(bpy)Cl2/LiHMDS catalytic system has been applied to a scope of four non-activated heteroaryl cores (1A-D) combined with eleven (hetero)aryl halide electrophiles (2a-k) as coupling partners (Figure S1). Under the standard reaction conditions shown in the Scheme 2, twenty organic polymers have been obtained with yields ranging from 34 to 99%, all having intrinsic porosity and optical properties.

The dimensionality playing a crucial role in the properties of porous semiconductors, [43] the choice of the molecular structure and composition of the targeted polymers was made to design three-dimensional porous networks, leading to a more limited number of combinations than expected from the total number of substrates. Indeed, the combination of ditopic benzodithiophene 1A with ditopic electrophiles 2a-d would have led to linear polymeric networks rather than to porous three-dimensional polymers as targeted here, explaining the restricted series of products 3Ae to 3Ak. Also the napthodithiophene 1B and 2,4,6tri(2-thienyl)-1,3,5-triazine 1D were selected as extended ditopic and tritopic heteroarenes for comparison with model benzodithiophene 1A and benzotrithiophene 1C, respectively, in combination with model tritopic iodide 1,3,5-tris(4iodophenyl)benzene 2e. The amount of coupling partners involved in the reaction has been adjusted in each case to get an equimolar ratio of C2-H protons in heteroarenes 1 and halogenides in aryl electrophiles 2.

All reactions gave rise to solids that are insoluble in common solvents including dichloromethane, dioxane and toluene. Importantly we could not detect any soluble short-chain oligomers in solution using GC analysis. Most of compounds 3 are obtained with isolated yields higher than 90%, as determined from the solid product weight considering the expected polymer structure, thus demonstrating the high efficiency of the nickel system to catalyze the C-H arylation of (poly)thiophene heteroaryls (Scheme 2). Elemental analysis and NMR spectroscopy data were supporting the expected polymers structures 3 (see Supporting Information). Indeed, solid-state NMR analysis showed the presence of thiophene-based and (hetero)aromatic repeating units and molecular C:N and C:S ratio confirm the expected composition. Moreover, the degree of cross-linking above 90% (except for 3Ci and 3Cj with ca. 70%), as determined from residual halogen content (Table S3), shows the ability of the nickel bipyridine system to give access to highly reticulated networks. Polymerization reactions performed in the absence of Ni catalyst or LiHMDS base did not lead to product formation. The use of a single substrate, either heteroarene 1A or aryl electrophiles 2a or 2e, under standard conditions reported in Scheme 2, did not give any homocoupled oligomers or polymeric solid. The aryl bromides 2d and 2j gave rise to quite moderate yields with 51% for 3Cd and 33-34% for 3Aj and 3Cj, respectively, most likely due to the lower reactivity of aryl bromides compared to iodides under Ni-catalyzed C-H arylation conditions, in line with report for molecular C-H arylation. [24] In addition, all the obtained polymers 3Ae to 3De have been evaluated by nitrogen physisorption at 77K (Figures S2-S4). While most of the polymers in series 3A, 3B and 3D show type II/H3 isotherms characteristic for non-porous nanoparticles, most of the materials in series C are characterized by type Ib/H4 shaped isotherms. Thus, the apparent surface areas present wide variations depending mostly on the geometry of the coupling partner involved (Table 1). In the 3A series, 3Af and 3Ak stands out for their significant porosity with apparent surface areas of approx. 670 m 2 /g and 250 m 2 /g respectively, which most likely arises from ridged tridimensional tri-and tetratopic node 2f and 2k. A similar trend is observed in the 3C series with the highest porosity measured for 3Cf (~1070 m 2 /g) and 3Ck (~680 m 2 /g). Overall, the 3C series gathers porous polymers with high porosity, including 3Cb (~180 m 2 /g), 3Cg (260 m 2 /g) and 3Ch (300 m 2 /g). [a] Calculated from nitrogen physisorption isotherms using the BET method.

[b] Calculated from UV-Vis spectra (n.d. = not determined).

In particular 3Ce having an apparent surface area of approx. 370 m 2 /g underlines the correlation between rigid tritopic building blocks and porosity, as the other materials in the series e (3Ae, 3Be and 3De) made with ditopic or flexible units show apparent surface area lower than 50 m 2 /g. The importance of rigid building blocks to achieve materials with a high porosity is demonstrated by 1 H and 13 C solid-state NMR spectroscopy. Indeed 3Be (62% yield, 50 m 2 /g) is highly reticulated with the lowest amount of unreacted CAr-I groups as deduced qualitatively from 13 C NMR spectra (Figure S10). Moreover deconvolution of the 1 H NMR data using DMFit, [44] indicates the lowest amount of residual HMDS groups for 3De (99% yield, 50 m 2 /g, see Figures S11 &S12), The HMDS groups are most likely stemming from trapped Ni(HMDS)x species. [25,45] In comparison, NMR spectroscopy indicates for 3Ce (99% yield, 370 m 2 /g) the presence of an higher amount of remaining CAr-I groups than in 3Be but similar amounts of CAr-I groups compared to 3Ae (99% yield, 50 m 2 /g) and 3 De and significantly less HMDS species than in 3Ae and 3Be (for further discussion see SI). Analogous nitrogen-and sulfur-containing porous polymers made by Pd-catalyzed Stille cross-coupling apparent surface areas between 80 and 700 m 2 /g have been reported. [38,39] In particular the analogues of 3Ah and 3Ch, reported previously by Bojdys and coworkers as SNP-BDT2 and SNP-BTT2, show similar apparent surface areas of 90 and 400 m 2 /g, respectively. [38] The nickel used herein therefore appears as an appealing alternative to precious-metal catalyst counterparts.

As expected from related literature, all these polymer materials possess indirect band gaps. [46] Indirect optical band gaps have been determined from UV-Vis spectroscopy analysis of the polymers 3Ae to 3De (Figures S5-S7). The indirect band gaps range from 2.0 to 2.7 eV, falling in the range of previously reported analogous polymers (Table 1). [38,39] Together these results show the versatility of this nickelcatalyzed synthetic pathway to get porous light harvesting materials with controlled donor-acceptor configuration. Such materials like 3Ch, made with benzotrithiophene (donor) and triazine (acceptor) moieties having an accessible surface area of 300 m 2 /g and an indirect band gap of 2.58 eV, would be promising candidates for water splitting application, for example. [39,47]

Conclusion

In conclusion, we demonstrated the wide applicability of a molecular Ni-based C-H arylation catalytic system for the direct arylation polymerization of di-and tri-thiophenyl compounds. Twenty highly cross-linked conjugated porous polymers were obtained with intrinsic porosity and genuine optoelectronic properties. This methodology based on an earth-abundant catalyst opens perspective for the synthesis of a library of conjugated porous polymers based on thiophene units, aiming at facilitating and accelerating the discovery of novel optoelectronic materials.

Experimental Section

For a typical synthesis exemplified for 3Ce, 32.

  Ni-catalyzed C-H arylation for heterobiaryl synthesis 21 b) Ni-catalyzed DArP for cross-linked polymer synthesis

Scheme 1 .Scheme 2 .

 12 Scheme 1. From heterobiaryls to conjugated porous polymers with controlled topology through Ni-catalyzed C-H arylation.

  1 mg (0.125 mmol) benzo[1,2-b:3,4-b':5,6-b'']trithiophene 1C, 87.3 mg (0.125 mmol) 4,4''diiodo-5'-(4-iodophenyl)-1,1':3',1''-terphenyl 2e, 142.3 mg (0.825 mmol) lithium bis(trimethylsilyl)amide (LiHMDS) and 2.7 mg (9.4 μmol)Ni(bpy)Cl2 were added inside a glovebox to a flame dried glass reactor equipped with a magnetic stirring bar. The glass reactor was sealed with a Teflon septum. Outside the glovebox, 5 ml of anhydrous dioxane were added and the reactor was placed in an oil bath preheated to 100°C. The reaction was stirred at 100°C for 20 h. The obtained solid was purified by Soxhlet extraction first using MeOH for 1 day and then CHCl3 for 3 days. The yellow solid was dried at 80°C under vacuum.

Table 1 .

 1 Textural and optical properties of the polymer series 3.

	compound	Surface Area	Indirect band gap
		SBET / m 2 •g -1	Eg / eV [b]
	3Ae	10	2.25
	3Af	673	2.27
	3Ag	37	n.d.
	3Ah	47	2.31
	3Ai	34	2.52
	3Aj	39	2.23
	3Ak	250	2.49
	3Be	50	2.32
	3Ca	25	2.63
	3Cb	182	2.54
	3Cc	65	2.16
	3Cd	15	2.04
	3Ce	371	2.54
	3Cf	1066	2.40
	3Cg	260	2.44
	3Ch	300	2.58
	3Ci	37	2.54
	3Cj	27	2.34
	3Ck	678	2.68
	3De	25	2.12

Acknowledgements

This work has been carried out within the H-CCAT project that has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 720996. The authors also thank the IRCELYON scientific services, and Laurent Veyre from CP2M laboratory (Lyon) for the access to UV-Vis spectrometer.