Abdulkadir Çelikkanat
email: abdcelikkanat@gmail.com

Fragkiskos D Malliaros
email: fragkiskos.me@gmail.com

Learning Node Embeddings with Exponential Family Distributions

Representing networks in a low dimensional latent space is a crucial task with many interesting application in graph learning problems, such as link prediction and node classification. A widely applied network representation learning paradigm is based on the combination of random walks with the traditional Skip-Gram approach, modeling center-context node relationships. In this paper, we emphasize on exponential family distributions to capture rich interaction patterns between nodes in random walk sequences. We introduce the generic exponential family graph embedding (EFGE) model, that generalizes random walk-based network representation learning techniques to exponential family conditional distributions. Our experimental evaluation demonstrates that the proposed technique outperforms well-known baseline methods in two downstream machine learning tasks.

, can be reinterpreted under the EFGE model.

Introduction

Graphs or networks have become ubiquitous as data from diverse disciplines can naturally be represented as graph structures. A recent paradigm in network analysis, known as network representation learning (NRL) [START_REF] Hamilton | Representation learning on graphs: Methods and applications[END_REF], aims at finding vector representations of nodes (i.e., node embeddings), in such a way that the structure of the network and its various properties are preserved in the lower dimensional representation space. That way, after obtaining the embeddings, the learned features can further be used in any downstream machine learning task, such as classification and prediction.

Random walk-based methods have become a prominent line of research [START_REF] Perozzi | DeepWalk: Online learning of social representations[END_REF][START_REF] Grover | Node2Vec: Scalable feature learning for networks[END_REF], being inspired from the field of natural language processing (NLP) [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF]. Typically, those methods sample a set of random walks on the input graph, treating them as the equivalent of sentences in natural language, while the nodes visited by the walk are considered as words. Then, widely used NLP models, such as the Skip-Gram model [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF], are used to learn node latent representations, examining simple co-occurrence relationships of nodes within the set of random walk sequences.

Nevertheless, Skip-Gram models the conditional distribution of nodes within a random walk based on the softmax function, which might prohibit to capture richer types of interaction patterns among nodes that co-occur within a random walk. In this work, we argue that considering more expressive conditional probability models to relate nodes within a random walk sequence, might lead to more informative representations. We capitalize on exponential family distribution models to capture interactions between nodes. The main contributions of the paper can be summarized as follows:

• We demonstrate that the proposed exponential family graph embedding models generally outperform widely used baseline approaches in various learning tasks on graphs.

Preliminary Concepts

Random walk-based methods. For a graph G = (V, E), a sequence of nodes w = (w 1 , ..., w L ) ∈ W will be called a walk if (w l-1 , w l ) ∈ E for every 2 ≤ l ≤ L and W will represent the set of walks of length L. The context sequence of a node w l , referred to as center, in the random walk w ∈ W is defined as N w γ (w l ) := (w l-γ , . . . , w l-1 , w l+1 , . . . , w l+γ ). Random walk-based node embedding methods [START_REF] Perozzi | DeepWalk: Online learning of social representations[END_REF][START_REF] Grover | Node2Vec: Scalable feature learning for networks[END_REF][START_REF] Nguyen | BiasedWalk: Biased sampling for representation learning on graphs[END_REF] generate a set of node sequences by simulating various (e.g., uniform, biased) random walk strategies. Node representations are then learned by optimizing the relationship between each center and context node pair in the generated sequences under the Skip-Gram model. We typically learn two embedding vectors α[v] and β[v] for each node v ∈ V, where β[v] corresponds to the vector if the node is interpreted as center and α[v] denotes the vector if it is considered as context (in the experiments, we will only consider α[v] to represent the embedding vector of v).

Exponential families. A class of probability distributions is called exponential family, if they can be expressed as p(x) = h(x) exp (ηT (x) -A(η)), where h is the base measure, η are the natural parameters, T is the sufficient statistic and A(η) is the log-partition function [START_REF] Andersen | Sufficiency and exponential families for discrete sample spaces[END_REF]. Different choices of base measure and sufficient statistics lead us to obtain different distributions. For instance, the base measure and sufficient statistic of Poisson distribution are h(x) = 1/x! and T (x) = x. We use the natural parameter η v,u to design a set of network representation learning models [START_REF] Rudolph | Exponential family embeddings[END_REF]; it is defined as the product of context and center vectors,

η v,u := f α[u] β[v]
, where f is called the link function.

Proposed Approach

We define the generic objective function to learn node embeddings in the following way:

L(α, β) := arg max Ω w∈W 1≤l≤L v∈V log p(y l w l ,v ; Ω), (1)
where y l w l ,v is the observed value indicating the relationship between center w l and context node v. Instead of restricting ourselves to the Sigmoid or Softmax functions in order to model the conditional probability in the objective function of Eq. (1), we assume that each y w l ,v follows an exponential family distribution. That way, the objective to learn node embeddings Ω = (α, β) can be rewritten as:

arg max Ω w∈W 1≤l≤L v∈V log h(x w l ,v ) + η w l ,v T (x w l ,v ) -A(η w l ,v ) . (2)
As we can observe, Eq. ( 2) generalizes Skip-Gram-based models to exponential family distributions. Therefore, the EFGE models have the additional flexibility to utilize a wide range of exponential distributions, allowing them to capture more complex types of node interactions.

We examine three particular instances of the EFGE model, in particular, we utilize the Bernoulli, Poisson, and Normal distributions leading to the corresponding EFGE-BERN, EFGE-POIS and EFGE-NORM models. For illustration, two dimensional embeddings of Dolphins network [START_REF] Lusseau | The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations[END_REF] are depicted in Fig. 1 in the Appendix. As we can observe, the proposed EFGE-BERN and EFGE-POIS models learn representations differentiating nodes with respect to their communities.

The EFGE-BERN Model

Our first model is the EFGE-BERN model, in which we assume that each y w l ,v follows a Bernoulli distribution, which is equal to 1 if node v appears in the context set of w l in the walk w ∈ W. It can be written as

y w l ,v = x l-γ w l ,v ∨ • • • ∨x l-1 w l ,v ∨ x l+1 w l ,v ∨ • • • ∨x l+γ w l ,v
, where x t+l w l ,v indicates the appearance of v in the context of w l at the position t (-γ ≤ t ≤ γ). We express the objective function of the EFGE-BERN model, L B (α, β), by dividing Eq. ( 2) into two parts with respect to the values of x w l ,v :

w∈W 1≤l≤L   v∈Nγ (w l ) log p(y w l v )+ v ∈Nγ (w l ) log p(y w l v )   = w∈W 1≤l≤L     -γ≤j≤γ u:=wj log p(x l+j w l u )+ -γ≤j≤γ u: =wj log p(x l+j w l u )     = w∈W 1≤l≤L     -γ≤j≤γ u:=wj log σ(η w l u )+ -γ≤j≤γ u: =wj log σ(-η w l u )     ,
where σ(η) is the sigmoid function defined as 1/(1 -exp(-η)) and the identity map is chosen for the link function, so η v,u is the product of vectors α[v] and β[u].

Relationship to negative sampling. In Lemma 1, we show that the log-likelihood L B (α, β) of the EFGE-BERN model in fact converges to the objective function of negative sampling given in Eq. ( 3). Lemma 1. For large values of k, the log-likelihood function L B converges to

w∈W 1≤l≤L -γ≤j≤γ log p(x l+j w l ,w l+j ) + k s=1 E u∼q -log p(x l+j w l ,u ) . (3)
Proof. Please see the appendix.

The EFGE-POIS Model

Let y w l ,v be a value indicating the number of occurrences of node v in the context of w l . We assume that y w l ,v follows a Poisson distribution, with the mean value λw l ,v being the number of appearances of node v in the context N w γ (w l ). Similar to the previous model, it can be expressed as

y w l ,v = x l-γ w l ,v + • • • + x l-1 w l ,v + x l+1 w l ,v + • • • + x l+γ w l ,v
, where x l+t w l ,v ∼ P ois(λ w l ,v ) for -γ ≤ t ≤ γ. That way, we obtain λw l ,v = γ j=-γ λ l+j w l ,v , since the sum of independent Poisson random variables is also Poisson. By plugging the exponential form of the Poisson distribution into Eq. ( 1), and following a similar strategy as in the EFGE-BERN model, the equation can be split into two parts for the cases where y w l ,v > 0 and y w l ,v = 0. That way, the negative sampling strategy (given in Eq. ( 3)) can be adopted:

w∈W 1≤l≤L -γ≤j≤γ u:=wj -log(x w l ,u !)+η w l ,u x w l ,u -exp(η w l ,u ) + -γ≤j≤γ u: =wj -exp(η w l ,u ) .
Relationship to overlapping community detection. It can be seen that the objective function of the widely used BIGCLAM overlapping community detection method by Yang and Leskovec [START_REF] Yang | Overlapping community detection at scale: A nonnegative matrix factorization approach[END_REF], can be obtained by unifying the objectives of the EFGE-BERN and EFGE-POIS models. Lemma 2. Let Z w l ,v be independent random variables following Poisson distribution with natural parameter η w l ,v defined as

log(β[w l ] • α[v]).
Then, if the model parameter π w l ,v is defined as p(Z w l ,v > 0), the objective function of EFGE-BERN model becomes equal to

w∈W 1≤l≤L v∈Nγ (w l ) log 1 -exp -β[w l ] • α[v] - v ∈Nγ (w l ) β[w l ] • α[v] .
Proof. Please see the appendix.

The EFGE-NORM Model

We consider each y w l ,v as an edge weight indicating the relationship between w l and v. We assume that

x l+t w l ,v ∼ N (1, σ 2 + ) if v ∈ N γ (w l )
, and x l+t w l ,v ∼ N (0, σ 2 -) otherwise. Hence, we obtain that

y w l ,v ∼ N (μ, σ2
), where μ is the number of occurrences of v in the context if we follow a similar assumption y w l ,v = γ j=-γ x l+j w l ,v as in the previous models. The base measure of the model is exp(-x 2 w l ,u /2σ 2 )/ √ 2πσ for known variance, and η w l ,u is defined as exp(-α[u] β[w l ]):

w∈W 1≤l≤L -γ≤j≤γ u:=wj log h(x w l u )+x w l u η w l u σ + - η 2 w l u 2 + -γ≤j≤γ u: =wj log h(x w l u )+x w l u η w l u σ -- η 2 w l u 2 .

Experimental Evaluation

The performance of the proposed models is evaluated in node classification and link prediction tasks.

The details of the experimental set-up, baseline methods and datasets are provided in the Appendix. These results can qualitatively be explained by the fact that, the exponential family distribution models enable to effectively capture the number of occurrences of a node within the context of another one, while learning the embedding vectors. Of course, the structural properties of the network, such as the existence of community structure, might affect the performance as shown in the toy example of Fig. 1 in the Appendix. The existence of well defined communities at the Dolphins network, allows the EFGE-POIS model to learn more discriminative embeddings with respect to the underlying communities.

Link prediction. Table 2 shows the area under curve (AUC) scores for the link prediction task. Since the networks used in the node classification experiments consist of disconnected components, we perform the link prediction over the largest connected component. As it can be seen, the EFGE-NORM model is performing quite well on almost all different types of networks. Although NODE2VEC is quite effective having the same performance in two datasets, EFGE-NORM performs quite better in the remaining networks, with gain ranging from 0.04% up to 18.29%.

Conclusions

We introduced the EFGE models, proposing three instances (EFGE-BERN, EFGE-POIS, and EFGE-NORM) that generalize random walk approaches to exponential family. The benefit of these models stems from the fact that they allow to utilize conditional distributions over center-context node pairs, going beyond simple co-occurrence relationships. The experimental results have demonstrated that the proposed models are able to outperform widely used baseline methods.

A Appendix A.1 Source Code

The C+ implementation of EFGE models can be reached at: https://github.com/ abdcelikkanat/EFGE

A.2 Visualization

For the illustration purposes of the proposed models, Fig. 1 depicts two dimensional embedding vectors of the Dolphins network composed by two communities.

A.3 Proofs of Lemmas

In this section, the proof of lemmas are given. Lemma 1. For large values of k. the log-likelihood function L B converges to

w∈W 1≤l≤L -γ≤j≤γ log p(x l+j w l ,w l+j ) + k s=1 E u∼q -log p(x l+j w l ,u ) .
Proof. Let q -(•|w l ) be the true conditional distribution of a random walk method for generating non-context nodes defined over V. Then, it can be written that

w∈W 1≤l≤L -γ≤j≤γ log p(x l+j w l ,v l+j ) + k s=1 E u∼q -log p(x l+j w l ,u ) ≈ w∈W 1≤l≤L -γ≤j≤γ log p(x l+j w l ,v l+j ) + k 1 k k s=1 us∼q - log p(x l+j w l ,us ) = w∈W 1≤l≤L -γ≤j≤γ log p(x l+j w l ,v l+j ) + k s=1 us∼q - log p(x l+j w l ,us ) ≈ w∈W 1≤l≤L -γ≤j≤γ u:=v l+j log p(x l+j w l ,u ) + -γ≤j≤γ u: =w l+j log p(x l+j w l ,u ) =L B (α, β)
, where the second line follows from the law of large numbers for the sample size of k and k is selected as |V| -1 in the fourth line.

Lemma 2. Let Z w l ,v be independent random variables following Poisson distribution with natural parameter η w l ,v defined as log(β[w l ] • α[v]). Then, the objective function of EFGE-BERN model becomes equal to

w∈W 1≤l≤L v∈Nγ (w l ) log 1 -exp -β[w l ] • α[v] - v ∈Nγ (w l ) β[w l ] • α[v] ,
if the model parameter π w l ,v is defined by p(Z w l ,v > 0).

Proof. Let y w l ,v follow a Bernoulli distribution with parameter π w l ,v and it is equal to 1 if v ∈ N γ (w l ), and 0 otherwise. Then, the objective function L B (α, β) can be divided into parts as follows:

L B = w∈W 1≤l≤L v∈Nγ (w l ) log p(y w l ,vj ) + v ∈Nγ (w l ) log p(y w l ,v ) = w∈W 1≤l≤L v∈Nγ (w l ) log 1 -p(z w l ,v = 0) + v ∈Nγ (vi) log p(z w l ,v = 0) = w∈W 1≤l≤L v∈Nγ (w l ) log 1 -exp -exp(η w l ,v ) + v ∈Nγ (w l ) exp(-η w l ,v ) = w∈W 1≤l≤L v∈Nγ (w l ) log 1 -exp -β[w l ] • α[v] - v ∈Nγ (w l ) β[w l ] • α[v] .

A.4 Description of Datasets

Here we provide a detailed description of networks that we have used in our study and (Table 3 shows various statistics of the networks):

• CiteSeer [3] is a citation network obtained from the CiteSeer library, in which each node corresponds to a paper and the edges indicate reference relationships among papers. The labels represent the subjects of the paper.

• Cora [START_REF] Sen | Collective classification in network data[END_REF] is another citation network constructed from the publications in the machine learning area; the documents are classified into seven categories.

• DBLP [START_REF] Perozzi | Don't walk, skip!: Online learning of multi-scale network embeddings[END_REF] is a co-authorship graph, where an edge exists between nodes if two authors have co-authored at least one paper. The labels represent the research areas.

• AstroPh [START_REF] Leskovec | Graph evolution: Densification and shrinking diameters[END_REF] is another collaboration network built from the papers submitted to the ArXiv repository for the Astro Physics subject area, from January 1993 to April 2003.

• HepTh [START_REF] Leskovec | Graph evolution: Densification and shrinking diameters[END_REF] network is constructed in a similar way from the papers submitted to ArXiv for the High Energy Physics -Theory category.

• GrQc [START_REF] Leskovec | Graph evolution: Densification and shrinking diameters[END_REF] is our last collaboration network which has been constructed from the e-prints submitted to the category of General Relativity and Quantum Cosmology.

• Facebook [START_REF] Leskovec | Learning to discover social circles in ego networks[END_REF] is a social network extracted from a survey conducted via a Facebook application.

|V|

A.5 Baseline Methods

We evaluate the three proposed EFGE models against five state-of-the-art NRL techniques:

• DEEPWALK [START_REF] Perozzi | DeepWalk: Online learning of social representations[END_REF] generates a set of node sequences by choosing a node uniformly at random from the neighbours of the node it currently resides.

• NODE2VEC [START_REF] Grover | Node2Vec: Scalable feature learning for networks[END_REF] relies on a biased random walk strategy, introducing two additional parameters which are used to determine the behaviour of the random walk in visiting nodes close to the one currently residing at. We simply set these parameters to 1.0.

• LINE [START_REF] Tang | LINE: Largescale information network embedding[END_REF] learns embeddings that are based on first-order and second-order proximity (each one of length d/2).

• HOPE [START_REF] Ou | Asymmetric transitivity preserving graph embedding[END_REF] is a matrix factorization method which aims at extracting feature vectors by preserving higher order patterns of the network (in our experiments, we have used the Katz index).

• NETMF [START_REF] Qiu | Network embedding as matrix factorization: Unifying DeepWalk, LINE, PTE, and Node2Vec[END_REF] aims at factorizing the matrix approximated by the pointwise mutual information of center and context pairs.

A.6 Experimental Setup

For node classification, we aim at predicting the correct labels of nodes having access to a limited number of training labels (i.e., nodes with known label). In our experiments, we split the learned embedding vectors into varying sizes of training and test sets, from 4% up to 90% in order to better evaluate the models. We perform our experiments applying an one-vs-rest logistic regression classifier with L 2 regularization.

In the link prediction task, the goal is to predict the missing edges or to estimate possible future connections between nodes. For this experiment, we randomly remove half of the edges of a given network, keeping the residual network connected. Then, we learn node representations using the residual network. The removed edges as well as a randomly chosen set of same number node pairs form the testing set. For the training set, we sample the same number of non-existing edges following the same strategy to have negative samples, and the edges in the residual network are used as positive instances. Since we learn embedding vectors for the nodes of the graph, we use the extracted node representation to build edge feature vectors using Hadamard product operator. In all experiments, we have used the logistic regression classifier with L 2 regularization over the networks listed in Table 3.

In our experiments, we have chosen walk length L = 10, number of walks N = 80 and window size γ = 10 for all models and the variants of EFGE model are fed with the same node sequences produced by NODE2VEC. The size of embedding vectors are chosen as 128 for all methods.

A.7 Optimization

For the optimization step of our models, we adopt Stochastic Gradient Descent (SGD) [START_REF] Bottou | Stochastic gradient learning in neural networks[END_REF] to learn latent representations Ω = (α, β). Except for the large networks, we start the initial learning rate from 0.025, and then it is linearly decreased with respect to the number of nodes which have been processed so far. We set the minimum step size to 0.0001 and we do not allow it to fall below this value. Since it is computationally very expensive to compute gradients for each node pairs, we take advantage of the fact that we have formulated the objective function of each model in a such way that it could be divided into two parts according to the values of x ij 's; thus, we adopt the negative sampling strategy with the sampling size k = 5 in all the experiments. We generate negative samples from the whole vertex set with respect to the number of occurrences of nodes in the generated walks raised to the power of 0.75, similar to Ref. [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF].

A.8 Parameter Sensitivity

We have performed further sensitivity analysis experiments, to better understand the impact of the various parameters in the performance of the proposed models.

Figure 1 :

 1 Figure 1: The Dolphins network composed by 2 communities and learned embeddings for d = 2.

Figure 2 :

 2 Figure 2: Influence of dimension size over CiteSeer network.

Figure 3 :

 3 Figure 3: Influence of window size γ for the CiteSeer network.

Figure 4 :

 4 Figure 4: Influence of standard deviation for the EFGE-NORM model.

Table 1 :

 1 Micro-F1 scores for node classification.

		4%	6%	8%	10% 30% 50% 70% 90%
	DEEPWALK 0.460 0.489 0.505 0.517 0.566 0.584 0.595 0.592
	NODE2VEC 0.491 0.517 0.530 0.541 0.585 0.597 0.601 0.599
	LINE 0.387 0.423 0.451 0.466 0.532 0.551 0.560 0.564
	HOPE 0.205 0.210 0.204 0.219 0.256 0.277 0.299 0.320
	NETMF 0.496 0.526 0.540 0.552 0.590 0.603 0.604 0.608
	EFGE-BERN 0.493 0.517 0.536 0.549 0.588 0.603 0.609 0.609
	EFGE-POIS 0.514 0.537 0.551 0.562 0.595 0.606 0.611 0.613
	EFGE-NORM 0.525 0.542 0.553 0.561 0.596 0.606 0.612 0.616
				a: CiteSeer
		4%	6%	8%	10% 30% 50% 70% 90%
	DEEPWALK 0.689 0.715 0.732 0.747 0.802 0.819 0.826 0.833
	NODE2VEC 0.714 0.743 0.757 0.769 0.815 0.831 0.839 0.841
	LINE 0.544 0.590 0.633 0.661 0.746 0.765 0.774 0.775
	HOPE 0.302 0.299 0.302 0.302 0.301 0.302 0.303 0.302
	NETMF 0.716 0.748 0.767 0.773 0.821 0.834 0.841 0.844
	EFGE-BERN 0.720 0.743 0.759 0.767 0.808 0.823 0.834 0.838
	EFGE-POIS	0.733 0.746 0.759 0.765 0.802 0.814 0.820 0.825
	EFGE-NORM 0.743 0.760 0.770 0.780 0.810 0.824 0.827 0.839
				b: Cora
		4%	6%	8%	10% 30% 50% 70% 90%
	DEEPWALK 0.585 0.600 0.608 0.613 0.626 0.628 0.628 0.633
	NODE2VEC 0.600 0.611 0.619 0.622 0.636 0.638 0.639 0.639
	LINE 0.580 0.590 0.597 0.603 0.618 0.621 0.623 0.623
	HOPE 0.378 0.379 0.379 0.379 0.379 0.379 0.378 0.380
	NETMF 0.589 0.596 0.601 0.605 0.617 0.620 0.623 0.623
	EFGE-BERN 0.598 0.610 0.617 0.622 0.634 0.638 0.638 0.638
	EFGE-POIS 0.605 0.614 0.620 0.624 0.635 0.637 0.636 0.638
	EFGE-NORM 0.614 0.622 0.624 0.628 0.637 0.640 0.642 0.641
				c: DBLP
	D EE PW AL K	N OD E2 V EC	L IN E	H O PE	N ET M F	E FG E -B ER N	E FG E -P OI S	E FG E -N OR M
	Citeseer 0.770 0.780 0.717 0.744 0.742 0.815 0.834 0.828
	Cora 0.739 0.757 0.686 0.712 0.755 0.769 0.797 0.807
	DBLP 0.919 0.954 0.933 0.873 0.930 0.950 0.950 0.955
	AstroPh 0.911 0.969 0.971 0.931 0.897 0.963 0.922 0.973
	HepTh 0.843 0.896 0.854 0.836 0.882 0.898 0.885 0.896
	Facebook 0.980 0.992 0.986 0.975 0.987 0.991 0.991 0.992
	GrQc 0.921 0.940 0.909 0.902 0.928 0.938 0.937 0.940

Table 2 :

 2 AUC scores for link prediction.Node classification.Table 1 shows the classification performance on three different networks. The experiments show that the proposed EFGE-POIS and EFGE-NORM models perform quite well, outperform most baselines especially on limited training data. The percentage gain for Micro-F 1 score of our best model with respect to the highest baseline score varies from 0.61% up to 5.33% for CiteSeer and from 0.22% to 2.44%

for DBLP. The highest gain of EFGE-NORM model against the best performing baseline is around 3.80% over Cora.

Table 3 :

 3 Statistics of networks used in the experiments. |V|: number of nodes, |E|: number of edges, |K|: number of labels and |C|: number of connected components.

			|E|	|K|	|C|	Avg. Degree Density
	CiteSeer	3,312	4,660	6	438	2.814	0.0009
	Cora	2,708	5,278	7	78	3.898	0.0014
	DBLP 27,199 66,832	4	2,115	4.914	0.0002
	AstroPh 17,903 19,7031	-	1	22.010	0.0012
	HepTh	8,638	24,827	-	1	5.7483	0.0007
	Facebook	4,039	88,234	-	1	43.6910	0.0108
	GrQc	4,158	13,428	-	1	6.4589	0.0016


ouverte pluridisciplinaire HAL, est destinée au dépôt

Chromogranins as molecular coordinators at the crossroads between hormone aggregation and secretory granule biogenesis

Ophélie Carmon, Fanny Laguerre, Lydie Jeandel, Youssef Anouar, Maite

Montero-Hadjadje

To cite this version:

Ophélie Carmon, Fanny Laguerre, Lydie Jeandel, Youssef Anouar, Maite Montero-Hadjadje. Chromogranins as molecular coordinators at the crossroads between hormone aggregation and secretory granule biogenesis. Chromogranins: from cell biology to physiology, Springer, pp.39-48, 2017, 978-3-319-58338-9. hal-02336001


Saber Malekmohammadi

Catherine Rosenberg

Razvan Stanica

On the Use of Wide Channels in WiFi Networks

Keywords: WiFi, Channel Allocation, Power Management

An increased density of access points is common today in WiFi deployments, and more and more parameters need to be configured in such networks. In this paper, we question current industrial guidelines for both residential and enterprise scenarios. More precisely, we investigate the joint channel, power, and carrier sense threshold allocation problem in IEEE 802.11ac networks, showing that the current practice, which is to use narrower channels at maximum power when the deployment is dense, yields much worse performance than a solution using the widest possible channel with a much lower power.

I. INTRODUCTION

Classically, deploying a WiFi network in the 2.4 GHz band consists in choosing the best 20 MHz channel for each access point (AP) from an interference point of view. However, this problem drastically changes with the evolution of the technology. Nowadays, the administrator of an IEEE 802.11ac wireless local area network (WLAN) has to select not only the channel width for each AP, but also the specific channel to use when several possibilities exist for a given width. The transmission power and the the carrier sense threshold (CST) of each AP can be modified, resulting is a complex joint channel, power and CST allocation problem.

In this paper, we explore whether it is better, in terms of overall performance for each AP, to use wide channels, in spite of the increased interference and collision region which appear in this case, or narrow channels in order to reduce interference and decrease the collision region. The widely accepted approach in the literature on WLAN configuration [START_REF]Cisco Meraki, High Density Wi-Fi Deployments Guide[END_REF][2] [START_REF]Very High-Density 802.11ac Networks Planning Guide[END_REF] is to allocate narrower channels and lower AP transmission power as the network density (in terms of APs) increases. Moreover, in residential WiFi networks, where the deployment and configuration of the APs is not coordinated, the default practice is to use narrow 20 MHz channels and the maximum authorised transmission power. In our work, we question these approaches and study the interplay between channel bandwidth, transmission power and the value of CST on the performance of the network.

Our contributions can be summarized as follows. We consider a WLAN operating on a single 160 MHz band, which can be divided in 8/4/2/1 non-overlapping 20/40/80/160 MHz channels. Through an extensive simulation campaign, we focus on a scenario with densely deployed APs, each serving a relatively low number of users with important downlink traffic. In these conditions, we first show that the current default settings for residential APs are underperforming. Indeed, our results show that the performance obtained on the single 160 MHz channel with low transmission power is significantly higher than the one observed when using the maximum transmission power, irrespective of the channel allocation. We also notice that, in such dense deployments, some APs see degraded performance, depending on the WLAN topology. Our tests show that, by using higher transmission power and CST for the poor APs, these APs can be brought up to level.

II. RELATED WORK

Channel allocation, power management and CST adjustment have been studied in centrally coordinated WiFi networks. In [START_REF] Hills | Large-Scale Wireless LAN Design[END_REF] and [START_REF] Rodrigues | On the Design and Capacity Planning of a Wireless Local Area Network[END_REF], channel assignment and AP placement problems were investigated, assuming the same fixed width for all the channels and fixed transmission power and CST for all APs. A channel allocation heuristic considering the traffic load at the MAC layer is formulated in [START_REF] Leung | Frequency Assignment for IEEE 802.11 Wireless Networks[END_REF]. However, the study assumes fixed channel bandwidth, transmission power and CST value. A Dynamic Sensitivity Control (DSC) algorithm is proposed in the IEEE 802.11ax amendment, and [START_REF] Afaqui | Evaluation of Dynamic Sensitivity Control Algorithm for IEEE 802.11ax[END_REF] shows that, by dynamically adjusting the CST threshold in a dense network, its aggregate throughput can be considerably improved. Again, it is assumed that the channels have a fixed bandwidth and transmission power values are fixed.

The most complete solutions in the area of WLAN management come from Cisco Systems: a centrally controlled power allocation algorithm (TPC) [START_REF]Cisco Systems, Transmit Power Control (TPC)[END_REF], and a dynamic channel assignment (DCA) algorithm [START_REF]Cisco Systems, Dynamic Channel Assignment (DCA)[END_REF]. In DCA, the central controller gathers information from all the APs in the network and calculates a goodness metric for each channel and for each AP, based on the received signal strength indicator (RSSI). DCA finds the AP with the lowest current goodness and tries to improve it, by assigning it a different, more suitable channel. This process is performed iteratively by the controller, but the general principle remains to assign narrower channels to APs in the network as the deployment density increases.

The TPC algorithm assigns (possibly different) transmission power values (P t ) to APs, aiming to maximize their coverage area and minimize their interference potential. Like DCA, TPC runs in a central controller, and its objective is to assign to each AP a transmission power T x ideal , computed as follows:

T x ideal (i) = T x max -(RSSI 3rd (i) -T P C threshold ), (1)
where T x max is the maximum power allowed by regulators, T P C threshold is a customizable power threshold, and RSSI 3rd (i) is the RSSI of the third largest interfering AP in the network, as measured by AP i.

DCA and TPC are executed by a central controller but independently from each other [9]. Other WiFi actors [START_REF]Cisco Meraki, High Density Wi-Fi Deployments Guide[END_REF] use a similar method for channel allocation and power management. In summary, most of the works in the literature study the effect of only one of the channel bandwidth, transmission power and CST parameters on WLAN performance, and, to the best of our knowledge, there is no prior work that studies the interplay of the three parameters together.

III. BENCHMARKS

Our main question in this work can be formulated as follows: given an already deployed WiFi network, what is the best combination of channel bandwidth (W j ), channel identity, total transmission power (P j ) and channel sensing threshold (CST j ) to allocate to AP j? We begin by discussing two state of the art channel allocation and power management schemes for enterprise WLANs and we take them as benchmarks.

1) Benchmark 1 -Using narrower channels as the deployment density increases: One common approach ([1], [START_REF]Cisco Systems, Dynamic Channel Assignment (DCA)[END_REF], [START_REF]Very High-Density 802.11ac Networks Planning Guide[END_REF]) is to assign narrower channels to APs in dense deployments. This way, APs get less bandwidth, but also less contention. Assuming that all APs use the same channel width, the same CST and their maximum power budget, the only two parameters need to determine are the value W of the channel width and the frequency reuse pattern. The works above show that, under these assumptions, it is better to use W = 20 MHz when the network is dense, while W can be larger for sparser networks. Note that none of these studies proposes a clear criteria to define a dense network in terms of AP. This approach is representative of a residential WLAN scenario, where an AP usually uses default parameters or a dynamic approach based on local measurements.

2) Benchmark 2 -Decoupled channel allocation and power control: Some studies [9] simplify the (W j , P j , CST j ) allocation problem by assuming the same fixed CST for all APs and by performing a dynamic channel assignment (e.g. DCA [START_REF]Cisco Systems, Dynamic Channel Assignment (DCA)[END_REF]) and a transmit power control (e.g. TPC [START_REF]Cisco Systems, Transmit Power Control (TPC)[END_REF]), separately from each other. The two algorithms are performed sequentially by a central controller connected to all APs, but it is obvious that changes in power allocation brought by TPC will yield changes in the results of DCA and vice-versa. While TPC allocates different powers to different APs, we can compute a lower bound on all the allocated powers by using the lowest possible value for T P C threshold (-80 dBm in [START_REF]Cisco Systems, Transmit Power Control (TPC)[END_REF]). In the following, we use as a second benchmark the combination of static DCA along with this lower bound provided by TPC. This approach is representative of enterprise and campus WiFi networks, where such a centralization is possible.

IV. NETWORK MODEL

We use the ns-3 simulator for studying realistic WiFi networks of different densities. For simplicity, we only consider downlink transmissions in this work. However, we note that this does not imply a zero uplink traffic, since the STAs transmit an acknowledgement message to the corresponding AP for every frame received correctly. We computed all simulation results with 99% confidence intervals but, since the obtained intervals are very tight, we do not show them in the figures, for better readability.

We consider WiFi networks consisting of 64 identical APs located on four floors of a building. On each floor, there are 16 APs, each of which is located in one square room, as shown in Figure 1. APs and users in each floor are mounted 1 m above that floor. The AP in each room is located randomly in a square of length 5 m, concentric with the room. The horizontal distance between two adjacent floors is set to 4 m. The distance between the centers of adjacent rooms (located on the same floor) is denoted as L. Therefore, the density of the system in terms of AP can be changed by changing the value of L, which allows us to easily switch between sparser (e.g. residential) and denser (e.g. enterprise) scenarios. There are 4 users located inside each room, associated to the AP in the same room (see Figure 1) and placed randomly at a distance up to L/2 (m) from the AP.

We assume a full buffer UDP traffic model with a uniform arrival rate equal to 230 Mbps; i.e., the APs always have data to transmit to their users. The overall performance metric that we use is the geometric mean (GM) of the throughput of all the users existing in the network. The higher the GM, the more efficient and fair is the network [START_REF] Fooladivanda | Joint Resource Allocation and User Association for Heterogeneous Wireless Cellular Networks[END_REF].

We use the ns-3 Log Distance Propagation Loss Model and add wall penetration loss effects on top of that by using the ns-3 Building Class. Between two adjacent rooms, there is a wall penetration loss equal to 8 dB. The APs antenna gain is equal to 12 dBi. The antenna gain on the user side is equal to 0 dBi and the number of spatial streams used by the APs is set to one. Whenever channel bonding is used by the IEEE 802.11ac APs, we assign them the same primary channel.

Regarding the transmission power, every country imposes a limit for the total RF power that can be radiated by antennas in the subband that is used. In this study, we consider a maximum power of 23 dBm (200 mW). Therefore, when using a channel with bandwidth 20, 40, 80 and 160 MHz, the maximum power that can be transmitted over each 20 MHz subchannel, denoted as P 20 t , is equal to 23, 20, 17 and 14 dBm, respectively. A minimum transmission power, denoted as P 20 t,min , can also be computed, considering that each AP needs to have a coverage range which is at least equal to the distance to the farthest corner of its room. Therefore, given a channel of a specific width allocated to an AP, the values between the computed P 20 t,min and the maximum P 20 t allowed by local regulations represent valid range of power values for a given AP.

V. UNIFORM AP SETTINGS

We first allocate the same channel bandwidth, the same P 20 t value and the same CST to all the APs in our simulated network, not unlike a residential deployment of APs with default settings. We consider the case where L = 40 m, which corresponds to a network of low AP density. In Figure 2, we show the results as a function of the total P t for different CSTs, while in Figure 3 we show the same results as a function of P 20

t , but for only the default CST of -82 dBm.

The first comment based on Figure 2 is that using the widest channel (160 MHz) provides the highest overall performance. The second comment is that the peak performance is obtained for a total transmit power of 15 dBm. As for the CST parameter, it only shows an impact when the APs use a high transmission power, mainly because, at low power values, the APs sense each other transmissions much less. Consequently, decreasing or increasing their CST does not affect their throughput much.

The best observed performancen in terms of GM is around 84 Mbps and is obtained for W = 160 MHz and for a relatively low power, i.e., P 20 t ≈ 6 dBm (which is in the valid range of powers). Note that the lower channel widths give much lower peaks. Benchmark 1 would use the maximum power P t and then would also select W = 160 MHz, resulting in a performance of only 35 Mbps. To compare with Benchmark 2, we show five vertical lines in Figure 3. The leftmost one shows the minimum power value necessary for coverage, i.e., P 20 t,min . The four vertical lines at the right of the figure show the lower bounds for the output values of the TPC algorithm for each channel width. The best possible values that could be selected by Benchmark 2 are W = 160 MHz and P 20 t = 11 dBm, yielding an overall performance of 53 Mbps, better than Benchmark 1 but still 58% lower than the peak we observe.

We then increase the density of the network to L = 15 m and we report the results in terms of P 20 t in Figure 4, when CST has the default value. As in the previous case, we show the five vertical lines depicting the performance of the two benchmark solutions. To conclude this part, in our scenarios, using a wide channel with low power is significantly more effective than using a narrow channel with any power, irrespective of the density of the network. This challenges the current default settings in residential WiFi deployment, which follow settings similar to Benchmark 1.

VI. NON-UNIFORM AP SETTINGS

While the results shown in the previous section demonstrate impressive gains in terms of GM throughput, in this section we further explore the distribution of performance among the APs. More precisely, when using the widest available channel (160 MHz), all the APs in the network share the same resources, suffering from co-channel interference and cochannel contention. So far, we tried to mitigate the interference and reduce the contention by adjusting power, while keeping the same CST for all APs, achieving a significant improvement in the network overall performance. Going a step further, we show in Figure 5 the GM of the throughput obtained on each of the 64 APs (four users associated to each AP), when L = 15 m. In these results, the APs are all allocated the widest channel (160 MHz), use the corresponding best power value (corresponding to the peak point in the previous section) and the default CST value. As it can be seen in Figure 5, some APs see much lower performance than others: APs 23, 26, 34, 36, 42 and 43, i.e., those located in the middle of the network. Specifically, we say that an AP is poor if it receives less than half of the arithmetic mean of the GMs of all APs (the dashed line in Figure 5 shows this value). The reasons why the APs located in the middle of the enterprise network are poor are twofold. The first one is that these APs have to contend with a lot of surrounding APs, while this is not the case for APs at the edges. Therefore, one suggestion for helping the poor APs is to increase their CST so that they can get access to the shared channel more easily. The second reason is the higher amount of co-channel interference that the users of these APs receive. Hence, one other suggestion for improving the performance of these APs is to increase their transmission power values slightly (the other APs keep the power value determined for the case with equal parameters) so that they can combat the higher interference that they experience.

To test these ideas, we consider two values for the CST of the poor APs: -79 dBm and -76 dBm (recall that the default value of the CST used so far was -82 dBm). We also increase their transmission power value by 3 dBm and 6 dBm with respect to rich APs, which are allocated their peak point power value. We consider all the possible combinations for changing the CST and power value of the poor APs and report these results in Table I. The table contains the overall GM throughput GM tot , the overall arithmetic mean throughput, AM tot , and the minimum of the GM throughput of each AP.

Table I shows that, although we only slightly increase the overall GM (about 6%), adjusting the CST and the TX power for the poor APs significantly increase the minimum GM throughput. Therefore, our results show that, in a centralized WLAN, unfairness among APs can be mitigated by fine tuning the CST and TX power of the APs with poor performance.

VII. CONCLUSION With the increased traffic demand, WLANs become denser and denser in terms of AP deployment. This raises new challenges in terms of resource allocation and several elements remain unaccounted for in the state of the art. In this context, we studied the channel allocation, power management and CST adjustment problems jointly in IEEE 802.11ac networks. We observed that, when allocating channels in a densely deployed WiFi network, using the maximum allowed transmission power by APs is not appropriate. This questions the current approach in residential WiFi networks, where default settings with high transmission powers are used. Our results also show that using homogeneous CST and power settings in a network yields unfairness among APs. We observe that modifying the CST of poor APs alleviates this unfairness problem and brings their performance up to level.

Fig. 1 .

 1 Fig. 1. One generic floor of the simulated network (four floors overall).

Fig. 2 .

 2 Fig. 2. The network overall performance vs. Pt for different channel widths and different CSTs, for L = 40 m.

Fig. 3 .

 3 Fig. 3. The network overall performance vs. P 20 t for different channel widths for the default CST, for L = 40 m.

Fig. 4 .

 4 Fig. 4. The network overall performance vs. P 20 t for different channel widths, for L = 15 m.

Fig. 5 .

 5 Fig. 5. The GM throughput of the 64 APs in the network for the power yielding the best overall GM and L = 15 m. The dashed line represents half of the arithmetic mean of the GM throughput of the APs.

TABLE I PERFORMANCE

 I UNDER NON-UNIFORM AP SETTINGS WHEN USING OPTIMAL SETTINGS. Cpoor+ (RESP. Ppoor ) IS THE INCREMENT OF CST (RESP. TX POWER) FOR POOR USERS.

	Cpoor+ Ppoor + GMtot AMtot GMmin
	none	none	22.8	29.51	3.92
	3	none	23.18	29.72	9.23
	6	none	24.24	30.05	9.47
	none	3	22.27	29.14	7.21
	3	3	21.04	28.58	7.49
	3	6	21.14	28.56	7.52


Marine Hoquante

Morgane Sanselme

Ivo B Rietveld

Gérard Coquerel

Disappearing conglomerates, assessment of the threat

HAL is

INTRODUCTION

Biological systems demonstrate predominantly one-handedness. For the pharmaceutical industry, it means that the biological properties in a pair of enantiomers can be very different [START_REF] Viedma | Chiral Symmetry Breaking During Crystallization: Complete Chiral Purity Induced by Nonlinear Autocatalysis and Recycling[END_REF][START_REF] Oketani | Practical Role of Racemization Rates in Deracemization Kinetics and Process Productivities[END_REF] .

Accordingly, chiral separation is an important field of research. There are several approaches to access pure chiral molecules, such as: stereoselective synthesis, chromatography, biocatalytic synthesis and crystallization. Among the latter, deracemization is a promising resolution method at an industrial scale, because it leads to quantitative recovery of the starting racemic substrate as a single enantiomer [START_REF] Viedma | Chiral Symmetry Breaking During Crystallization: Complete Chiral Purity Induced by Nonlinear Autocatalysis and Recycling[END_REF][START_REF] Oketani | Practical Role of Racemization Rates in Deracemization Kinetics and Process Productivities[END_REF][START_REF] Suwannasang | Use of Programmed Damped Temperature Cycles for the Deracemization of a Racemic Suspension of a Conglomerate Forming System[END_REF][START_REF] Rougeot | Ultrasound-Enhanced Deracemization: Toward the Existence of Agonist Effects in the Interpretation of Spontaneous Symmetry Breaking[END_REF] . It combines crystallization of a conglomerate forming system with simultaneous in situ racemization in the liquid phase [START_REF] Noorduin | From Ostwald Ripening to Single Chirality[END_REF][START_REF] Sögütoglu | Ripening: A Reliable Crystallisation Method to Reach Single Chirality[END_REF] . The main limitation of this process is that it only applies to conglomerate forming systems, i.e. 5-10 % of the racemic crystals [START_REF] Jacques | Racemates and Resolutions[END_REF][START_REF] Viedma | 22 -Crystallization of Chiral Molecules[END_REF] . Since racemization in the liquid is a requirement for the deracemization process, auxiliary racemizing agents are often needed [START_REF] Noorduin | Complete Chiral Symmetry Breaking of an Amino Acid Derivative Directed by Circularly Polarized Light[END_REF][START_REF] Engwerda | Attrition-Enhanced Deracemization of the Antimalaria Drug Mefloquine[END_REF][START_REF] Van Der Meijden | Attrition-Enhanced Deracemization in the Synthesis of Clopidogrel -A Practical Application of a New Discovery[END_REF] . In this work, we focused on conglomerates with enantiomers that are able to interconvert in solution under UV irradiation without any additional racemizing agent. A photoracemizable conglomerate forming system is required to prove the concept of photoderacemization.

1-1'-Binaphtyls [START_REF] Zhang | Photoracemization of Optically Active 1,1'-Binaphthyl Derivatives: Light-Initiated Conversion of Cholesteric to Compensated Nematic Liquid Crystals[END_REF] and particularly 1-1'-Bi-2-naphtol (BINOL) and some of its derivatives are known to photoracemize in solution [START_REF] Solntsev | Excited-State Proton Transfer in Chiral Environments: Photoracemization of BINOLs[END_REF][START_REF] Flegel | Photochemistry of 1,1′-Bi-2-Naphthol (BINOL) -ESIPT Is Responsible for Photoracemization and Photocyclization[END_REF] . In 2016 Maria et al [START_REF] Maria | Solid State Investigation of BINOL and BINOL Derivatives: A Contribution to Enantioselective Symmetry Breaking by Crystallization[END_REF] reported the crystal structure of 2'benzyloxy-1-1'-binaphtalene-2-ol (BINOL-OBn, Figure 1) and investigated the phase diagram of binary enantiomer mixtures. The system crystallizes in a P2 1 2 1 2 1 space group (refcode UBULUB, CCDC 1443741) meeting the crystallographic requirement to reach homochirality in a deracemization process. Previously, Takahashi et al. reported the melting points of the pure enantiomer and of the racemic mixture, respectively 122°C and 103°C implying the presence of a eutectic transition at the racemic composition [START_REF] Takahashi | An Expedient Route to Some Monoalkyl Ethers of Enantiomerically Pure Bi-β-Naphthol[END_REF] . Because of its conglomerate, which makes the system amenable for resolution by crystallization, BINOL-OBn seemed to be a good candidate for photoderacemization. BINOL-OBn is an atropisomer, which means that it exhibits axial chirality. The rotation of the bond connecting the two naphthyl groups is inhibited because of the steric hindrance caused by the two oxygen groups so that two conformers can be identified and isolated.

RESULTS AND DISCUSSION

Preparation and characterization

Pure (R) enantiomer, (S) enantiomer and racemic BINOL-OBn were synthesized according to a previously reported procedure [START_REF] Carrilho | New Binaphthyl-Based C3-Symmetric Chiral Hemilabile Monophosphite Ligands: Synthesis and Characterization of Their Platinum Complexes[END_REF] . The synthesis of racemic BINOL-OBn first gave rise to a viscous oil and the crystallization was not straightforward despite the absence of an apparent purity problem. However, after a month at room temperature, a stable racemic compound appeared. This new racemic compound was characterized by means of Differential Scanning Calorimetry (DSC), Infrared spectroscopy (IR), X-Ray powder diffraction (XRPD), Second Harmonic Generation (SHG) and single crystal X-ray diffraction (SCXRD). The XRPD patterns of the enantiopure and of the racemic crystals do clearly not overlap (Figure 2). Moreover, the infrared spectrum of the racemic compound (Figure 3) shows three bands for the OH stretching vibrations at ν OH = 3413 cm -1 , ν OH = 3482 cm -1 and ν OH = 3529 cm - 1 implying that there are three hydroxyl groups with different interactions in the structure, whereas for the enantiopure structure a single OH stretching band exists at ν OH = 3515 cm -1 . The differences between the spectra and diffraction patterns prove the existence of two different phases of which the (rac)-BINOL-OBn has never been observed before. The centrosymmetry of the crystals was investigated by Second Harmonic Generation (SHG). SHG signal only occurs for crystals with non-centrosymmetric space groups and is therefore a fast method to recognize such type of systems [START_REF] Bučar | Disappearing Polymorphs Revisited[END_REF] . While irradiating at 900 nm, no SHG signal at 450 nm was observed confirming the heterochiral packing of this new phase.

Constructing the binary phase diagram

Considering the appearance of the racemic compound, the previously reported phase diagram needs to be revised. Pure enantiomer, racemic compound and their mixtures were analyzed by DSC (heating rate = 5 K/min). The data are available in the supplementary information. Mixtures with different enantiomeric compositions were prepared by manual grinding assisted by a small amount of solvent and the results are summarized in Table 1 and2 The liquidus of the racemic compound was calculated using its melting point (115.6 ± 0.5 °C) and enthalpy (79 ± 5 J.g -1 ) assuming ideal behavior (Prigogine-Defay). The eutectic transition between the enantiomer and the racemic compound is obtained through the intersection of the two equations at 109 ± 1°C and it was experimentally found at 108.7 ± 0.5°C indicating no significant deviation from ideality. The metastable eutectic liquid between the pure enantiomers corresponds to the racemic composition (x metastable eutectic = 0.5) with a melting temperature experimentally found at 92 ± 0.5°C, equal within error to the temperature obtained by the intersection of the Schröder equation (T calculated eutectic = 93 ± 1°C) for the liquidus lines of the R and S enantiomers. The phase diagram for the BINOL-OBn system based on this fit and confirmed by experimental points, is shown in Figure 4. To test the stability of the conglomerate versus the racemic compound, 'artificial' conglomerate was prepared mixing powdered crystals of (R) and (S) enantiomers rigorously respecting the <1-1> composition. A cross-seeding experiment was conducted at -10°C in diethyl ether. After 48 hours, the diffractogram showed that the suspension was exclusively composed of the racemic compound indicating that it is the stable phase at -10°C (Figure 5). Therefore BINOL-OBn is not amenable to resolution by crystallization under the given conditions. Taking into consideration the cross-seeding experiment and the fact that the racemic compound has a melting point of 115.6 ± 0.5°C with a melting enthalpy of 79 ± 5 J/g, whereas the eutectic temperature of the conglomerate is found at 92°C with a melting enthalpy of 73 ± 5 J/g, the racemic compound is clearly the more stable phase at room temperature. It is even likely that this stability is 'monotropic' in nature i , considering that the melting enthalpy of the racemic compound appears to be 6 J/g higher than that of the conglomerate. Nonetheless, strictly speaking the enthalpies have overlapping experimental errors, so a stable temperature domain for the conglomerate below -10°C cannot entirely be excluded (through a eutectoid invariant).

It is surprising however, that the conglomerate has been observed in the first place and even by two groups independently [START_REF] Maria | Solid State Investigation of BINOL and BINOL Derivatives: A Contribution to Enantioselective Symmetry Breaking by Crystallization[END_REF][START_REF] Takahashi | An Expedient Route to Some Monoalkyl Ethers of Enantiomerically Pure Bi-β-Naphthol[END_REF] , because the newly identified racemic compound melts more than 20 degrees higher than the racemic conglomerate and is the more stable form at room temperature. The complexity of the structure of the racemic compound with three molecules possessing different conformations in the asymmetric unit (Z' = 3) might justify this behavior.

Structural characterization and discussion

Single crystals of (rac)-BINOL-OBn were obtained by crystallization from chloroform/nhexane and the crystalline structure was resolved by single crystal X-ray diffraction.

i strictly speaking this is not polymorphism since the following irreversible transformation (ΔG < 0): <S> + <R>  <RS> involves three solid phases. Nevertheless, by analogy with a genuine monotropy we use that term. As shown in the crystallographic data collected in Table 3, in the crystal lattice of the racemic compound, three independent molecules with different conformations can be found in the asymmetric unit (see Figure 6) as inferred from the infrared spectrum (Figure 3). When considering the energies of isolated molecules in vacuum, one of the conformations displays a higher energy than the others (about 1.3 times higher for so-called MolA represented in yellow in Figure 6). The other two conformations (MolO and MolB) form dimers together through hydrogen bonds (Figure 7). These dimers exhibit π -π interactions with adjacent dimers along the c axis giving rise to periodic bond chains (PBCs) in this direction (Figure 8). The interaction between adjacent PBCs along the b and a axes is ensured by MolA through π -π interactions.

The packing diagram is shown in Figure 9.

Whereas the pure enantiomer structure contains only π -π and Van der Waals interactions, the racemic compound exhibits hydrogen bonding too. The cost in energy required to put one third of the molecules in the conformation A and the global lower density seems balanced by the gain in stability created by hydrogen bonds. However, kinetically this new phase is not favored because of the lower probability for the molecules to occur in three different conformations during crystallization, which may in particular be difficult for conformation A. This could explain why a metastable conglomerate has been observed before the racemic compound. The belated crystallization of the racemic BINOL-OBn is reminiscent of the molecules discussed in the papers by Bernstein on disappearing polymorphs mostly involving molecules that can adopt different conformations (i.e. conformational freedom) [START_REF] Dunitz | Disappearing Polymorphs[END_REF][START_REF] Bučar | Disappearing Polymorphs Revisited[END_REF][START_REF] Cruz-Cabeza | Conformational Polymorphism[END_REF] . This case shows that the lack of hydrogen bonds in the conglomerate and the flexibility of the molecule are telltale signs of the possible crystallization of a racemic compound. It is possible to anticipate such a threat by challenging the stability of the conglomerate phase implementing the following procedure:

(i) Introduce as many variations of crystallization as possible with oscillatory physical stimuli (temperature cycling, spells of ultrasounds, rubbing effects on the inner wall of the container, different solvents and supersaturation, etc.). (ii) Use different routes for the synthesis of the racemic mixture. Indeed, different impurity profiles could change the nucleation and crystal growth rates drastically. (iii) Seed with racemic compounds of related chiral molecules. (iv)

Compute possible racemic compounds especially with common space groups (e.g. P2 1 /c; P-1;

C2/c) and with Z'>1 and different conformations and assess their energies versus the energy of the pure enantiomer. Above all, the time is essential here as a metastable form can be kinetically favored [START_REF] Mccrone | Polymorphism. In Physics and Chemistry of the Organic Solid State[END_REF] .

So far, the 'reappearing' conglomerate has not been observed. Considering its relative stability, it will not be easy to obtain it again, even if it has been demonstrated that it is possible to obtain and to work on a metastable phase once a more stable one has been observed 23 . To do so, the right experimental conditions must be found (e.g. a solvent or an impurity that inhibits the formation of nuclei of the stable form) [START_REF] Brandel | Impact of Molecular Flexibility on Double Polymorphism, Solid Solutions and Chiral Discrimination during Crystallization of Diprophylline Enantiomers[END_REF][START_REF] Davey | The Isolation of a Metastable Conglomerate Using a Combined Computational and Controlled Crystallization Approach[END_REF][START_REF] Harfouche | Enabling Direct Preferential Crystallization in a Stable Racemic Compound System[END_REF] .

CONCLUSION

This work demonstrates that pure racemic BINOL-OBn should not be a suitable system for deracemization because, contrary to earlier reports, a racemic compound appears to be more stable than the conglomerate. The melting point of the racemic compound exceeds by 20°C the eutectic temperature of the conglomerate. However, there are possibilities to find another conglomerate with an appropriate solvent or coformer 27 . As with any crystallization process, it is difficult to predict such behavior and chemists should be aware of the possibility that a more stable form may appear. The risk may be more pronounced if the crystal structure mainly depends on Van der Waals interactions, whereas H-bond donors and acceptors are available in the molecule in particular when combined with molecular flexibility. EXPERIMENTAL (R), (S), and racemic Binol were acquired from Alfa Aesar. The conversion of 1-1'binaphtalene-2-ol to 2'-benzyloxy-1-1'-binaphtalene-2-ol was carried out in a one-step synthesis according to a previously reported procedure [START_REF] Carrilho | New Binaphthyl-Based C3-Symmetric Chiral Hemilabile Monophosphite Ligands: Synthesis and Characterization of Their Platinum Complexes[END_REF] with a yield between 78 to 89 % (up to 5g). The Second Harmonic Generation (SHG) SHG analysis was carried out with an Insight X3 single laser with automated dispersion compensation (Spectra-Physics) and a TCS SP8 MP confocal microscope (Leica Microsystems) at 900 nm. The laser was controlled with the LASX Leica software.

InfraRed (IR)

Infrared spectra were obtained with an Alpha Platinum ATR spectrometer from Bruker. The program Opus was used for the data treatment.

Figure 1 .

 1 Figure 1. Molecular structure of the two enantiomers of 2'-benzyloxy-1-1'-binaphtalene-2-ol

Figure 2 .Figure 3 .

 23 Figure 2. X-Ray powder diffraction patterns of a) racemic and b) enantiopure BINOL-OBn

Figure 4 .

 4 Figure 4. Binary phase diagram of BINOL-OBn based on the experimental data in Tables 1 and

Figure 5 .

 5 Figure 5. Diffractograms of a) (R)-BINOL-OBn, b) (rac)-BINOL-OBn, c) cross-seeding

Figure 6 .Figure 7 .Figure 8 .Figure 9 .

 6789 Figure 6. The three independent molecules in (rac)-BINOL-OBn asymmetric unit (MolA in

1 H

 1 NMR data was in accordance with the literature. Details are shown in the supplementary material.Differential Scanning Calorimetry (DSC)DSC experiments were performed using a Netzsch DSC 214 Polyma apparatus on powder samples. Each DSC run was completed in aluminum pans with pierced lid and the atmosphere was regulated by a nitrogen flux (40mL/min). The Netzsch Thermal Analyses Proteus Software was used to treat the data. Onset temperatures were used as the melting points.X-Ray Powder Diffraction (XRPD)XRPD analyses were performed using a D8-Discover diffractometer from Bruker equipped with a θ/θ goniometer. The incident X-ray beam wavelength is monochromatic Cu Kα radiation (λ=1.5418 Å). The diffraction patterns were recorded between 3 to 30° by steps of 0.02° with 0.5 second per step and 20 rotations of the sample holder per minutes. The EVA software of Bruker was used for the treatment of the data.Single Crystal X-Ray Diffraction (SCXRD)The crystal structure was determined by single crystal diffraction on a SMART APEX diffractometer (with Mo Kα1 radiation: λ=0.71073Å). The cell parameters and the orientation matrix of the crystal were preliminary determined by using the SMART Software. Data integration and global cell refinement were performed with the SAINT Software. Intensities were corrected for Lorentz, polarization, decay and absorption effects (SAINT and SADABS Software) and reduced to FO 2 . The program package WinGX3 was used for space group determination, structure solution and refinement.

Table 1 :

 1 . Temperature of solidus and liquidus for binary mixtures between the pure enantiomer and the racemic compound of BINOL-OBn x (R)-BINOL-OBn T solidus /K T liquidus /K Enthaply of fusion /J.g -1

	1		393.8 a	77 ± 5 a
	0.919	380.2	391.2
	0.872	381.4	388.1
	0.791	381.6	386.0
	0.752	382.1	382.4
	0.656	382.1	386.9
	0.57	383.1	388.4
	0.5		388.8 a	79 ± 5 a
	a Results are the mean of n = 4 experiments. Temperatures are given with an uncertainty of ±
	0.5K.

Table 2 .

 2 Temperature of solidus and liquidus for metastable binary mixtures between the pure

	enantiomers (R)-and (S)-BINOL-OBn
	x (R)-BINOL-OBn T solidus /K T liquidus /K Enthaply of fusion /J.g -1
	1		393.8 a	76.9 ± 5 a
	0.183	364.6	385.7
	0.38	366.2	376.6
	0.5	365.1		72.7 ± 5

a Results are the mean of n = 4 experiments. Temperatures are given with an uncertainty of ± 0.5K.

Table 3 .

 3 Crystal data for (rac)-BINOL-OBn compared to the pure enantiomer

	(rac)-BINOL-OBn	(S)-BINOL-OBn 15

Funding Sources

This research is part of a PhD thesis funded by the French ministry of research.

Energy calculations

The energies of the isolated molecules in vacuum were calculated with the software Material Studio using the Dreiding forcefield and the Gasteiger method to take into account the charge distribution over the molecules.

ASSOCIATED CONTENT

Supplementary Information

The following files are available free of charge.

Synthesis of 2'-benzyloxy-1-1'-binaphtalene-2-ol racemic, R and S enantiomer, [START_REF] Viedma | Chiral Symmetry Breaking During Crystallization: Complete Chiral Purity Induced by Nonlinear Autocatalysis and Recycling[END_REF]