Ayan Kumar Bhowmick
email: ayankb@iitkgp.ac.in

Koushik Meneni

Maximilien Danisch
email: maximilien.danisch@lip6.fr

Jean-Loup Guillaume
email: jean-loup.guillaume@univ-lr.fr

Bivas Mitra

LouvainNE: Hierarchical Louvain Method for High Quality and Scalable Network Embedding *

Keywords: CCS CONCEPTS, Computing methodologies → Machine learning algorithms, • Mathematics of computing → Graph algorithms Network embedding, scalability, real-world graph algorithms

Network embedding, that aims to learn low-dimensional vector representation of nodes such that the network structure is preserved, has gained significant research attention in recent years. However, most state-of-the-art network embedding methods are computationally expensive and hence unsuitable for representing nodes in billion-scale networks. In this paper, we present LouvainNE, a hierarchical clustering approach to network embedding. Precisely, we employ Louvain, an extremely fast and accurate community detection method, to build a hierarchy of successively smaller subgraphs. We obtain representations of individual nodes in the original graph at different levels of the hierarchy, then we aggregate these representations to learn the final embedding vectors. Our theoretical analysis shows that our proposed algorithm has quasi-linear runtime and memory complexity. Our extensive experimental evaluation, carried out on multiple real-world networks of different scales, demonstrates both (i) the scalability of our proposed approach that can handle graphs containing tens of billions of edges, as well as (ii) its effectiveness in performing downstream network mining tasks such as network reconstruction and node classification.

INTRODUCTION

Representation learning on graphs, or network embedding [START_REF] Cui | A survey on network embedding[END_REF][START_REF] Goyal | Graph embedding techniques, applications, and performance: A survey[END_REF], involves a mapping of nodes in the graph to a low-dimensional vector space, such that the topological structure of the network is preserved. Such learned embedding vectors can be efficiently used as features for carrying out various network mining tasks. Most of the existing network embedding approaches employ random walks on graphs, matrix factorization techniques and deep learning architectures [START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF][START_REF] Qiu | Network embedding as matrix factorization: Unifying deepwalk, line, pte, and node2vec[END_REF][START_REF] Wang | Structural deep network embedding[END_REF] to represent nodes. However, these methods are computationally expensive for networks containing billions of edges.

Recently, few embedding methods [START_REF] Chen | Harp: Hierarchical representation learning for networks[END_REF][START_REF] Liang | MILE: A Multi-Level Framework for Scalable Graph Embedding[END_REF][START_REF] Ma | Hierarchical Taxonomy Aware Network Embedding[END_REF] that have proposed hierarchical approaches to learn node embeddings have been developed. For instance, Ma et al. [START_REF] Ma | Hierarchical Taxonomy Aware Network Embedding[END_REF] captures the latent hierarchical taxonomy denoting categories of different granularity in the learned vertex representations. HARP [START_REF] Chen | Harp: Hierarchical representation learning for networks[END_REF] and MILE [START_REF] Liang | MILE: A Multi-Level Framework for Scalable Graph Embedding[END_REF] repeatedly coarsens the original graph into a series of smaller graphs. Next, vertex representations of the coarsened graphs are learned using state-of-the-art embedding approaches, followed by a refinement step to obtain the final node embeddings of original graph. However, graph convolution network and gradient updates introduce a high computational overhead. Recently proposed RandNE [START_REF] Zhang | Billionscale Network Embedding with Iterative Random Projection[END_REF] is faster than other approaches on large-scale networks, however it compromises with the embedding quality unless its numerous parameters are accurately tuned. Another recent method is ProNE [START_REF] Zhang | ProNE: Fast and Scalable Network Representation Learning[END_REF] which learns high quality embeddings, but do not scale well for networks with billions edges.

In this paper, we leverage the notion of community structures present in real networks [START_REF] Fortunato | Community detection in graphs[END_REF] as an effective mechanism to compute node embeddings. The nodes present in a single community have similar types and are densely connected among themselves [START_REF] Mark | Finding and evaluating community structure in networks[END_REF]. Hence, placing those nodes closely in the embedding space may facilitate multiple graph mining tasks such as node clustering and node classification, as well as network reconstruction and link prediction. Partitioning the graph recursively into a series of coarsened subgraphs (communities) can help to capture similarity between nodes at different levels of proximity: the recursive partitioning essentially creates a hierarchy of communities in the network, where (a) the nodes present in the same community at the top level of the hierarchy indicates a cluster of similar nodes with higher-order structural relationships and (b) communities lower down in the hierarchy preserve the neighborhood relationship between connected pairs of nodes. Hence, generating the embedding vectors for a node from its presence in the communities at various levels of the hierarchy preserves the embedding quality and makes it suitable for various graph mining tasks. Side by side, one may use state-of-the-art fast community detection algorithms to make this process scalable for large-scale networks [START_REF] Günce Keziban Orman | Qualitative comparison of community detection algorithms[END_REF].

The major contribution of this paper is to develop a fast and scalable network embedding framework, LouvainNE, which is able to generate high quality embeddings for networks containing tens of billions of edges. First, we define the problem of scalable network embedding and explain the scope of community structures present in the network to efficiently learn node embedding vectors (Section 2). Leveraging these insights, we develop LouvainNE, a framework for node embedding based on hierarchical community detection. The development of LouvainNE involves three major steps: (a) A hierarchical clustering method is proposed to build a hierarchy of subgraphs. We use the Louvain algorithm [START_REF] Vincent D Blondel | Fast unfolding of communities in large networks[END_REF] to recursively coarsen a large graph into smaller communities and construct the hierarchy of subgraphs. (b) Next, we generate the level specific node embeddings for each subgraph in the hierarchy. We propose two different approaches to embed each subgraph (i) stochastic embedding (ii) standard embedding. (c) Finally, we combine the obtained embeddings at different levels of the hierarchy into the final embedding of individual nodes in the graph (Section 3). We introduce various real-world network datasets and describe the state-of-the-art network embedding methods used as baselines (Section 4). We perform extensive evaluation of the quality of the learned embedding vectors using LouvainNE on various downstream graph mining tasks such as network reconstruction and node classification. Our evaluation demonstrates the effectiveness of LouvainNE in learning high quality embedding vectors, which significantly outperform state-of-the-art methods, especially for large-scale datasets (Section 5). We explore the different variants of LouvainNE implementations as well as investigate the effect of tuning the model hyperparameters on the performance of Lou-vainNE (Section 6). Finally, we perform scalability evaluation on multiple real-world networks with up to 20 billions of edges which shows that LouvainNE scales linearly with the size of the network, while state-of-the-art algorithms fail to generate embeddings in a reasonable amount of time (up to 5 days). We present the related work in Section 8.

PROBLEM STATEMENT AND KEY IDEA

In this section, we first formulate the problem of scalable network embedding. Next, we provide the detailed intuition behind the proposed methodology.

Notations and Problem Definition

Let G = (V, E) denote an undirected graph where

V = {v 1 , v 2 , . . . , v N } is the set of N = |V | nodes and E is the set of M = |E |
undirected edges in G. In this paper, our objective is to find a mapping function f that learns a low-dimensional embedding vector y v of dimension d for every node v ∈ G where d << N is the predefined number of dimensions of the embedding. Mathematically, we learn the function f :

V → R d such that ∀v ∈ V, f (v) = y v .
The function f is learned in such a way that it preserves various network properties in the embedding space. For instance, (a) preserving the neighborhood proximity between connected node pairs (b) keeping the similar albeit non-adjacent nodes relatively close to each other (c) ensuring that dissimilar and non-adjacent nodes are placed far away (d) preserving higher-order structural relationships in the embedding space (e) representing the graph topology, are essential requirements of learning embedding vectors.

Apart from generating high quality node embedding vectors, our aim is to ensure that the developed algorithm is fast and highly scalable for large-scale networks. It should be able to learn embedding vectors in a network consisting of a few billion nodes and edges efficiently in a reasonable time, without compromising the quality of embedding for effectively performing various downstream network mining tasks.

Embedding and community: Key idea behind LouvainNE

In real-world networks, the distribution of links is inhomogeneous, with high concentrations of links within specific groups of nodes, and low concentrations between these groups [START_REF] Clauset | Finding community structure in very large networks[END_REF]. This feature of real networks results in the formation of community structure, where nodes inside the community are more densely connected, than with the rest of the network [START_REF] Yang | Defining and evaluating network communities based on ground-truth[END_REF]. Communities in a network are groups of nodes which probably share common properties and/or play similar roles within the network. We envision the notion of community as a useful tool to compute node embeddings. Indeed, it is intuitive to place nodes in the same community closely in the embedding space, compared to the nodes in different communities. The node embedding vectors learned from this community structure may facilitate to cluster the similar nodes in the graph, as well as infer the node types effectively. Moreover, dense connections within community will also help to preserve the higher-order structural relationships. Unfortunately, vanilla community structure does not capture the neighborhood property of the connected node pairs. However, if we construct a hierarchy of subgraphs by repeatedly identifying smaller sub-communities from the larger communities in the network, this will help to preserve the local proximity between node pairs in the network, since the neighbors of a node are more likely to be in the same sub-community lower down in the hierarchy. This will effectively preserve the neighborhood relationship between two nodes and will help to accurately reconstruct the original network. Finally, the presence of a wide variety of fast community detection algorithms in the literature [START_REF] Günce Keziban Orman | Qualitative comparison of community detection algorithms[END_REF] capable of detecting communities in massive-scale graphs in a very short time, paves the way of utilizing community structure for developing scalable node embedding algorithms. In this paper, we leverage these aforesaid observations to develop LouvainNE.

DEVELOPMENT OF LouvainNE

LouvainNE involves three broad steps for learning the embeddings in large-scale networks. First, we propose a fast method to create a hierarchy of partitions of the original graph. Next, we generate level specific node embeddings for each partition in the hierarchy. Finally, we compute embedding vectors of each individual node of the original graph by combining the embeddings of all the partitions that contain this node. The detail follows.

Hierarchy construction

We use a graph partitioning algorithm to compute a partition P of the set of nodes V of a graph G = (V, E), i.e., to compute a set of nonempty subsets of V such that every node v ∈ V belongs to exactly one of these subsets. Then, for each set of nodes S ∈ P, we build the subgraph G[S] induced in G by S and repeat the partitioning process on G[S] for every set S. This induced subgraph construction and partitioning is repeated recursively until the partitioning algorithm gives a trivial singleton partition that cannot be further decomposed. This happens for instance if the considered graph is a single node or a clique. In the end, when a single set of nodes is obtained, we create a partition of singletons {v} for each node v in that set. This procedure is general and any graph partitioning algorithm could be used. However a fast and reliable algorithm must be used to ensure the quality and scalability and we therefore use here the Louvain algorithm [START_REF] Vincent D Blondel | Fast unfolding of communities in large networks[END_REF]. This recursive procedure can be represented as a tree, which we depict in Figure 1 (left for a graphical representation of the nested partitions and middle for the corresponding tree). The constructed tree has the following properties: (i) each tree-node contains a subset of nodes of the input graph, (ii) the root node contains all nodes in the graph, (iii) the children of a given tree-node contain the sets of nodes corresponding to the partition of the subgraph induced by the set of nodes in the tree-node and (iv) each leaf contains a single node and there is a single leaf for every node of the original graph.

We detail this procedure in Algorithm 1. The procedure is recursive and calls itself in line 10. The input consists in the graph of interest and a partition function, called in line 3. We used the Louvain algorithm to partition a graph. The output consists in the hierarchical tree. if C contains a single set of nodes S then 5:

for each node v in S do 6:

output leaf {v}

Generating level specific embedding vectors

Once the hierachy is obtained, we compute an embedding vector for each one of the tree-nodes except the root node. We propose two embedding techniques.

(1) Standard embedding. Given any tree-node, including the root but except the leafs, we construct a weighted undirected meta-graph, where the nodes are the children of the considered tree-node and the weighted-edge between two children S 1 and S 2 is given by:

w S 1 S 2 = |E S 1 S 2 | |S 1 | • |S 2 |
.

Here E S 1 S 2 denotes the set of edges between the sets of nodes S 1 and S 2 in the original graph. The normalisation

1 |S 1 | • |S 2 |
allows to obtained more relevant weights when the sets S are of different sizes (two large sets of nodes may have a large number of edges between them compared to two small sets). We note that other weight functions can be used, we defer this study to future work. Once the meta-graphs are obtained, we apply a standard graph embedding algorithm (for instance, node2vec [START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF] or DeepWalk [START_REF] Perozzi | Deepwalk: Online learning of social representations[END_REF]) to obtain an embedding vector in dimension d of each one of these metagraphs independently.

(2) Stochastic embedding. For each tree-node, except the root, we simply generate a random vector of dimension d from the standard normal distribution Normal(0, 1) with zero mean and unit variance. If d is large enough, then any two random vectors will be roughly at the same distance (curse of dimensionality) which is a good property for what we do next. Interestingly, here, we somehow leverage the curse of dimensionality. In the case of Standard embedding, we note that the size of the meta-graphs are much smaller compared to the size of the original graph G, we can thus learn node embeddings of high quality in a very short time. We also note that a given edge of the original graph intervenes only in a single one of the obtained meta-graph (as a contribution to a weighted edge). This makes the overall time to compute all the embeddings no slower than the time that it would require to compute the embedding of the original graph with the chosen standard embedding method if the chosen standard embedding method has a linear running time. And it makes it much faster if it has running time slower than linear, say a quadratic running time.

In the case of Stochastic embedding, we observe that the embedding step does not directly depend on the structure of the input graph and it only depends on the tree. The input graph can thus be omitted as input for that step, which takes as input the tree only. This step is thus extremely fast as it only consists in generating a random vector for each tree-node.

Combining embeddings at different levels

Finally, we compute the embedding vectors y v for each node v ∈ G such that (i) the pairwise neighborhood relationship is preserved and (ii) similar nodes are close to one-another in the embedding space.

We assume that the leaf {v} (where v is the node of interest) of the tree is at depth h. We denote y t v for 1 ≤ t ≤ h the embedding vectors of the h tree-nodes that lie on the path from the root the the leaf {v} (excluding the root, but including the leaf).

We combine these h different embedding vector components in order to obtain the final embedding vector y v of node v. For that purpose, we introduce a parameter α (0 < α < 1) that regulates the weighting of the embedding vector components at different hierarchical levels t. We compute the final embedding vector as follows:

y v = h t =1 α t -1 y t v (1)
Equation 1 ensures that the contribution of the vector components y t v gradually diminishes with increase in t as we move down the hierarchy. Hence, two nodes which belong to the same community very low down in the hierarchy will be placed extremely close to each other in the embedding space, compared to two nodes

A E D Z M L J P O N G B F R Q C T I X Y K H S1 S12 S11 S112 S111 S2 S3 S32 S33 S31 S1 V S2 S3 S11 S12 S33 S32 S31 S112 S111 L M P J O G N E D A Z B F R Q C T I X Y K H S1 S11 S12 E D A Z S112 S111 L M P J O G N S2 B F R S3 S33 S32 S31 H X Y K Q C T I
Figure 1: From graph to hierarchical tree to embedding that are part of the same community at a top level of the hierarchy but belong to different communities lower down in the hierarchy. Nevertheless, in the second case too, the embedding vectors will be placed relatively close. This helps to preserve the neighborhood relationship between nodes in the embedding space, as well as place similar nodes (part of the same community) relatively close in the embedding space.

We illustrate this step of converting a hierarchy into an embedding in Figure 1 (middle and right). For illustration purposes, we set the number of dimensions to 2 and represent the non-leaf treenodes in shaded grey at the coordinates given by Equation 1, but where the sum is truncated at the depth of the tree-node.

EXPERIMENTAL SETUP

In this section, we describe the datasets and the state-of-the-art algorithms used for evaluating the proposed LouvainNE algorithm.

Datasets

We use datasets of different scales for our experiments, see Table 1 for detailed statistics. We consider Blogcatalog as the moderatescale and Youtube and Flickr as the large-scale datasets. We will consider larger graphs in Section 7.

(a) Blogcatalog (BC): This dataset contains the network of social relations [START_REF]Dataset : BlogCatalog3[END_REF] between bloggers on the Blog catalog website with labels reflecting their categories of interest (say technology).

(b) Youtube dataset: This is a social network [START_REF]IMC 2007 Data Sets[END_REF] consisting of all user-to-user links in the Youtube video sharing website. The labels denote groups that are subscribed to by users on Youtube such as gaming.

(c) Flickr dataset: This is a network dataset [START_REF]IMC 2007 Data Sets[END_REF] where nodes are images shared on Flickr. Edges are formed between images sharing common metadata in Flickr such as same location, submitted to same gallery, sharing common tags, images taken by friends, labels such as animal etc.

Dataset

Competing algorithms

We consider the following recent state-of-the-art methods for evaluating the performance of LouvainNE:

(a) HARP: This is a meta-strategy [START_REF] Chen | Harp: Hierarchical representation learning for networks[END_REF] for embedding graphs preserving higher-order structural features. It employs a hierarchical embedding approach where it first coarsens the graph using edge & star collapsing, followed by applying standard embedding to learn representations on the coarsest graph. Finally, it refines the embeddings from the coarsest to the finest graph. We used the implementation of the authors in [START_REF] Chen | Harp: Hierarchical representation learning for networks[END_REF] with the default parameters: the walk length= 10, number of walks= 40 and window size= 10; node2vec is used to embed the coarsest graph with the return parameter p and in-out parameter q, selected by grid search over the values {0.25, 0.5, 1, 2, 4}.

(b) MILE: [START_REF] Liang | MILE: A Multi-Level Framework for Scalable Graph Embedding[END_REF] developed a framework similar to [START_REF] Chen | Harp: Hierarchical representation learning for networks[END_REF] where it scales up the performance of a standard embedding method even for large graphs. The proposed methodology includes a hybrid matching strategy to coarsen the graph and a graph convolution network to compute the refined final embeddings of the original graph. We use the authors' implementation with default parameters: the number of coarsening levels= 2, learning rate= 0.001. We use DeepWalk as the base embedding method and GCN is used to refine embeddings with self-loop weight= 0.05.

(c) RandNE: [START_REF] Zhang | Billionscale Network Embedding with Iterative Random Projection[END_REF] proposed a simple random projection based network embedding approach using an iterative projection procedure, that helps to learn embeddings for billion-scale networks. We used the default parameter settings recomanded by the authors: with q = 1, weights=[1, 0.1] and using the adjacency matrix for network reconstruction. For classification, we have used the transition matrix with parameter values q = 3 and weights=[1, 10 2 , 10 4 , 10 5 ].

(d) NetSMF: [START_REF] Qiu | Netsmf: Large-scale network embedding as sparse matrix factorization[END_REF]relies on sparse matrix factorization and on spectral sparsification of a dense random-walk matrix polynomial. We have used the implementation provided by the authors with default parameter settings of context window size T = 10, b = 1 and number of samples M = 10 3 × T × m where m is the number of edges in the graph.

(e) ProNE: [START_REF] Zhang | ProNE: Fast and Scalable Network Representation Learning[END_REF] developed a fast and scalable network embedding framework, which relies on spectral propagation to enhance the quality of learned embeddings. We used the authors' implementation with default parameter settings where term number of the Chebyshev expansion k = 10, µ = 0.1 and θ = 0.5. Unless otherwise specified, we have used the embedding dimension d = 128 for all the methods including LouvainNE.

EMBEDDING QUALITY EVALUATION

In this section, we evaluate the quality of the learned node representations obtained from LouvainNE against the one obtained from the competing algorithms based on standard downstream tasks.

Network reconstruction

In this task, the learned embedding vectors are expected to well reconstruct the graph [START_REF] Ou | Asymmetric Transitivity Preserving Graph Embedding[END_REF]. A good network embedding algorithm should ensure that the embedding vectors can preserve the original network topology, such that they can be used to reconstruct the network. Precisely, we aim to predict the links in the original network by ranking the node pairs based on the similarity of their learned embedding vectors. The larger the similarity between embedding vectors of a node pair, the higher is the probability for the node pair to be connected by a link.

Evaluation procedure. We use the following two standard evaluation metrics for the task of network reconstruction:

(a) precision@K: This metric [START_REF] Wang | Structural deep network embedding[END_REF] is measured as the fraction of node pairs in the top-K ranked pairs that are connected by an edge. In order to rank the node pairs, we compute the Euclidean distance between embedding vectors of all node pairs and rank them in increasing order of Euclidean distance (smaller Euclidean distance denotes larger similarity). For computing precision@k for network reconstruction, we rely on the following two ways depending on graph size:

1. Classical evaluation: For a graph G of size N , we compute the Euclidean distance for all N 2 pairs in the graph and sort these pairs in increasing order of Euclidean distance. Then we compute precision@K for top-K pairs over the entire ranked list of N 2 pairs. We follow this method to compute precision@K only for the moderate-scale network of Blogcatalog.

2. Scalable evaluation: The classical evaluation becomes infeasible for the large-scale datasets of Youtube and Flickr since the total number of node pairs in these networks are of the order of 10 13 . Hence, we perform graph sampling to reduce the number of nodes and edges in the network and perform network reconstruction considering only the node pairs in the sampled graph. For graph sampling, we rely on the forestfire technique [START_REF] Hu | A survey and taxonomy of graph sampling[END_REF] to obtain a subgraph of the original graph. We fix the size of the sampled subgraph as the randomly chosen 1% nodes of the original graph.

We then rank the node pairs in this sampled subgraph based on the computed Euclidean distance in increasing order. We compute precision@K considering only the node pairs belonging to this subgraph. We repeat this process 100 times and report the average precision@K. We apply this evaluation technique on the large-scale networks of Youtube and Flickr.

(b) Area Under the Curve (AUC): This metric [START_REF] Fawcett | An introduction to ROC analysis[END_REF] measures the probability that a randomly selected adjacent pair of nodes (positive sample) is ranked higher than a randomly selected nonadjacent pair of nodes (negative sample), in terms of similarity between their respective embedding vectors. As the exact AUC is resource intensive to compute due to the size of the considered networks, we compute an estimation of it by sampling. We sample an adjacent pair of nodes and a non-adjacent pair of nodes; we check whether the sampled adjacent pair is more similar than the sampled non-adjacent pair. Here the similarity between a pair of nodes is measured in terms of Euclidean distance between the learned embedding vectors of the two nodes (the lower the Euclidean distance, higher is the similarity). We repeat this procedure a large number of times (100 times the number of edges in the corresponding graph) and report the estimated AUC.

Results. To show the effectiveness of our proposed LouvainNE method on network reconstruction task, we plot the precision@K over increasing values of K for various datasets in Figure 2. We observe that LouvainNE outperforms the state-of-the-art methods across all datasets in terms of the precision@K, though the method ProNE performs comparably to LouvainNE on Blogcatalog and Youtube datasets. Moreover, in Table 2, we show that the proposed LouvainNE achieves good AUC scores outperforming all state-of-the-art methods except ProNE. Importantly, LouvainNE has comparable performance to ProNE in terms of AUC for the largescale Youtube and Flickr datasets. This superior performance of LouvainNE for the network reconstruction task is due to the fact that it relies on the construction of hierarchy of subgraphs based on successively applying community detection on the original graph (Section 3.1). This ensures that the majority of a node's neighbors will continue to belong to the same community as the given node lower down in the hierarchy. Subsequently, the embedding and combining steps ensure that a node is placed very close to its neighbors in the embedding space. Hence, connected node pairs will have highly similar embedding vectors compared to disconnected pairs, thereby preserving the pairwise neighborhood relationships in embedding space.

Node classification

In this section, we evaluate the quality of embeddings generated by LouvainNE for node classification. Since a node can have one or multiple ground truth labels in all our used datasets, the resultant problem of classifying nodes is a multi-label classification problem.

Evaluation procedure. We apply the generated embedding vectors as features in a supervised learning framework to classify a node into the corresponding ground truth label(s). Specifically, we use the classifier chain technique [START_REF] Read | Classifier chains for multi-label classification[END_REF] of performing multi-label classification, that takes into account label correlations by constructing a chain of binary classifiers, equal to the total number of ground truth labels. We implement a Logistic Regression model to train this multi-label classifier. We randomly sample 80% of the vertices as the training set and evaluate the classifier performance on the remaining vertices that form the test set (20%). We predict the label(s) of a node in the test set and report the Micro-F1 and Macro-F1 scores averaged over 100 iterations.

Results. In Tables 3 and4, we observe that the proposed LouvainNE outperforms all state-of-the-art methods for the large-scale Youtube and Flickr datasets. However, for the moderate scale Blogcatalog dataset, NetSMF (Micro-F1) and HARP (Macro-F1) performs slightly better. This stems from the variations in the structural characteristics across the different networks, which impact the outcome of the multi-label node classification task. The superior performance of LouvainNE on the node classification task can be attributed to the graph embedding step of LouvainNE (Section 3.2) that assigns the same embedding vector to all nodes in a single community (reflecting nodes of similar labels) at a given hierarchy level. This ensures that the resultant embedding vectors of nodes belonging to the same community, obtained after the graph combining step (Section 3.3), will be placed very close to each other in the embedding space.

DRILLING DOWN LouvainNE

In this section, we explore the variants of LouvainNE implementations as well as investigate the impact of the model parameters on the performance of LouvainNE.

Embedding variants of LouvainNE

We compare the performance of variants of LouvainNE with different level specific embedding implementations in Section 3.2. We implement node2vec [START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF] and DeepWalk [START_REF] Perozzi | Deepwalk: Online learning of social representations[END_REF] as Standard embedding techniques as well as the Stochastic embedding and evaluate the performance on downstream tasks. First, we compare the AUC of network reconstruction across the model variants in Table 5. Next, we compare the performance of multi-label node classification in Table 6. We observe that all the variants of LouvainNE perform pretty uniformly across the tasks. Nevertheless, DeepWalk implementation of Standard embedding is the best performing model variant for the moderate scale Blogcatalog dataset, whereas Stochastic embedding is the best performing variant of LouvainNE for the large-scale Youtube dataset.

Parameter tuning

Here we investigate the impact of (i) varying the maximum number of levels (the maximum depth) h max in the hierarchy and (ii) varying the parameter α, while combining embeddings at different levels (Section 3.3), on the performance of LouvainNE. In that section, we only consider the stochastic variant of LouvainNE.

(i) Effect of varying maximum depth h max : We vary maximum depth h max from 1 to 9 (as 9 is the maximum number of levels encountered while building the hierarchy on Youtube and Blogcatalog). In that, we modify the stopping condition in the recursive Algorithm 1 by adding, line 4: "or if the depth is equal to h max ". We then generate the embedding vectors y t u using Stochastic embedding and perform the combining of embedding vector components. In Figure 3 (top-left) and Figure 3 (bottom-left), we compare the respective AUC for network reconstruction and Micro-F1 for multi-label node classification using the embedding vectors obtained by LouvainNE over different values of h max on Blogcatalog and Youtube. We observe that AUC is pretty low for h max = 1 for both datasets, since the top level communities (due to their size) fail to discriminate (in terms of learned embedding vectors) between the connected and disconnected node pairs. Similar observation holds true in case of Micro-F1 for node classification. As t increases, the performance improves gradually since the embedding vectors can better distinguish between neighbors and non-neighbors inside a community, as we go down in the hierarchy.

(b) Effect of varying α: We vary the value of α in the range [10 -5 , 1] and generate node embeddings for each value of α. We plot the AUC for network reconstruction using the embedding vectors obtained for different values of α for Blogcatalog and Youtube in Figure 3 (top-right). We observe that AUC is pretty high for low values of α and decreases as α increases. This reveals that lower values of α enable the learned embeddings to better preserve the neighborhood relationships. It is rather intuitive, as for very small values of α and given a node, the nodes closer to that node will be the ones sharing the same community at the lower hierachical level. We also plot the Micro-F1 for multi-label node classification as a function of α in Figure 3 (bottom-right). Here we observe that, contrarily to the problem of network reconstruction, the lower value of α is not the better for classification. We indeed observe a cap shape where medium values of α, say α = 0.01, perform much better than low or high values. This may be due to a trade-off between (i) the fact that low values of α are better as they put nodes sharing the same community at a low hierachical level relatively close to one-another and (ii) the smoothness of variation in the vectors required by the subsequent machine learning algorithm used to predict the label (logistic regression).

SCALABILITY EVALUATION

In this section, we push LouvainNE to its limits in order to demonstrate that it can deal with the largest publicly available real-world graphs. In order to evaluate our implementation of LouvainNE (with the Stochastic embedding variant), we gathered three real-world graphs with at least one billion edges (detailed statistics are presented in Table 7). We carried out these experiments on a Linux machine equipped with a processor Intel Xeon CPU E5-2660 @ 2.60 GHz and with 512 GB of RAM DDR4 2133 MHz. We implemented LouvainNE efficiently in C 1 .

We observed that none of the existing implementations of stateof-the-art methods (including ProNE that shows comparable performance to LouvainNE on downstream tasks) can compute an embedding of such large graphs using such a computer withing a reasonable amount of time. We thus re-implemented RandNE efficiently in C2 , in order to have a competitive baseline to compare against and evaluate the scalability performance of LouvainNE.

We present the results in Table 7 where we report the time to compute the hierarchy of LouvainNE, as well as the overall running time of LouvainNE and RandNE for the embedding dimensions d = 16, 128 and 512. We observe that when d is very small then RandNE is faster than LouvainNE, but for medium or large values of d LouvainNE is faster. For d = 128, LouvainNE is nearly 5 times faster than RandNE and for d = 512, LouvainNE is nearly 10 times faster. Complexity analysis: The hierarchy construction step by applying Louvain [START_REF] Vincent D Blondel | Fast unfolding of communities in large networks[END_REF] takes O(M • H ) time, where M is the number of links (Louvain has a linear running time) and H is the height of the hierarchical tree (note that we observe in practice that H is very small, say around 10 or less). We note that this first step does not depend on the number of dimensions d of the embedding. The generation of level specific embedding vectors of dimension d and combining embeddings take O(d • N ) time (where N is the number of nodes), leading to an overall time complexity of O(M • H + d • N ). The memory complexity is in O(M • H ) in the worst case (we store the input graph (in an adjacency array datastructure), as well as a subgraph for each level of the hierarchy).

The time complexity of RandNE is in

Θ(N • d 2 + M • d) [54]
. RandNE is thus significantly slow when d is large on large graphs (both terms N • d 2 and M • d are problematic). This is demonstrated experimentally in Table 7.

We also note that, the construction of the hierarchy takes a significant part of the overall running time of LouvainNE (say, 50% for d = 512). In case of competing hierarchical approaches such as HARP and MILE, either the graph coarsening step is complicated (for HARP) or the graph combining step based on GCN is slow (for MILE). This results in significantly increasing the overall running time, which makes these competing approaches unsuitable for embedding large-scale networks. Finally, we evaluate the running time of LouvainNE as a function of the number of edges M. In order to do so, we relied on a publicly available Twitter graph (users and follow links) crawled in 2012 [START_REF] Gabielkov | Studying Social Networks at Scale: Macroscopic Anatomy of the Twitter Social Graph[END_REF]. We ignore edge orientations since this is a directed graph. This graph has around 500 million users and the undirected version has 20 billion edges. We sampled subgraphs of different sizes in terms of number of edges and we measure the running time of LouvainNE for d = 128 on each of these subgraphs comparing it against RandNE. We present our results in Fig. 4 where we observe that the running time of LouvainNE and RandNE scales linearly with the number of edges, but that LouvainNE is about 5 times faster than RandNE.

We conclude that our implementation of LouvainNE is faster than our implementation of RandNE for medium values of d and much faster for large values of d. However, we note that RandNE (without the Gram-Schmidt process to obtain orthogonal projections) is embarrassingly parallel and offers a good degree of parallelism. LouvainNE can also be made parallel, but we defer the study of its parallelism to future work.

RELATED WORK

Earliest approaches on network embedding [START_REF] Belkin | Laplacian eigenmaps and spectral techniques for embedding and clustering[END_REF][START_REF] Sam | Nonlinear dimensionality reduction by locally linear embedding[END_REF][START_REF] Tenenbaum | A global geometric framework for nonlinear dimensionality reduction[END_REF] relied on various dimensionality reduction techniques for mapping nodes to a low-dimensional vector space from high-dimensional adjacency matrix such that adjacent nodes get placed close to each other in the embedding space. However, these methods, with time complexity in ≈ O(|V | 2 ), were only suitable to process small graphs.

Recently, most of the existing state-of-the-art network embedding approaches that are based on random walks [START_REF] Dong | metapath2vec: Scalable representation learning for heterogeneous networks[END_REF][START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF][START_REF] Mehdi Keikha | Community aware random walk for network embedding[END_REF][START_REF] Thomas | Semi-supervised classification with graph convolutional networks[END_REF][START_REF] Perozzi | Deepwalk: Online learning of social representations[END_REF][START_REF] Leonardo Fr Ribeiro | struc2vec: Learning node representations from structural identity[END_REF][START_REF] Zhou | Scalable graph embedding for asymmetric proximity[END_REF] are inspired by the word2vec model [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF] for learning representations of words. These methods rely on generating large number of walks for training the model, thus considerably slowing down the learning process. On the other hand, network embedding approaches employing matrix factorization techniques [START_REF] Cao | Grarep: Learning graph representations with global structural information[END_REF][START_REF] Donnat | Spectral graph wavelets for structural role similarity in networks[END_REF][START_REF] Huang | Label informed attributed network embedding[END_REF][START_REF] Ou | Asymmetric Transitivity Preserving Graph Embedding[END_REF][START_REF] Qiu | Network embedding as matrix factorization: Unifying deepwalk, line, pte, and node2vec[END_REF][START_REF] Wang | Community preserving network embedding[END_REF] are inherently slow since they learn node representations using a dense objective matrix, often with millions of rows and columns making them both memory and computationally infeasible. In addition, there also exist some embedding methods that learn node representations on graphs mainly capturing local neighborhood information using the knowledge of vertex-vertex connections [START_REF] Tang | Line: Large-scale information network embedding[END_REF][START_REF] Wang | Graphgan: Graph representation learning with generative adversarial nets[END_REF][START_REF] Wang | Paired restricted boltzmann machine for linked data[END_REF] or capturing the non-linearity of graphs by employing deep learning methods that needs to optimize huge number of parameters [START_REF] Cao | Deep Neural Networks for Learning Graph Representations[END_REF][START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF][START_REF] Hamilton | Inductive representation learning on large graphs[END_REF][START_REF] Vincent | Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion[END_REF][START_REF] Wang | Structural deep network embedding[END_REF][START_REF] Wang | Signed network embedding in social media[END_REF]. Besides these, algorithms have been developed for embedding nodes in heterogeneous networks [START_REF] Dong | metapath2vec: Scalable representation learning for heterogeneous networks[END_REF], signed networks [START_REF] Kumar Bhowmick | On the Network Embedding in Sparse Signed Networks[END_REF][START_REF] Kim | SIDE: Representation Learning in Signed Directed Networks[END_REF][START_REF] Wang | Attributed signed network embedding[END_REF][START_REF] Yuan | Sne: signed network embedding[END_REF] and attributed networks [START_REF] Wang | Attributed signed network embedding[END_REF][START_REF] Yang | Network representation learning with rich text information[END_REF]. However, all these methods mainly capture the local structural relationships as well as suffer from high computational complexity.

More recently, there have been few works on hierarchical representation learning [START_REF] Du | Galaxy Network Embedding: A Hierarchical Community Structure Preserving Approach[END_REF][START_REF] Ma | Hierarchical Taxonomy Aware Network Embedding[END_REF]. Probably, the closest approach to ours are HARP [START_REF] Chen | Harp: Hierarchical representation learning for networks[END_REF] and MILE [START_REF] Liang | MILE: A Multi-Level Framework for Scalable Graph Embedding[END_REF]. However, these approaches relies on coarsening the original graph by repeatedly aggregating nodes based on structural similarity followed by embedding the coarsest graph using a state-of-the-art method (Deepwalk, node2vec etc.). The embeddings of the coarsest graph are then successively refined to get the final embeddings of the original graph. Our approach differs in the way we construct the hierarchy of communities as well as the embedding and combining steps to obtain the final embeddings from this hierarchy.

Few works have relied on spectral graph sparsification techniques [START_REF] Qiu | Netsmf: Large-scale network embedding as sparse matrix factorization[END_REF][START_REF] Zhang | ProNE: Fast and Scalable Network Representation Learning[END_REF] and random projection [START_REF] Zhang | Billionscale Network Embedding with Iterative Random Projection[END_REF] for obtaining scalable network embedding. However, none of these approaches except (our re-implementation of) RandNE [START_REF] Zhang | Billionscale Network Embedding with Iterative Random Projection[END_REF] scales to networks with tens of billions of edges.

CONCLUSION

We leverage the notion of community structure to develop Lou-vainNE, a scalable graph embedding framework relying on three steps: (a) constructing a hierarchy of subgraphs (b) computing level specific embeddings for each subgraph in the hierarchy and (c) combining these level specific embeddings. We use the Louvain algorithm to recursively partition the graph and obtain the hierarchy.

We have shown that LouvainNE leads to high-quality embeddings, for downstream graph mining tasks, relatively to the stateof-the-art. We have shown that our implementation of LouvainNE is able to process graphs with tens of billions of edges, that its running time scales linearly with the number of edges and that it is much faster than its fastest competitor (our re-implementation of RandNE).

For future work, we would like to study the parallelism of Lou-vainNE, use partitioning algorithms other than Louvain, as well as

7 : else 8 : 9 :

 789 for each cluster S in C do output internal tree-node S 10: recpart(G[S])

Figure 2 :

 2 Figure 2: precision@K for network reconstruction

Figure 3 :

 3 Figure 3: Parameter tuning of LouvainNE: (top-left) AUC as a function of the maximum depth; (top-right) AUC for network reconstruction as a function of α; (bottom-left) Micro-F1 as a function of the maximum depth; (bottom-right) Micro-F1 for node classification as a function of α.

Figure 4 :

 4 Figure 4: Linear time complexity of LouvainNE and RandNE

Table 1 :

 1 Dataset statistics

		# Nodes # Edges # Labels
	Blogcatalog	10312	333983	39
	Youtube	1138499 2990443	47
	Flickr	1715255 15555042	195

Table 2 :

 2 AUC scores for network reconstruction

	Dataset	HARP RandNE MILE NetSMF ProNE LouvainNE
	Blogcat.	0.733	0.712	0.702	0.725	0.747	0.683
	Youtube	0.883	0.818	0.907	0.932	0.917	0.939
	Flickr	0.806	0.797	0.899	0.902	0.913	0.908
	Dataset	HARP RandNE MILE NetSMF ProNE LouvainNE
	Blogcat.	0.316	0.308	0.264	0.334	0.323	0.306
	Youtube	0.305	0.303	0.304	0.307	0.296	0.307
	Flickr	0.384	0.385	0.386	0.356	0.361	0.389

Table 3 :

 3 Micro-F1 scores for node classification (f) LouvainNE: Our proposed solution. Unless otherwise specified, we use the Stochastic embedding variant and set the value of the parameter α = 0.01.

Table 4 :

 4 Macro-F1 scores for node classification

	Dataset	HARP RandNE MILE NetSMF ProNE LouvainNE
	Blogcat.	0.190	0.182	0.150	0.179	0.160	0.167
	Youtube	0.153	0.189	0.155	0.190	0.192	0.189
	Flickr	0.251	0.249	0.251	0.248	0.250	0.254

Table 5 :

 5 AUC for network Reconstruction of the variants of LouvainNE: Stochastic, Node2vec and DeepWalk.

	Dataset		Stochastic Node2vec DeepWalk
	Blogcatalog	0.683	0.691	0.695
	Youtube		0.939	0.923	0.925
	Dataset		Stochastic	Node2vec	DeepWalk
	Blogcatalog 0.306 (0.167) 0.311 (0.162) 0.314 (0.164)
	Youtube	0.307 (0.189) 0.305 (0.186) 0.305 (0.187)

Table 6 :

 6 Performance for node classification of LouvainNE (Stochastic, Node2vec and DeepWalk) in terms of Micro-F1 (Macro-F1 in parentheses)

Table 7 :

 7 Running time for the hierarchy construction step of LouvainNE and overall running time of LouvainNE and RandNE

	d = 16	d = 128	d = 512

Publicly available implementation: https://github.com/maxdan94/LouvainNE

Publicly available implementation: https://github.com/maxdan94/RandNE. We do not perform the Gram-Schmidt process to obtain orthogonal projections as it takes important additional time and memory. We use q = 2 in the experiments.

investigate other level specific embedding methods and combining steps. Can we use the hierarchy directly as input to machine learning algorithms?

This research was partially supported by the DST -CNRS funded Indo -French collaborative project "Evolving Communities and Information Spreading" and French National Agency (ANR) under the JCJC project LiMass http://bit.ly/LiMass.


Dr Shrikant Kunjir

Dr Mireia Rodriguez-Zubiri

Dr Vincent Coeffard

François-Xavier Felpin

Patrick Giraudeau

Dr Jonathan Farjon
email: jonathan.farjon@univ-nantes.fr

Prof. F.-X Felpin

Prof P Giraudeau

Merging gradient-based methods to improve benchtop NMR spectroscopy: A new tool for flow reaction optimization

Emerging low cost, compact NMR spectrometers that can be connected in-line to a flow reactor are suited to study reaction mixtures. The main limitation of such spectrometers arises from their lower magnetic field inducing a reduced sensitivity and a weaker spectral resolution. For enhancing the spectral resolution, the merging of Pure-Shift methods recognized for line narrowing with solvent elimination schemes was implemented in the context of mixtures containing protonated solvents. One more step was achieved to further enhance the resolution power on compact systems, thanks to multiple elimination schemes prior to Pure-Shift pulse sequence elements. For the first time, we were able to remove up to 6 protonated solvent signals simultaneously by dividing their intensity by 500 to 1700 with a concomitant spectral resolution enhancement for signals of interest from 9 to 12 as compared to the standard 1D 1 H. Then, the potential of this new approach was shown on the flow synthesis of a complex benzoxanthenone structure.

Introduction

Better understanding chemical reactions is a long standing goal of synthetic chemists. Detecting and monitoring the molecular changes in a complex reaction mixture allows optimizing the reaction conditions for better conversions and selectivities. NMR is undoubtedly one of the most versatile techniques to identify and quantify components of reacting mixtures in real time.

High-field NMR is able to monitor real time reactions in a flow setting [1], but it is relatively difficult to implement due to the bulky environment and operating costs of high-field spectrometers.

Fortunately, a new generation of benchtop NMR spectrometers has emerged, which are compact [2], cryogen free, low cost, and can be easily coupled to a reactor. However, the online monitoring of chemical reactions remains difficult due to the inherent limitations of low field NMR: a low sensitivity and a weak spectral resolution. The performance of benchtop NMR spectrometers has been significantly improved by including gradient coils, making it possible to implement high-resolution pulse sequences of the kind typically used at high field to study complex reacting mixtures. Thus, 2D ultrafast methods were applied to reaction monitoring and authentication [3]. DOSY experiments were also reported and applied to the demixing of drugs formulations [4]. More recently, Pure-Shift (PS) methods [5] were implemented for the first time on a benchtop spectrometer, allowing significant line-narrowing in complex mixtures of small-to medium-sized molecules [6].

In addition to these advanced pulse-sequence developments, the study of reaction mixtures with possibly various mixed protonated solvents make it crucial to selectively suppress multiple and intense undesired peaks from solvents or other compounds in order to detect the smallest ones coming from analytes of interest. Towards this aim, gradient-based solvent elimination schemes have been compared on a benchtop spectrometer [7] and applied to reaction monitoring [8,9]. These developments have shown that benchtop NMR could be made more efficient by capitalizing on methodological developments made at high field. But while such proof-of-concept studies have shown the potential of gradient-based pulse sequences to improve the performance of benchtop spectroscopy, such potential could be pushed even further by combining tailored pulse sequence blocks to optimize the detection of signals of interest in a complex mixture. In this paper, we describe and evaluate the combination of several of the above-mentioned approaches for multiplying the resolutive power of compact instruments to better detect analyte 1 H signals in mixtures of protonated solvents. First, we assess the efficiency of merging several solventsuppression blocks like multiple WET180 [10] either in a successive or interleaved fashion (see Figure 1). WET180 will be combined to NOESY 1D [11] able to remove the faraway contribution for making n WET180NOE (see Figure 1). Moreover, multiple WATERGATE W5: m W5 [12] (see Figure 1) is also evaluated and these two multi-suppression families are compared in terms of signal resolution and sensitivity. Such multi-elimination schemes have never been implemented on a benchtop NMR instrument in the context of reaction mixtures, and can help removing numerous unwanted intense protonated signals like for those from solvents. Then, in order to further improve the resolution of relevant analyte signals, a novel approach is implemented, based on the combination of multiple solvent eliminations with PS techniques (Figure 1). Among 1 H homodecoupling sequences of interest, we chose to investigate the potential of TSE PSYCHE (Triple Spin Echo Pure Shift Yielded by Chirp Excitation) [13] recognized as the best compromise between sensitivity losses and resolution gain for small sized molecules [6]. Merging TSE PSYCHE with multi-elimination sequences forms a novel approach to push further the capabilities of each block taken individually. x,-x,x,-x,x,-x,x,-x; 2:x,x,x,x,x,y,y,y,y,y; 3:x,x,y,y,x,x,y,y4:x,x,x,x, x,x,x,x; acq:x,-x,-x,x,x,x,x,-x; The gradients Gi have a trapezoidal shape and are expressed in % of the maximum gradient coil amplitude (16 G.cm -1 ). Each non-encoding gradient depicted by a Gaussian shape has a 1 ms duration is followed by a 500 s stabilization delay. An additional gradient pulse (75%) was used during the recycling time to get rid of unwanted coherences. B A double WET180 scheme with n=2 either in interleaved (2 WET180int) or successive (2 WET180) manner is using four Gaussian pulses having 81.4°, 101.4°, 69.3° and 19° angles prior to the excitation [7] G1:G2:G3:G4=90:45:22.5:11.25, to remove solvent signals. C A multiple W5 scheme (m W5) [12] directly after the excitation can be used as an excitation sculpted way to remove 1 or 2 signal(s) with one W5 block. Successive W5 have been used and phase of the first half of each W5: x,x,-x,-x,x,x,-x,-x and -x,-x,x,x,-x,-x,x,x for the second half of W5. In this work Gm=1 to 4 have been used. For 4 W5, G1:G2:G3:G4=42:33:22:45. For 3 W5, G1:G2:G3=42:33:22. For 2 W5, G1:G2 =42:33. For 1 W5, G1=42. D An additional NOESY 1D block tunable with the mixing time tmix can be used to remove the fareway solvent signal in exchange with the one in the sensitive volume [11]. Single and double black rectangle are depicting 90° and 180° pulses respectively.

With this aim, two criteria were chosen to evaluate the performance of these multi-pulse sequences. Sensitivity is evaluated through Signal-to-Noise ratios (SNR), and resolution is evaluated by the linewidth at 5% height, which is more adapted to multiplets than at 50% height. An initial assessment of these methods is described on a model mixture with two protonated solvents. Then, the best ultra-resolved strategy is applied to the optimization of the flow synthesis of a benzoxanthenone, an analogue of natural product carpanone, as a case study.

Results and discussion

Evaluating multiple suppression techniques

In order to assess the different techniques for removing the different solvent signals, an indole derivative was used as a model and dissolved in dimethylformamide (DMF) and ethyl acetate, (EA) (see Experimental Part) usually used in synthetic chemistry and have different 1 S1). In this context, different solvent suppressing techniques were evaluated, first alone then followed by a TSE PSYCHE scheme to further enhance the spectral resolution (see Figure 1).

The following strategies were evaluated to remove the six signals coming from the protonated solvents: WATERGATE W5 was implemented with either 2, 3 or 4 blocks, [12] as well as two different 2 WET180NOE methods (Figure 1). For 2 WET180NOE schemes, two selective bands were considered: one for the H N of DMF and a second larger one for methyl groups of DMF and EA (see Experimental part).

We first optimized the 2 W5 pulse sequence (Figure 1), with one block focusing on DMF signals and the other one on EA signals. This scheme was not found optimal for removing the six protonated signals as expected (Figure S2). Then, 3 W5 was implemented by dedicating one W5 block for the two signals of DMF, the second for the singlet (CH3CO) and the quadruplet (CH2) of EA, and the third one was dedicated to the triplet (CH3) of EA. Under these conditions, the six signals were efficiently reduced at 1 -2 % of their average SNR as compared to 1D 1 H (see Figure 3.A and S3.B for details). Moreover, SNRs for the indole derivative remained almost unaffected with 98% of the 1D averaged SNR. The 4 W5 version with an additional W5 block for DMF -2 W5 blocks are used for removing it -gave close results as compared to 3 W5 in terms of solvent elimination performances despite smaller averaged SNR (see Figure 3.A and S3.A), certainly due to the longer elimination scheme during which T2 relaxation induces signal losses. In comparison, the 2 WET180NOE pulse sequence induced a much larger loss in sensitivity (80-85%) with slightly better SNRs for the shorter interleaved version which is less affected by T2 related sensitivity losses (see Figure 1 and Figure 2). Even if WET-based methods are expected to be more selective for solvent elimination due to their more tunable Gaussian pulses (see Experimental Section), the m W5 technique shows some sensitivity losses -from 2 to 19%-presumably due to T2 relaxation effects but also to the m W5 nulling area impacting nearby indole signals. Thus, a careful optimization of m W5 nulling delays is essential to prevent any signals of interest to be reduced or removed. The nulling width is dictated by inter pulse delays. For example for 2 W5 (see Fig. S2), the methyl 1 H signal of indole has 92% of the 1 H SNR (see Fig. S3.A). It is slightly impacted by the 2 W5 centered on the methyl signals of DMF. The difference in chemical shift between DMF and indole methyls is close to 0.7 ppm. Thus, an estimated nulling width of ± 0.7 ppm -60 Hz at 43 MHz -is a realistic value for inter pulse delays in the range of 4 ms (see table 2).

In terms of resolution, m W5 techniques efficiently reduce the linewidth at 5% height by an average factor from 4 to 5 compared to 1D without solvent suppression (see blue arrows in Figure S2 and4). These narrowing effects are due to the important reduction of DMF and EA signals whose tails were strongly overlapped with those of the indole derivative signals on the 1 H spectrum (see blue arrows in Figure S1). Moreover, 4 W5 provides cleaner in-phase lineshapes than 2 and 3 W5 (see Figure S2), probably due to the compensation of J-modulation effects over several W5 blocks. Overall, m W5 methods perform better than 2 WET-based methods that allow narrowing the peaks at 5% height by a factor of 3 as compared to 1D spectra (see Figure 4).

Combination of multisuppression with Pure-Shift

To go on improving the signal dispersion at low field, we merged multiple solvent elimination schemes with a TSE PSYCHE sequence (Figure 1), which was previously found as the best compromise for Pure-Shift spectroscopy at 43 MHz [6]. Pure-Shift methods are known to induce sensitivity losses, which have been well characterized [14]. Here, SNRs for indole were found better for 4 W5-TSE PSYCHE close to 3% on average of the 1 H 1D (see Figure 3.B and Figure S3.B). Under these conditions, DMF and EA signals were reduced in a range of 500 to 1700 as compared to 1 H (see Figure 3.B and S4.B). In comparison, 2 and 3 W5-TSE PSYCHE allowed reaching better solvent suppression from 580 to 3333 but with worse averaged SNRs close to 0.7% of the 1 H spectrum since aromatics signals were very low sensitive (see Figure 2 with high zoom factors and S4 for more details).

2 WET180NOE-based TSE PSYCHE led to worse SNRs for indole from 0.7% for the successive version to 1.6% of the 1 H for the interlaved version better for sensitivity as already shown without the TSE PSYCHE block (see Figure 3 A,B). 2 WET180NOE versions gave access to comparable elimination performances to 4 W5-TSE PSYCHE. Moreover, 2 WET180NOE versions showed some decoupling artefacts flanking the aromatics from the indole derivative (Figure 2). These artefacts come either from non-ideal decoupling leading to side bands and / or from small imperfections of chunk reconstruction from the different experiments of the pseudo 2D scheme (see Figure 1).

Regarding resolution based on the linewidth at 5% height, all Pure-Shift experiments allowed reducing the linewidth by 2 to 3 linewidth compared to 1D W5 leading to indole peaks 11 times narrower than on the 1D spectrum (Figure 4). Thus, the novel merging m W5 with TSE PSYCHE allowed multiplying resolution gain of each scheme taken separately. Moreover, in the perspective of monitoring applications, the repeatability of 4 W5-TSE PSYCHE was evaluated to 2% over integrals for indole after 5 replicates.

Sensitivity considerations

As described above, the utilization of multiple elimination schemes could impact differently the sensitivity for signals of interest. For low concentrated samples, m W5 pulse sequences with conventional 1 H detection are the best solutions for the sake of sensitivity with significant resolution gains from 4 to 5 but with some J-modulations (see Figure S2). In the case of concentrated samples where sensitivity is not an issue and where efficient line narrowing is needed, the best alternative remains m W5-TSE PSYCHE but with only 1 to 3% of the 1D sensitivity.

Application to the flow synthesis of benzoxanthenone 5

Benchtop NMR already showed its potential in the context of flow chemistry as an essential element of self-optimizing autonomous flow reactors [9,15,16]. Thus, to enrich essential NMR developments on compact apparatus, we were interested in showing the potential of multi-suppression methods combined with TSE PSYCHE for analyzing complex chemical mixtures composed on bigger-sized molecules at low field. Regarding solvent suppression, we focused on m W5 methods that gave optimum results on the model samples. Thus, the synthesis of benzoxanthenone 5 by a cobaltcatalyzed oxidative dimerization of the starting adduct 1, previously reported [17] through a flow process, was carried out (see Experimental Part). As detailed on Figure 5 the compact NMR apparatus equipped with a flow cell is receiving the reactional mixture from the flow reactor. The flow synthesis of 5 was previously optimized [15] but to show the potential of the new NMR experiments, 4 different experimental conditions with different yields were investigated (see Table 1). First, signal assignments were previously obtained for the starting compound 1 and the final compound 5 separately with the help of a 2D TOCSY map (see Figure S5 and for the derived assignment Table S1). Then, the performance of the pulse sequence elements introduced in this paper was evaluated. Ilustrative examples are shown in Figure 6, where comparisons are performed between TSE PSYCHE versus standard 1 H for reactant 1, and between 1 W5 TSE PSYCHE and standard 1 H for product 5. For compound 1, most of the signals are resolved in the 1 H spectrum: Ha, Hd and Hc. The effect of the homodecoupling can be seen at a glance on the signal of 1 where the Ha doublet is turned into a singlet in the Pure-Shift spectrum but with some artefacts coming from an imperfect decoupling (Figure 6.A). Moreover, signals He and Hf can be separated thanks to the Pure-Shift approach. On product 5, the dimer of 1, it was essential to reduce the huge CHCl3 signal, and 1 W5 was used prior to the TSE PSYCHE scheme (see Figure 6.B). With this pulse sequence, it became possible to better visualize methyl groups Hi and Hk thanks to JHH collapsing. Other low intense Hj, Hl, Hm and Hn signals could be detected and were better separated despite the loss of sensitivity due to the homodecoupling (see section 2.2 for this point). The aromatic area of product 5 was better resolved thanks to the combination of 1 W5 and TSE PSYCHE. In order to show the potential of such 1 W5-TSE PSYCHE pulse sequence it was further tested on the complete reaction mixture composed of 1, 2, 3, 4 and 5. S1)

Potential of m W5-TSE PSYCHE

1 W5-TSE-PSYCHE was found optimum to enhance the spectral resolution for this complex reacting mixture (see Figure 7). 1 W5 was prefered over 2 W5 for removing huge signals of tBuOOH and CHCl3 since significant sensitivity losses from 20 to 40% (data not shown) arose from T2 evolution during the double W5 echo. Moreover, 1 W5 was sufficient enough to remove the two unwanted intense signals. S1).

Monitoring several 1 H sites during a reaction is essential to visualize the structural changes in any moieties of the different molecules within the reaction mixture. Thus, the potential of 1 W5-TSE PSYCHE to detect different compounds in a reaction, was tested through 4 different reaction conditions (see Figure 7). The presence of only the starting product 1 without catalyst (0% yield), is confirmed with signals Ha, He and Hh (in red Figure 7). When the oxidizing agent tBuOOH is added in default (0.2 eq) a low concentrated amount of product 5 (27% yield) is visible (see signal Hu in blue in Figure 6) as well as signals from starting reactant 1. When the loading of tBuOOH is increased to 1.09 eq, yields of 70 and 91% at 5 and 25 min of residence time, respectively, were recorded (see Table 1 & Figure 5), blue assigned signals Hi, Hk, Hs, are better recognized for product 5 as well as the red assigned signal Ha that remains confirming the presence of the starting product 1 (Figure 6). 1 W5-TSE PSYCHE allows reducing the linewidth of 1 H signal by a range from 3 to 25 (see Figure S6 andS7).

Although it retains only 1 to 4% of the 1 H sensitivity, sensitivity remains sufficient to detect the signals of interest (see Figure S7).

Conclusion

We introduced a new toolbox of solvent-suppression Pure-Shift methods for the investigation of complex systems in reaction monitoring on compact low field instruments. An improved spectral resolution was reached thanks to the novel merging of gradient-based schemes by combining multiple W5 prior to the TSE PSYCHE. First, a model mixture composed on an indole compound with DMF and EA was investigated to show the potential of the new techniques. In this case, 4 W5-TSE PSYCHE sequence allowed the reduction by 500 to 1700 of unwanted signals and to enhance the resolution by 11 as compared to 1D 1 H while retaining between 2 to 3% of the 1D 1 H SNR. The merging of PS techniques with multiple elimination schemes multiply the performance of either PS or m W5 independently implemented.

The potential of this approach to better appreciate the composition of reaction mixtures for optimizing the conditions of flow synthesis was shown on a complex benzoxanthenone natural product analogue.

Here, the total time for each experiment was between 7 to 14 min due to the pseudo 2D mode of the TSE PSYCHE scheme. Efforts to reduce the overall time is under progress with fast repetition techniques [18] and Non Uniform Sampling or NUS [19] in order to further accelerate the NMR measurement time and to make it compatible with the monitoring of faster chemical events. Thus, combining resolution-enhanced schemes could open new horizons for emerging applications of benchtop NMR on concentrated samples such as edible oils or biolfuids like urine.

Experimental Section

Sample preparation

Model mixture A 500 mM methyl indole-4-acetate solution was prepared by dissolving 473 mg in 2.5 ml DMF and 2.5 ml of EA. A peristaltic pump was used to generate the continuous flow-1 mL.min -1for the studied samples.

Starting reactant 1 and product 5

For evaluating the performance of the new implemented techniques a sample of 0.35 M of 1 in CDCl3 and 0.18 M of 5 in CHCl3 were prepared.

Synthesis of benzoxanthenone 5

The experimental setup consisted in two streams as depicted in Figure 5 The first stream, equipped with a PEEK injection loop (1 mL), was loaded with a solution of methoxypropenylphenol 1 (0.35 M), cobalt catalyst 3 (4.5 x 10 - 3 M) and p-xylene 2 (0.35 M) as the internal standard in CHCl3.

The second stream contained a solution of tBuOOH 4 (1.09 or 0.2 eq) in CHCl3 which was continuously pumped. Both streams were pumped with two independent pumps at the required flow rate, and met in a stainless steel T-shaped piece (internal volume: 0.57 μL). The resulting mixture was introduced in a PEEK reactor coil (5 mL, 0.75 mm i.d.) heated at 32 °C. The reactor outlet was connected to the by-pass system consisting of an NMR flow-cell and PEEK tubing for a total volume of 2 mL. The flow rate was calibrated and optimized with sample solvents. All 2 WET and W5 1D spectra were recorded in 144 scans with a repetition time of 3 s for the indole in DMF system, leading to an overall experiment duration of 7 min and in 336 scans with a repetition time of 3 s for the indole in DMF/EA system leading to an overall experiment duration of 14min for sake of comparison with PS techniques.

In the TSE PSYCHE pulse sequence, a double saltire CHIRP pulse of 2 x 30 ms was used with a bandwidth of 2.5 kHz and a smoothing of 10%, applied simultaneously with a 0.8 G.cm -1 gradient. An angle of 25° was chosen as a compromise between sensitivity and decoupling sidebands intensity. Moreover two 5 ms CHIRP pulses with 2.5 kHz bandwidth and 10% smoothing were applied simultaneously with a gradient of 1.6 G.cm -1 to suppress zero quantum coherences.

Finally, solvent suppression schemes along with TSE PSYCHE experiments also known as broadband homonuclear decoupling were implemented as pseudo 2D experiments by repeating a 1D while incrementing the duration t1/2. The free Induction decays (FIDs) were divided into short periods called "chunks" (33ms was found as optimal) in order to prevent the J-coupling evolution. All the chunks were then stored and combined to reconstruct a hybrid 1D FID that underwent a Fourier transform.

For DMF/ EA indole all 2D PS matrices were acquired with 8192×16 points and 16 scans per experiment with 3 s for the repetition time, lasting 14 min. The experimental time is the same for all the PS experiments on a given sample, in order to compare their respective sensitivity. Conventional 1D 1 H experiments were also recorded with 8192 points within the same duration (336 scans) for the sake of comparison.

For benzoxanthenone 5 and starting reactant 1 alone or under reaction, all 2D PS matrices were acquired with 8192×16 points and 16 scans per experiment with 3 s for the repetition time, lasting 14 min. Quantitative 1D NMR experiment also recorded with 3 s repetition time and with 16 scans for calculating yields from para-xylene. T1 times have been measured with T1 max = 1s, a repetition time of 3s was chosen as a compromised between quantitative conditions (3 x T1 max ) and short acquisition times: 1 min NMR data processing: All the NMR experiments were recorded within the programming interface Spinsolve Expert. Signal processing were achieved with Mnova 12.0. It appeared essential to carefully adjust the apodization function for each technique studied here, in order to find the best compromise between resolution and sensitivity. For 1 H 1D standard NMR, Traficante function (Traf) and an exponential function both with a 1.0 Hz line broadening factor have been chosen. For all other techniques decribed above, the best compromise for Traficante function (Traf) and an exponential function is 0.3 Hz as a line broadening factor. Phase correction was done manually and automatically or via pre-recorded zero and first-order parameters. For the model mixture, spectra were referred with the solvent signal H N DMF at 8.00 ppm. For reaction mixtures,spectra were referred with the xylene methyls at 2.32 ppm. For all spectra, an additional baseline correction was achieved through a Whitthaker smoother function.

Entry for the Table of Contents

Figure 1 .

 1 Figure 1. Pulse sequences for solvent eliminated Pure-Shift 1 H experiments used in this work are composed on different blocks. A Triple echo based method (TSE PSYCHE) with G1:G2:G3:G4:G5:G6=57:10:21:5:33:10, 1:x,-x,x,-x,x,-x,x,-x; 2:x,x,x,x,x,y,y,y,y,y; 3:x,x,y,y,x,x,y,y4:x,x,x,x, x,x,x,x; acq:x,-x,-x,x,x,x,x,-x; The gradients Gi have a trapezoidal shape and are expressed in % of the maximum gradient coil amplitude (16 G.cm -1 ). Each non-encoding gradient depicted by a Gaussian shape has a 1 ms duration is followed by a 500 s stabilization delay. An additional gradient pulse (75%) was used during the recycling time to get rid of unwanted coherences. B A double WET180 scheme with n=2 either in interleaved (2 WET180int) or successive (2 WET180) manner is using four Gaussian pulses having 81.4°, 101.4°, 69.3° and 19° angles prior to the excitation[7] G1:G2:G3:G4=90:45:22.5:11.25, to remove solvent signals. C A multiple W5 scheme (m W5)[12] directly after the excitation can be used as an excitation sculpted way to remove 1 or 2 signal(s) with one W5 block. Successive W5 have been used and phase of the first half of each W5: x,x,-x,-x,x,x,-x,-x and -x,-x,x,x,-x,-x,x,x for the second half of W5. In this work Gm=1 to 4 have been used. For 4 W5, G1:G2:G3:G4=42:33:22:45. For 3 W5, G1:G2:G3=42:33:22. For 2 W5, G1:G2 =42:33. For 1 W5, G1=42. D An additional NOESY 1D block tunable with the mixing time tmix can be used to remove the fareway solvent signal in exchange with the one in the sensitive volume[11]. Single and double black rectangle are depicting 90° and 180° pulses respectively.

Figure 2 .

 2 Figure 2. Different solvent eliminating methods combined with a TSE PSYCHE pulse sequence applied on the indole derivative in DMF/EA. Pulse sequences names are mentioned (see Figure 1) as well as the vertical zoom factors used for each spectra. * is an impurity and  are artefacts. Blue arrows indicate the linewidths that were used to evaluate resolution enhancements at 5% height.

Figure 3 .

 3 Figure 3. Averaged SNR as compared to 1D 1 H for A) 1D solvent eliminated methods and B) combined with TSE PSYCHE version for the indole derivative, DMF and EA solvents.

Figure 4 .

 4 Figure 4. Averaged linewidth at 5% height obtained for different solvent elimination methods for indole derivative in DMF/EA as compared to 1 H 1D spectra.

Figure 5 .

 5 Figure 5. On-flow synthesis of benzoxanthenone 5, from the methoxypropenylphenol derivative 1 triggered by the tert-butylhydroperoxide 4 and catalyzed by cobalt catalyst 3 (condition 1 is mentioned see table1). p-xylene is added as an internal reference for determining yields. After filling tubings and the flow-cell (2 mL), NMR spectra were acquired under stopped-flow conditions.

Figure 6 .

 6 Figure 6. Comparison of standard 1 H (top) and TSE PSYCHE (bottom) for A) the starting reactant 1 in CDCl3 with residual (CH2)2Cl2 , and of standard 1 H (top) and 1 W5-TSE PSYCHE (bottom) B) the final product 5 in CHCl3. * show homodecoupling artefacts. See ESI for signal assignments (TableS1)

Figure 7 .

 7 Figure 7. Comparison of 1 W5-TSE PSYCHE spectra for the 4 investigated reaction conditions specified on the left. Assignment of the starting 1 (in red), the internal reference (in green) and final 5 product (in blue) is added for showing a better signal recognition. * show homodecoupling artefacts. See ESI for signal assignment (TableS1).

  Combining gradient-based techniques on a benchtop spectrometer efficiently removes up to 6 intense solvent signals and enhances more than x 10 the resolution of relevant signals thanks to the novel m W5 TSE PSYCHE experiment. This strategy is applied to the flow synthesis of a complex benzoxanthenone in mixture. S. Kunjir, M. Rodriguez-Zubiri, V. Coeffard, F.-X. Felpin, P. Giraudeau and J. Farjon* Page No. -Page No. Merging gradient-based methods to improve benchtop NMR spectroscopy: A new tool for flow reaction optimization

Table 1 .

 1 Reaction conditions for the synthesis of benzoxanthenone 5

	Reaction	Flow rate	Residence	Eq.	of	Catalyst	Yield a
	Conditions	(ml/min)	time	tBuOOH	(%)	(%)
			(min)
	1	0.2	25	1.09		1.3	91
	2	1.2	5	1.09		1.3	70
	3	0.2	25	0.2		1.3	27
	4	0.2	25	1.09		0	0
	[a] derived from quantitative 1D 1 H by using MeO signal (13)
	for 5 and Me signal (1) for p-xylene

m W5: Multiple excitation sculpting W5 echoes flanked by gradients have been also used (see Figure

1

). Nulling delays dm of each mW5 block have been finely tuned in order to periodically remove signals each 1/(2xdm) Hz. All optimized delays dm are summarized in table 2.

Table 2 .

 2 Nulling dm delays in s for multiple W5 schemes (m W5) implemented for removing solvent signals of the investigated mixtures.

	Indole in		H N	CH2	2xCH3	CH3CO	CH3
	DMF	m W5	(DMF)	(EA)	(DMF)	(EA)	(EA)
	/ EA
		2 W5	2225	5494	2225	5494
		3 W5	2225	5494	2225	5494	500
		4 W5 [a]	2225	5494	2225	5494	500
	Product
	5	m W5	CHCl3
		1 W5	2000
	Reaction
		m W5	tBuOOH	CHCl3
	mixtures
		1 W5	1914
			[a] The 2 first W5 of the 4 W5 scheme are
			dedicated to remove DMF signals

Acknowledgements

S. K. warmly acknowledges the French National Center for Scientific Research (CNRS) for a one year postoctoral fellowship in the field of the Emergence@INC2019 ARES-TATION project. S. K. also acknowledges Dr. Vasudevan Natarajan for assistance with the flow apparatus. J. F. warmly thanks his partner Sandrine Bouchet for an unfailing precious assistance.

Keywords

: benchtop NMR • multiple solvent suppression • Pure-Shift NMR • flow chemistry • resolution enhancement


domain: chim phys

text:
Diego Magaldi

Maria Ulfa

Sébastien Péralta

Fabrice Goubard

Thierry Pauporté

Thanh-Tuân Bui
email: tbui@cyu.fr

Keywords: Hole transporting materials, perovskite solar cell, carbazole, arylamine, oligomer

come

Introduction

Generally the active component of a perovskite solar cell (PSC), composed of an inorganic or hybrid perovskite material layer acting as the light absorber, is sandwiched between two charge transporting layers. The hole transporting layer (HTL) extract holes at the HTL/perovskite interface and transport them towards the cathode while the electron transporting layer (ETL) permits the electrons extraction and transportation. It has been clearly established that the incorporation of HTL and ETL in PSC's device structure is highly important for the devices to be efficient and stable. [1] In this context, accurate design of HTL is of major interest. The HTL is usually made from a single hole transporting material (HTM). However, in some case a mixture of different HTM can be employed. [2] Hole transporting capacity is key for achieving high photovoltaic efficiency. Usually, some additives and/or chemical dopants are incorporated to increase the charge transport capacity of the organic semiconductors. In addition, the EHOMO and ELUMO of the HTM must be compatible with the valence band maximum and conduction band minimum of the perovskite in order to provide the suitable driving force for charge transfer and block the electrons back transfer. Other researched characteristics include excellent thermal, morphological and photochemical stability, and good hydrophobic properties to protect the perovskite materials from moisture. The hole transporter can be either inorganic, organic or coordination compounds. [3,4] Organic HTMs are sub-classified into molecular and polymeric materials.

If molecular HTMs have well-defined molecular weight and structure and good synthetic reproducibility, the polymeric counterparts have important advantages in terms of processability, thermal and mechanical stability, and higher intrinsic hole mobility. [5] Among various organic HTM, carbazole-based HTMs is of particular interest thanks to their unique features suitable for developing photovoltaic materials. [6,7,[16][17][18][19][20][21][22][23][24][25][8][9][10][11][12][13][14][15] We have previously exploited carbazole platform to design HTMs for solidstate DSSCs [9] and PSCs. [6,11,26] Bulky dendritic carbazole compound, which take the best of both molecular and polymeric HTMs features, has been proved to be an effective design strategy, giving improved PSCs in terms of the stability. [6,7] Herein a new oligomeric cyclic carbazole based HTM (Scheme 1) is proposed which provide device performance comparable to that of Spiro-OMeTAD and also markedly improves the device stability.

Scheme 1. Molecular structures of investigated HTMs

The presence of the 2,4,6-trimethylaniline unit as hydrophobic and structuring agent is expected to favor the nano-structuration in HTM inducing a better conductivity in PSC device. This singularity has already been observed with PTAA as HTM in PSCs. [27] Moreover, methyl groups decrease the arrangement disorder between molecules due to lower repulsion of methyl groups and benzene rings, favoring also the carrier mobility within the HTM. [28] 2. Results and discussion The final step consisted in a Pd-catalysed Buchwald Hartwig amination reaction. The treatment of 2 with an equimolar equivalence of 2,4,6-triphenylaniline gave an inseparable mixture of oligomers. MALDI-TOF MS analysis suggests clearly the co-existence of tris, tetra and penta co-oligomers in the mixture. The theoretical DFT calculations have confirmed the presence of a mixture of different oligomers in the mixture. The obtained materials presented a good solubility in organic solvents commonly used for the deposition of HTM such as chlorobenzene, toluene, chloroform, etc. We then directly characterized this mixture for the purpose of application as HTM in PSCs.

Material Characterization

DFT Calculation

These oligomers (tris, tetra and penta cyclic compounds) were investigated by DFT. The optimized geometries and energies of the tetra-oligomer, the most abundant one, are given in the Figure 1 and the data for the other two compounds are provided in the ESI (Figures S1, S2 and Table S1). As can be observed for PCz1, their HOMO and LUMO were partially developed over the whole molecular structure. This is explained by the fact that the cyclic compounds have a distorted structure. All oligomers have nearly an identical optimized energy per repeated unit (Table S1, ESI). While the ELUMO levels are nearly identical for all the compounds, the tris oligomer has a deeper HOMO level compared to the tetra and penta oligomers.

Thermal, morphological and optoelectrochemical properties

PCz1 showed a good thermal stability with a Td > 340°C (Figure 2a) which is high enough for the applications in PSC. No melting point was observed until 280°C in differential scanning calorimetry (DSC) analysis (data not shown) suggesting that the material obtained is amorphous. However, the DSC curves revealed a complex behaviour due to the presence of different oligomers. The absorption properties of the PCz1 mixture recorded in dilute dichloromethane solution is shown in Figure 2b. It presents a strong absorption band centered at 338 nm and shoulders at lower energy attributed to an extended conjugated system including carbazole/tertiary amine ones. The simulated absorption spectra in CH2Cl2 of different oligomers are also displayed in Figure 2b with the maximum absorption wavelength are 331, 347 and 378 nm for tris, tetra and penta cyclic molecules, respectively. This is in good accordance with the assumption that PCz1 is a mixture composed mainly of these cyclic oligomers. PCz1 mainly absorbs in the UV spectral domain, which will not cause a competitive absorption effect with the perovskite materials. The optical bandgap (Eg = 2.74 eV) is calculated from the absorption onset wavelength (Figure 2b, 460 nm). This value is somewhat smaller than that of the Spiro-OMeTAD (3.00 eV). [9] Cyclic voltammetry (CV) was employed to investigate the redox behavior of PCz1 in dilute solution (Figure 2c). The cyclic voltammogram of PCz1 shows multiple oxidation processes. The first reversible oxidation could be attributed to the formation of radical cation of the carbazole moiety. PCz1 has an oxidation onset potential of -0.36 V versus Fc/Fc + , which is lower than that of Spiro-OMeTAD (-0.11 V/ Fc/Fc + ) measured under the same conditions. [9] These redox potentials were then employed to estimate their EHOMO and ELUMO in the solid state by using the ferrocene ionization energy as the standard.

EHOMO was estimated from the first oxidation potential while ELUMO was estimated from the EHOMO and Eg.

The EHOMO values of PCz1 and Spiro-OMeTAD are -4.74 eV and -4.99 eV, respectively, assuming that the ionization potential of Ferrocene is 5.1 eV versus vacuum. The corresponding ELUMO are -2.0 and -1.99 eV, respectively. The EHOMO level of PCz1 is thus higher than that of Spiro-OMeTAD meanwhile both compounds have very similar ELUMO levels. This suggests that, energetically, PCz1 could be suitable as an alternative HTM to Spiro-OMeTAD.

In order to improve the device stability against moisture in real operation, the HTM layer must protect the perovskite layer as far as possible. Hydrophilic HTM will be more susceptible to moisture ingress than the hydrophobic ones. Even if the permeability to liquid water/moisture can also depend on other factors such as thin film porosity, it remains that hydrophobicity should play a key role in moisture resistance for films of similar smoothness and compactness. We performed water contact angle measurements on the PCz1 layers studied in this work. LiTFSI and tert butylpyridine were incorporated into the film for mimicking the condition in PSCs. PCz1 film exhibited an average contact angle value of 93.7 ± 5.6° which correspond to rather high hydrophobic properties compared to Spiro-OMeTAD (70°) (Figure S3 in ESI). [29] It has been demonstrated that such hydrophobic properties are extremely beneficial to PSC stability. [30] A low surface roughness is an important requirement to decrease the current losses in PSC and get a better contact with the counter-electrode. For this purpose, the film forming property of doped PCz1 was examined by atomic force microscopy (Figure S4 in ESI). The PCz1 film showed a uniform and smooth surface with a mean surface roughness values of 0.97 nm. This low value should improve the contact with the perovskite layer and counter-electrode.

Photovoltaic application

PCz1 was then studied as HTMs using Cs0.08FA0.80MA0.12Pb(I0.88Br0.12)3 perovskite as the absorber material which has proved to be highly efficient for PSC application. [31,32] The device structure employed was: Glass-FTO/cTiO2/mpTiO2/perovskite/HTM/Au with cTiO2 a compact layer prepared by spray pyrolysis and mpTiO2 a mesoporous TiO2 layer prepared by spin coating. [33,[START_REF] Viana | Proc. SPIE[END_REF] As mentioned above, the EHOMO and ELUMO of PCz1 are close to that of Spiro-OMeTAD so PCz1 should be suitable and work as well as Spiro-OMeTAD. The best cell J-V curves are displayed in Figure 3a and the J-V curve parameters are gathered in Table 1. The initial performances of Spiro-OMeTAD cells were superior to those of the PCz1 cells. The former exhibited a better Voc and Jsc. Both cells had a rather large hysteresis between the reverse and the forward scan curves. This trend was observed for both the record cells and the averaged data (Table S2 in the ESI). Table 1 also report the J-V curve parameters after 7 weeks of aging (storage in the N2 filled glovebox). We remark that the efficiency of the Spiro-OMeTAD PSCs decreased during this period of time. On the other hand, the PCE of PCz1 cells increased significantly reaching a value of 18.04 % which is slightly superior to the initial PCE of the Spiro-OMeTAD PSC. Aging reduces the hysteresis in the case of the Spiro-OMeTAD device, while this parameter was slightly enlarged in the case of the PCz1 cell.

Nevertheless, some of us have shown recently that with the triple cation perovskite employed in this study, the systems have a slow response in time and that the steady-state PCE is close to the PCE determined in the reverse bias scan conditions. [31] Figure 3b compares the spectral response of the two kinds of cells. The external quantum efficiency (EQE) curves of fresh cells are very close in the visible and near infra-red regions. They reach high efficiency values, up to 85%. The spectra present an edge at about 780 nm due to the bandgap of the employed perovskite (1.60 eV). The main difference is found in the UV region with PCz1 cells having lower EQE(%) in the UV region which is related to the lower bandgap of PCz1 compared to Spiro-OMeTAD that could reduce the component due to the light reflection on the back electrode. Nevertheless, as the UV-component of the solar spectrum is very low, this feature is not detrimental for the performances and the photogenerated Jsc is close for these two devices.

Conclusion

In summary, novel carbazole-based oligomers were synthetized, characterized, and applied as hole transporting material in perovskite photovoltaic devices. The thermal, morphological, optical and electrochemical characterizations have shown that the PCz1 oligomer material is suitable for the role of HTM. Initial photovoltaic measurements gave a PCE of 16.84 % which increased over 6 weeks to reach a value of 18.04 %. This PCE is superior to the PCE of the benchmark Spiro-OMeTAD cell. Moreover, the former has been shown much more stable than the latter. Current study is being conducted to optimize the synthesis of this family of molecules. More compounds of this family will be prepared in order to better understand the molecular structure/photovoltaic performance relationship of this series of carbazole derivatives.

Materials and methods

The detailed descriptions of materials synthesis, characterization, computational DFT calculation, perovskite-based photovoltaic device fabrication and measurements are given in the Electronic Supporting Information.

2. 1

 1 Material synthesis PCz1 was synthetized in three steps as detailed in Scheme 2. Briefly, N-(4methoxyphenyl)carbazole (1) was produced by a copper-catalysed Ullmann amination reaction. 1 was then treated by N-bromosuccinimide giving 3,6-dibromo-N-(4-methoxyphenyl)carbazole (2).

Scheme 2 .

 2 Scheme 2. Synthesis of carbazole-based HTM PCz1

Figure 1 .

 1 Figure 1. Optimized geometry and HOMO LUMO distribution of the tetra-oligomer.

Figure 2 .

 2 Figure 2. (a) TGA curve of PCz1. (b) Absorbance of PCz1 in dichloromethane and calculated spectra for tris, tetra and penta cyclic molecules (CycloN, N=3, 4 and 5). (c) Cyclic voltammetry of PCz1 in dichloromethane.

Figure 3 .

 3 Figure 3. J-V and EQE curves of PSC devices (a) J-V curves of PCz1 and Spiro-OMeTAD cells, fresh and 7 weeks old; (b) EQE curves of PCz1 and Spiro-OMeTAD cells.

a 7

 7 weeks old cells. b Hysteresis index defined as: 𝐻𝐼(%) = 𝑃𝐶𝐸 𝑟𝑒𝑣 -𝑃𝐶𝐸 𝑓𝑜𝑟 𝑃𝐶𝐸 𝑟𝑒𝑣The aging of the devices and the stabilization of the perovskite layers and interfaces are important parameters for the development of this technology. We have followed the evolution of the J-V curve parameters and PCE of solar cells prepared with the two different HTMs and stored in a N2 filled glovebox. The results are shown in FigureS5in ESI. Clearly, the PCz1 cell presents a high stability with a PCE increase during the first weeks of aging. The improvement is mainly due to the enlargement of the Voc and, in a less extent, of the Jsc. On the other hand, the Spiro-OMeTAD PSCs present stable Voc and Jsc parameters while the FF decreases slightly. Overall, during the first 60 days, the PCE of the PCz1 cell increased by 3% while it decreased by 5.5% for the Spiro-OMeTAD cell. It can be explained by the rather large size and flexibility of the oligomer HTM compared to the molecular one. The perovskite is covered by a denser layer and then better protected against moisture and other degrading agents. We can also suppose that the alternation of mesitylene and anisole in the PCz1 groups allows the close packing of the molecules by the low repulsion derived of fewer methoxy groups on the molecule. The latter can be partially confirmed by the low roughness values of the deposited layer. This is in good agreement with higher hydrophobic properties of the PCz1 film compared to the Spiro-OMeTAD one. The enlargement of the Voc also suggests an improvement of the perovskite/PCz1 interface during the weeks following the cell preparation. Long-term stability and increment of the final PCE observed over time of 60 days is mainly due to the increase of final the voltage output from 0.98 V to 1.04 V. In parallel a small reduction of the current output going from 22.88 mA.cm -2 to 22.41 mA.cm -2 is observed. Accordingly, the final PCE went from 16.84 % to 18.04 % for the forward scan. It represents an increment of 7 % from the first PCE measurement. Many factors can contribute to the observed phenomena like the solid-state diffusion of the different components of HTL towards its self-stabilization.[START_REF] Lamberti | [END_REF] It is evident that the stabilization of the PCz1 cell signifies an amelioration of the interphase contact derived of the HTL nature.

Table 1 .

 1 Photovoltaic J-V parameters and PCE of best PSC cells prepared with PCz1 and Spiro-OMeTAD.

		HTM	Scan direction	Voc (V)	Jsc (mA.cm -2 )	FF (%)	PCE (%)	HI (%)
	Fresh cells	Spiro-OMeTAD	Reverse Forward	0.99 0.91	22.25 22.19	81.20 65.06	17.94 13.12	26.9
		PCz1	Reverse	0.95	21.88	80.87	16.84	30.1

Acknowledgements M.U. and D. M. acknowledges the Indonesian and Mexican governments for doctoral scholarships, respectively.

Conflicts of interest: the authors declare that there is no conflict of interest