Lamine Diop
email: diop.lamine3@ugb.edu.sn

Cheikh Talibouya Diop
email: cheikh-talibouya.diop@ugb.edu.sn

Arnaud Giacometti
email: arnaud.giacometti@univ-tours.fr

Arnaud Soulet
email: arnaud.soulet@univ-tours.fr

Pattern Sampling in Distributed Databases

Many applications rely on distributed databases. However, only few discovery methods exist to extract patterns without centralizing the data. In fact, this centralization is often less expensive than the communication of extracted patterns from the different nodes. To circumvent this difficulty, this paper revisits the problem of pattern mining in distributed databases by benefiting from pattern sampling. Specifically, we propose the algorithm DDSampling that randomly draws a pattern from a distributed database with a probability proportional to its interest. We demonstrate the soundness of DDSampling and analyze its time complexity. Finally, experiments on benchmark datasets highlight its low communication cost and its robustness. We also illustrate its interest on real-world data from the Semantic Web for detecting outlier entities in DBpedia and Wikidata.

Introduction

Many applications require storage and manipulation of distributed databases [START_REF] Özsu | Principles of distributed database systems[END_REF] like large-scale wireless sensor networks [START_REF] Shen | A distributed spatial-temporal similarity data storage scheme in wireless sensor networks[END_REF] or the Semantic Web [START_REF] Berners-Lee | The semantic web[END_REF]. In most cases, the centralization of data is very costly, in particular when the databases evolve continuously. Sometimes legal constraints also prevent this centralization [START_REF] Domadiya | Privacy preserving distributed association rule mining approach on vertically partitioned healthcare data[END_REF]. Thus, [START_REF] Zhang | Mining multiple data sources: local pattern analysis[END_REF] underlines the importance of extending knowledge discovery to distributed databases. In the context of the Semantic Web, Table 1 illustrates an example of Resource Description Framework (RDF) data distributed over four triplestores P = {D 1 , D 2 , D 3 , D 4 } accessible via SPARQL queries. In this context, the properties describing the entity identified by TId 1 (e.g., the singer "Youssou N'Dour") are spread over several fragments (i.e., DBpedia D 1 with the property A, and Wikidata D 2 with the properties B and C). There exist federated systems to execute SPARQL queries on multiple triplestores [START_REF] Gombos | Federated query evaluation supported by sparql recommendation[END_REF]. Unfortunately, SPARQL is not expressive enough to directly extract patterns like frequent itemsets. By relying on a basic communication model, this paper aims at extracting patterns from a distributed database P (including RDF data) as if the data were centralized (see P * in Table 1).

Few works in the literature [START_REF] Cheung | Efficient mining of association rules in distributed databases[END_REF][START_REF] Otey | Mining frequent itemsets in distributed and dynamic databases[END_REF][START_REF] Jin | Systematic approach for optimizing complex mining tasks on multiple databases[END_REF][START_REF] Kum | Sequential pattern mining in multi-databases via multiple alignment[END_REF] are dedicated to pattern mining in distributed databases. Unfortunately, they suffer from three major limitations. First, they exclusively address horizontally partitioned data (i.e, unlike the example P in Table 1, a transaction cannot be split into two fragments). Second, Table 1: Example of a distributed database P = {D 1 , D 2 , D 3 , D 4 } as these proposals focused on an exhaustive extraction of patterns, the volume of data exchanged between the different fragments can be very high. Finally, these proposals need computing capacity on each fragment for locally mining patterns, which is often impossible (for example, in the case of the SPARQL endpoints of the Semantic Web). To overcome these limitations, we propose to benefit from pattern sampling [START_REF] Hasan | Output space sampling for graph patterns[END_REF][START_REF] Boley | Direct local pattern sampling by efficient two-step random procedures[END_REF]. Pattern sampling consists in randomly drawing a collection of patterns with a probability proportional to their interest. This technique has a low computational cost, but it is also useful in many tasks such as classification [START_REF] Boley | Direct local pattern sampling by efficient two-step random procedures[END_REF], outlier detection [START_REF] Giacometti | Anytime algorithm for frequent pattern outlier detection[END_REF] or interactive data mining [START_REF] Dzyuba | Learning what matters-sampling interesting patterns[END_REF].

In this paper, we show how to sample patterns from a distributed database that can be partitioned both horizontally and vertically, without using the computing capacity of the different fragments. Our main contributions are as follows:

-We propose a generic algorithm called Distributed Database Sampling (DDSampling) which randomly draws a pattern from a distributed database proportionally to an interestingness measure combining frequency and length-based utility functions (including length constraints). -We demonstrate that DDSampling performs an exact sampling and analyze its complexity on average. Experiments show that DDSampling is very fast and that the communication cost of our proposal is much lower than that of data centralization for drawing a few thousand patterns. We also show that in the context of distributed databases, DDSampling is a fault-resistant algorithm against network and node failures. -We illustrate the interest of DDSampling on a use case by detecting outliers in two real-world triplestores: DBpedia and Wikidata. These experiments show the importance of using a maximum length constraint and that output space sampling is more efficient than input space sampling.

This paper is structured as follows. After the related work in Section 2, Section 3 introduces basic definitions and formalizes the problem of pattern sampling in distributed databases. We detail the algorithm DDSampling in Section 4 that exactly draws patterns as if the distributed database was centralized. Section 5 evaluates its performance on benchmarks datasets and illustrates the interest of pattern sampling to detect outliers in the Semantic Web.

Several approaches in the literature focused on frequent pattern mining in distributed databases. This task is complex because whatever the minimum frequency threshold user-specified on the distributed database (global frequency), it is not possible to constrain the local frequency on each fragment without communicating information between sites. In this context, [START_REF] Cheung | Efficient mining of association rules in distributed databases[END_REF] proposes the first method to extract all the globally frequent patterns by identifying the sites where the patterns are the most frequent and thus, reducing communication costs. More drastically, [START_REF] Otey | Mining frequent itemsets in distributed and dynamic databases[END_REF] proposes to save communication costs by limiting themselves to the collection of the maximal frequent patterns. To prevent each fragment from enumerating all its patterns, [START_REF] Jin | Systematic approach for optimizing complex mining tasks on multiple databases[END_REF] imposes a minimum local frequency threshold on each fragment. From the different local extractions, [START_REF] Kum | Sequential pattern mining in multi-databases via multiple alignment[END_REF] builds an approximate global collection of frequent patterns. A centralized pruning proposed by [START_REF] Zhu | Discovering relational patterns across multiple databases[END_REF] is based on the construction of a tree containing for each pattern all its occurrences (i.e., fragment/transaction pairs), which still requires a considerable volume of communications. More recently, [START_REF] Zhu | CLAP: Collaborative pattern mining for distributed information systems[END_REF] implements a decentralized pruning technique within the extraction on each fragment by exchanging Bloom filters. This approach significantly reduces computation time but the cost of communications remains too large. Indeed, for low support threshold, the volume of extracted patterns invariably generates an enormous communication cost much higher than that of data centralization. In addition, all these frequent pattern mining approaches are limited to horizontal partitioning of data, i.e. the same transaction cannot be distributed on two separate fragments. Finally, all the existing proposals require a computation capacity on each fragment, which is not always possible. For instance, the Semantic Web provides access to distributed data via SPARQL endpoints, but it is not possible to execute a pattern mining routine on these endpoints. For all these reasons, in this paper, we revisit the discovery of patterns in distributed databases in the light of pattern sampling. We will see that our approach does not require computation capacity on the fragments and it reduces the communication costs because all the patterns are not extracted.

Output space sampling methods [START_REF] Hasan | Output space sampling for graph patterns[END_REF][START_REF] Boley | Direct local pattern sampling by efficient two-step random procedures[END_REF] aim at drawing patterns with a probability distribution proportional to their interest. Most sampling techniques fall into two broad categories: stochastic methods [START_REF] Hasan | Output space sampling for graph patterns[END_REF] and multi-step methods [START_REF] Boley | Direct local pattern sampling by efficient two-step random procedures[END_REF]. In order to randomly walk from a pattern X to another, stochastic methods require to consider the global interest of all the neighboring patterns of X. For example, in the case of frequency, it would be necessary to know the global frequency of all the subsets and supersets of X, which would generate many communications. For this reason, we prefer to adopt a multi-step random method. This type of method has already been used for several interestingness measures (e.g., support or area [START_REF] Boley | Direct local pattern sampling by efficient two-step random procedures[END_REF] or exceptional measure [START_REF] Moens | Instant exceptional model mining using weighted controlled pattern sampling[END_REF]) and several data types like sequential data [START_REF] Diop | Sequential pattern sampling with norm constraints[END_REF]. Nevertheless, the context of distributed databases is an orthogonal challenge. In particular, we determine minimal information that the fragments must communicate to make an unbiased draw of patterns.

3 Problem Formulation 3.1 Pattern language and distributed databases Given a set I of distinct literals called items, an itemset (or pattern) is a subset of I. The language of itemsets corresponds to L = 2 I , and the size or length of an itemset X ∈ L, denoted by |X|, is its cardinality. In our approach, a transactional database D is a set of pairs (j, X) where j is the unique identifier of a transaction and X is an itemset in L, i.e. D ⊆ N × L. In the following, given a transactional database D, for every integer j ∈ N, D[j] represents the itemset of transaction j, i.e. D[j] = X if (j, X) ∈ D (otherwise, we consider that D[j] = ∅). Moreover, |D| is the number of transactions in D and ||D|| = j∈N |D[j]| defines the size of the transactional database D. For example, in Table 1, the transactional database D 1 contains 3 transactions of identifiers 1, 4 and 5. Besides, we have D 1 [START_REF] Hasan | Output space sampling for graph patterns[END_REF] = A, D 1 [START_REF] Cheung | Efficient mining of association rules in distributed databases[END_REF] = BE and D 1 [START_REF] Diop | Sequential pattern sampling with norm constraints[END_REF] = BC. Thus, we have

||D 1 || = 1 + 2 + 2 = 5.
Intuitively, a (transactional) distributed database is a set of transactional databases, also called fragments, where transactions do not overlap. More formally, a distributed database is defined as follows:

Definition 1 (Transactional distributed and centralized databases). A (transactional) distributed database

P = {D 1 , . . . , D K } is a set of transactional databases D k (k ∈ [1..K]) such that for each j ∈ N, we have D k [j] ∩ D l [j] = ∅ if k = l.
Then, the centralized version of P, denoted by P * , is the transactional database defined by:

P * = {(j, X) : X = K k=1 D k [j] ∧ X = ∅}.
For example, in Table 1, it is easy to see that P * is the centralized version of the distributed database P = {D 1 , . . . , D 4 }. In the following, we also say that P is a partitioning of the centralized database P * . In general, different types of partitioning are distinguished. On the one hand, a distributed database is a horizontal partitioning if every transaction is described in only one fragment, i.e. if for every j ∈ N, D k [j] = ∅ and D l [j] = ∅ implies that k = l. On the other hand, a distributed database is a vertical partitioning if every item is present in only one fragment, i.e. if for every x ∈ I, x ∈ j∈N D k [j] and x ∈ j∈N D l [j] implies that k = l. Finally, a partitioning is said to be hybrid if it is neither horizontal nor vertical. For example, in Table 1, P is a hybrid partitioning of P * . Indeed, the transaction 2 is described both in the fragments D 2 , D 3 et D 4 and the item A is both present in fragments D 1 and D 4 .

We consider only two forms of query can be sent to the fragment:

1. lengthOf primitive: Given a transaction identifier j and a fragment D k , the query lengthOf(j, D k ) returns the length of the transaction j in fragment k, i.e. lengthOf(j,

D k ) = |D k [j]|. In our example, we have lengthOf(4, D 1 ) = 2 since |D 1 [4]| = |BE| = 2. 2.
itemAt primitive: Given a position i, a transaction j and a fragment D k , the query itemAt(i, j, D k ) returns the i-th item of the transaction j in fragment D k (assuming an arbitrary order over the set of items I). In our example, considering the lexicographic order over I, we have itemAt(2, 4, D 1 ) = E.

Our communication model is generic since more complex queries can be reduced to these two basic primitives for describing the exchanges. For instance, advanced queries may directly obtain a transaction of items from a fragment corresponding to itemAt primitives. For the Semantic Web, for the primitive lengthOf, the SPARQL query SELECT (COUNT(DISTINCT ?p) AS ?length) WHERE {wd:Q210734 ?p ?o} returns the length of Youssou N'Dour's transaction (wd:Q210734) from Wikidata SPARQL endpoint (here, the length is 143 items). In the same way, the query SELECT DISTINCT ?p WHERE {wd:Q210734 ?p ?o} OFFSET 2 LIMIT 1 gives the second item of the transaction. Of course, it is possible to use the query SELECT DISTINCT ?p WHERE {wd:Q210734 ?p ?o} for having the entire transaction instead of using 143 itemAt queries. Unlike the two primitives that have 1 as communication cost, the communication cost of this query is 143.

Class of interestingness measures

Pattern discovery is based on interestingness measures that evaluate the quality of a pattern. One of the most popular of interestingness measures is the frequency which is an intuitive interestingness measure for experts and is an essential atomic element to build many other interestingness measures (like area or discriminative measures). The frequency of an itemset X ∈ L in the transactional database D, denoted by freq(X, D), is defined by: freq(X, D) = |{(j, T ) ∈ D : X ⊆ T }|. In Table 1, we have freq(DF, P * ) = 2 since the itemset DF is included in transactions 2 and 3.

It is also common to associate a utility to an itemset, and to combine the frequency of an itemset with its utility. For example, if we consider the utility function u(X) = |X|, we obtain the area measure: freq(X, D) × |X|. More generally, we consider the class of interestingness measures of the form freq(X, D) × u(X) where u exclusively depends on the length of itemsets: Definition 2 (Length-based utilities and measures). A utility u defined from L to R is called a length-based utility if there exists a function f u from N to R such that u(X) = f u (|X|) for each X ∈ L. Given the set U of lengthbased utilities, M U is the set of interestingness measures m u such that for every pattern X and database D, m u (X, D) = freq(X, D) × u(X) with u ∈ U.

We already see that the utility function defined for every pattern X ∈ L by u area (X) = |X| allows us to consider the area measure freq(X, D) × |X|. Obviously, let us notice that the utility function defined by u freq (X) = 1 enables us to consider the frequency as interestingness measure. Besides, the utility function defined by u ≤M (X) = 1 iff |X| ≤ M (0 otherwise) simulates a maximum length constraint. Indeed, with the induced interestingness measure freq(X, D) × u ≤M (X), an itemset with a cardinality strictly greater than M is judged useless (whatever its frequency). Dually, the utility function defined by u ≥m (X) = 1 iff |X| ≥ m (0 otherwise) simulates a minimum length constraint. Finally, the utility function defined by u decay (X) = α |X| with α ∈]0, 1[, named exponential decay, is useful for penalizing large itemsets but in a smooth way in comparison with u ≤M .

Pattern sampling in a distributed database

A pattern sampling method aims at randomly drawing a pattern X from a language L according to an interestingness measure f . X ∼ π(L) denotes such a pattern where π(.) = f (.)/Z is a probability distribution over L (with Z as normalizing constant). In this paper, our goal is to randomly draw patterns in a distributed database according to an interestingness measure in M U :

Given a distributed database P, an interestingness measure m ∈ M U , we aim at randomly drawing a pattern X ∈ L with a probability distribution π proportional to its interestingness measure m i.e., π(X) = m(X,P * ) Z where P * is the centralized version of P and Z = X∈L m(X, P * ) is a normalizing constant.

A naive approach could apply the classical two-step random procedure [START_REF] Boley | Direct local pattern sampling by efficient two-step random procedures[END_REF] after having centralized all the fragments of the distributed database P for building P * (using itemAt queries). The communication cost of this preliminary centralization would be very high. Indeed, it would be proportional to the size of P * , i.e.

||P * || = K k=1 ||D k ||.
Next section shows that it is only necessary to centralize the lengths of the transaction parts stored in the different fragments of P in order to draw an exact sample of patterns.

Decentralized Pattern Sampling

DDSampling algorithm

This section presents our algorithm called DDSampling (for Distributed Database Sampling), which randomly draws a pattern from a distributed database P proportionally to an interestingness measure m ∈ M U .

The key idea of our proposal is first to centralize only the lengths of the transaction parts contained in the different fragments. Indeed, this information requires a low communication cost and it enables us to draw a transaction identifier j according to its weight ω(j) and an itemset length proportionally to ω (j). Finally, we show how to emulate a decentralized sampling of a subset of D[j] of length without centralizing all the items of D[j]. Preprocessing phase. In this phase (see lines 1-2 of Algorithm 1), we first compute and store locally a matrix M that contains for every transaction j and every fragment D k of P, the length of the transaction j in D k .

Definition 3 (Weight matrix). Given a distributed database

P = {D 1 , . . . , D K }, let M ∈ R |P * |×|P| be the matrix defined for every j ∈ [1..|P * |] and k ∈ [1..K] by M jk = lengthOf(j, D k ).
In practice, it is important to note that the matrix M is computed offline before the drawing phase. In the following, M j• denotes the sum

M j• = K k=1 M jk .
It is easy to see that M j• represents the length of the transaction j in P * , i.e. M j• = |P * [j]|. For example, Table 2 presents the weight matrix M of the distributed database of Table 1. We can also check that

M 1• = 1 + 2 = |P * [1]|. j M Mj• M j• =0 ω freq (j) = ω freq (j) ω area (j) ω 2 (j) ω decay (j) 1 1 2 0 0 3 1 + 3 + 3 + 1 = 8 12 7
: i ∼ u([1..Mj•] \ ϑ) 9: k := min{p ∈ [1.
.K] : i ≤ p m=1 Mjm} 10:

i := k m=1 Mjm -i + 1 and x := itemAt(i , j, D k ) 11:

X := X ∪ {x} 12:

ϑ := ϑ ∪ {i} 13: od 14: return X Given a distributed database P and its associated weight matrix M , Property 1 shows how the weights ω(j) and ω (j) can be computed for each transaction j and length for any length-based utility function u: Property 1. Let P = {D 1 , . . . , D K } be a distributed database and u ∈ U a length-based utility. Given the weight matrix M associated with P,

for each transaction j ∈ [1..|P * |] and ∈ [1..M j• ], we have: ω (j) = X⊆P * [j]∧|X|= u(X) = Mj• × f u ( ). Moreover, we have ω(j) = Mj• =0 ω (j).
Due to space limitation, the proofs have been omitted. Intuitively, this property is valid because all the itemsets of length in a transaction j have the same utility. In Algorithm 1, this property is useful during the preprocessing phase to compute the weights of all the transactions. This preprocessing phase is illustrated in Table 2 with four length-based utility functions: u freq , u area , u ≤2 and u decay (with α = 0.1). For example, because u freq (X) = 1 for every X ∈ L, we have ω freq (1) = Drawing phase. In this phase, we can apply a direct generalization of the two-step random procedure proposed in [START_REF] Boley | Direct local pattern sampling by efficient two-step random procedures[END_REF] to draw itemsets with a probability proportional to their interest in the manner of the area measure. We start by drawing in Step 1 a transaction identifier j with a probability proportional to its weight ω(j) (see line 3 of Algorithm 1). The only difference is that the weights ω(j) are computed during the preprocessing phase using the weight matrix M and Property 1.

In Step 2, as the weights ω (j) of the transaction j are not stored during the preprocessing phase to reduce the storage cost, line 4 computes them for any length . After drawing the length of the itemset that will be returned (lines 4-5), DDSampling draws an itemset of length from the different fragments of P. At each iteration of the while loop (lines 7-13), the key idea is to draw without replacement the position i of an item in the transaction j (line 8) and to search the fragment D k that contains this item (line 9). Then, we compute the position i of this item in the fragment D k before querying the corresponding item x (see line 10). Finally, the item x is added to the itemset X (line 11) that will be returned (line 14) and the position i is added to the set ϑ (line 12) in order to avoid sampling the same position (and item) twice (see line 8). This process is repeated times in order to return an itemset X of length . For example, considering the toy example in Table 2, if we draw the position i = 2 in the transaction j = 1, we find that the involved fragment is D 2 since 2 > M 11 = 1 whereas 2 ≤ M 11 + M 12 = 3. Then, we compute i = 2 -1 = 1 and the item itemAt(1, 1, D 2 ) = B is added to the itemset X.

Theoretical analysis of the method

The following property states that DDSampling returns an exact sample of itemsets without centralizing the distributed database: Property 2 (Correction). Given a distributed database P = {D 1 , ..., D K } and a length-based utility function u ∈ U, Algorithm 1 draws an itemset X according to a distribution proportional to m u (X, D) = freq(X, D) × u(X) where D = P * . This follows from the fact that the three different draws (i.e., transaction j, length and the sampled itemset) take into account the number of itemsets occurring in each transaction weighted by the utility function. We now study the complexity of our method by distinguishing the two main phases: the preprocessing phase (where the matrix M is computed) and the drawing phase of itemsets. For each phase, we evaluate the complexity in time and in communication. As this storage cost is really low in practice, we store the matrix in memory. In an extreme case, the matrix could be stored in a database with a B-tree index to have quick access to the rows.

Communication complexity. In order to evaluate the communication cost, we simply count the number of queries lengthOf and itemAt required for the two main phases. First, it is easy to see that for the preprocessing phase, the construction of the weight matrix

M requires O(|D 1 | + • • • + |D K |) exchanges (using lengthOf queries), which is in general significantly lower than the cost of a complete centralization of P in O(||D 1 || + • • • + ||D K ||) = O(||P * ||) exchanges (using itemAt queries).
For the drawing of an itemset of length , it is clear that itemAt queries are necessary to return an itemset. Therefore, the average communication cost to draw an itemset is equal to the average length E[L] of an itemset returned by DDSampling. Given a distributed database P, we can easily see that j) is the probability to draw an itemset of length . Considering our toy example in Table 2 and These phenomena induce a bias in the performed drawings because we have to reject a pattern X as soon as an itemAt query fails during its drawing. In the case of network failure, let net be the probability that an itemAt query failed. Assuming that the failures are independent, we can show that P(reject

E[L] is equal to +∞ =1 P ( ) × where P ( ) = j∈[1..|P * |] ω (j) j∈[1..|P * |] ω(
) = P ( )• 1 -(1 -net ) . Thus, if net is a small number, we have P(reject) ≈ net • E[L] (where E[L]
is the average length of a sampled itemset). Now, in the case of node failure, let node be the probability that a node is down. If the distributed database is an horizontal partitioning of a centralized database, it is clear that P(reject) = node . Otherwise, if the distributed database is a vertical or hybrid partitioning of P * , assuming that the items are uniformly distributed over the nodes, we can show that P(reject) = P ( ) • 1 -(1node ) . Thus, if node is a small number, we have P(reject) ≈ node • E[L]. Nowadays, net and node are very small. Therefore, it follows that the rejection is negligible.

In this section, we evaluate the efficiency of our approach compared to a centralized solution (see Section 5.1) and its interest to find outliers in knowledge bases of the Semantic Web (see Section 5.2). Note that our prototype is implemented in Java and is available at https://github.com/DDSamplingRDF/ddsampling. git. All experiments are conducted on a 3.5 GHz 2 core processor with the Windows 10 operating system and 16 GB of RAM.

Efficiency and robustness of DDSampling

In our first experiments, we use 4 UCI datasets that we uniformly partition into K = 10 fragments to simulate distributed databases. In the case of horizontal partitioning (resp. vertical partitioning), each transaction (reps. each item) is randomly placed to a fragment with the same probability 1/K; in the case of hybrid partitioning, all the items of a transaction are randomly placed to a fragment (with the same probability 1/K). The first columns of Table 3 show statistical information about all datasets. In all experiments, we use an interval length constraint u ≥1 and u ≤M (with M = {3, 5}). This choice avoids drawing too much infrequent patterns, in particular for datasets containing long transactions. Execution times and communication costs. Table 3 indicates the execution times of our method by distinguishing the preprocessing and sampling phase, only in the case of vertical partitioning for M = 3. As expected, the preprocessing time (which can be prepared offline) increases with the size of the dataset. However, it is very small (less than 3 s). Regarding the sampling phase, whatever the dataset, it is always under 0.02 ms (per pattern). For the communication costs, we consider the three types of partionings. In the preprocessing phase, the communication cost corresponds to the number of lengthOf calls for constructing the weight matrix. This cost is naturally higher for hybrid and vertical partitionings since the items of a transaction may not be in the same fragment. In the drawing phase, the communication cost corresponds to the number of itemAt calls, and Table 3 shows the mean number of calls for drawing a pattern. This cost does not depend on the type of partitioning, but on the maximum length (M ∈ {3, 5}). Finally, we compare the communication cost between distributed and centralized approaches by evaluating the number N max of drawn patterns in the worst case (when M = 5 for vertical partitioning) that are necessary for the sampling approach to be as costly as data centralization. For all datasets, we can see that DDSampling can draw a few thousand patterns with a communication cost lower than that of a data centralization. Robustness. As seen in Section 4.2, network or node failures induce the rejection of some patterns. In this context, we evaluate the mean rejection rate by drawing 10,000 patterns by varying p and z. We repeat 100 times each experiment by randomly generating the dataset partition and changing down nodes. As rejection rates are independent of datasets, Figure 1 plots the average rejection rate and the standard deviation computed by averaging the results from the X⊆u}|) . Given a maximum length constraint M , we can show that fpof S (t, D) tends to the exact FPOF fpof ≤M (t, D) = X⊆t,|X|≤M freq(X,D) max u∈D ( X⊆u,|X|≤M freq(X,D)) when the size of S tends to infinity and S is sampled according to m(X, D) = freq(X, D) × u ≤M (X). Figure 2 depicts the FPOF distributions of all entities for Person and Organisation without constraint (M = ∞) or with a maximum length constraint M ∈ {1, 2, 3, 4, 5, 10}. We see that without constraint or with a large value for M (≥ 5), a large majority of FPOF values are equal to zero, which implies that it is impossible to distinguish outliers from normal entities. Indeed, without constraint, FPOF suffers from the long tail problem [START_REF] Diop | Sequential pattern sampling with norm constraints[END_REF]. Therefore, the use of a maximum length constraint is crucial for detecting outliers by means of FPOF. Consequently, we use M = 3 in the following experiments. Output space vs. input space sampling. This experiment compares input and output space sampling to determine which method is the best for the same budget (same number of patterns, same communication cost). For a pattern budget k, we start by drawing a sample S out of k patterns with DDSampling and we calculate its communication cost Cost out . Then, we draw a sample of transactions D requiring the same communication cost: Cost in = Cost out . Finally, we draw a sample S in of k patterns from D. Given a sample S, we evaluate the quality of its approximated FPOF by using the Euclidean distance (S, D) as error: (S, D) = t∈D ( fpofS (t, D) -fpof ≤M (t, D)) 2 . Figure 3 Fig. 3: Evolution of the Euclidean norm error for input/output space sampling 100 repeated samples with its standard deviation). Of course, both errors tend to zero when the sample size tends to infinity. But, it is clear that the convergence is faster and more stable with output space sampling (e.g., the FPOF quality of S out with 10k patterns equals that of S in with 100k patterns). This experience shows that output space sampling is more efficient at equal budget.

Qualitative evaluation. We manually analyze the 50 best and worse entities of Person according to the FPOF. It is interesting to note that all entities with the highest FPOF are real persons, and not outliers. On the contrary, only 36% of the entities with the lowest FPOF are real persons, and 64% of them can be considered as outliers. Indeed, 44% of entities are fictional characters and more importantly, 8% of them should be classified in Organisation (a sibling class of Person), and 12% of them should be classified in another class (e.g., Event).

Conclusion

This paper proposes the first pattern sampling method in a distributed database. It allows to consider different interestingness measures and interestingly, hybrid or vertical partitioning. As only transaction lengths are centralized, the communication costs of DDSampling are low because the exchange of items is done only when the patterns are drawn. The experimental study emphasizes this low

Table 2 : 1

 21 Weight matrix M and transaction drawing weights Algorithm DDSampling Input: A distributed database P = {D1, • • • , DK } and a length-based utility u ∈ U Output: An itemset X ∈ L randomly drawn w.r.t. freq(X, P * ) × u(X) // Preprocessing Phase 1: Compute M defined by M jk := lengthOf(j, D k ) for j ∈ [1..|P * |] and k ∈ [1..K] 2: Compute the weights ω defined by ω(j) := M j• =0 M j• × fu ( ) for j ∈ [1..|P * |] // Drawing Phase -Step 1: sampling of a transaction 3: Draw a transaction identifier j ∈ [1..|P * |] proportionally to ω: j ∼ ω(P * ) // Drawing Phase -Step 2: decentralized sampling of an itemset 4: Compute the weights defined by ω (j) := M j• × fu ( ) for every ∈ [0..Mj•] 5: Draw a length proportionally to ω (j): ∼ ω [0..M j• ] (j) 6: ϑ := ∅ and X := ∅ 7: while |X| < do 8

2 =0 3 = 1 + 3 + 3 = 7 .

 231337 utility function u area , we have ω area (1) =3 =0 3 × = (1 • 0) + (3 • 1) + (3 • 2) + (1 • 3) = 3 + 6 + 3 = 12 since u area (X) =for every pattern X of length . With the maximum length constraint, it is easy to see that ω 2 (1) = Finally, with the decay utility function u decay and α = 0.1, we have ω decay (1) = 3 =0 3 × 0.1 = (1 • 0.1 0 ) + (3 • 0.1 1 ) + (3 • 0.1 2 ) + (1 • 0.1 3 ) = 1 + 0.3 + 0.03 + 0.001 = 1.331.

  Time and space complexity. In the preprocessing phase, the weight matrix M is first computed with a complexity in time O(|P * | • |P|). Then, the weight ω(j) of all transactions j ∈ [1..|P * |] is computed in time O(|P * | • |I|) due to the use of the binomial function. Thus, the preprocessing phase is performed in time O(|P * | • (|P| + |I|)). The draw of itemsets is less expensive. First, the draw of a transaction identifier can be achieved in O(log(|P * |)). Then, the drawing of an itemset from this transaction is done in time O(|I|). Therefore, the drawing complexity of an itemset is in O(log(|P * |) + |I|). Besides, the space complexity only depends on the weight matrix dimension and it is in O(|P * | • |P|).

  the maximum length utility function u ≤2 , we have P (L = 1) = 3+6+3+6+3 7+11+7+11+7 = 21 43 and P (L = 2) = 3+4+3+4+3 7+11+7+11+7 = 17 43 . Thus, the average communication cost for drawing an itemset is equal to E[L] = ( 21 43 • 1) + (17 43 • 2) ≈ 1.28. Note that without length constraint (using u freq utility function), this cost is higher and equal to E[L] = ( 21 56 • 1) + ( 17 56 • 2) + ( 11 56 • 3) + ( 2 56 • 4) ≈ 1.71. Rejection rate. Two main problems arise in distributed databases: network communication errors (network failure) and node inaccessibility (node failure).

Fig. 2 :

 2 Fig. 2: Long tail problem of the FPOF distributions

Table 3 :

 3 Communication costs and execution times for 4 UCI partitioned datasets 4 UCI datasets. For network failures, the average rejection rate does not depend on a particular partitioning type and Figure1(left) presents the evolution of the average rejection rates according to p for M ∈ {1, 2, 3, 4, 5}. In accordance with the theoretical analysis, for a given p, we see that the rejection rate increases with M since E[L] increases with M . Moreover, for a given M , we see that the increase of the rejection rate is sub-linear because E[L] decreases with p. Nevertheless, it remains inferior to 50% if p is inferior to 0.1, which already is a level of network failure that is much higher than what is observed in practice. Figure1(right) presents the average rejection rates with the proportion of down nodes. First, as proved in Section 4.2, we observe that the average rejection rate is lower for horizontal partitioning than for hybrid or vertical partitioning (since E[L] ≥ 1). Second, the standard deviation of vertical partitioning is higher than that of other partitioning. Indeed, with vertical partitioning, the average rejection rate can be very low or very high depending on whether the most frequent items are placed or not into down nodes. However, as for network failures, we note that the rejection rate remains acceptable. DBpedia |I| Wikidata |t| min |t| max |t| avg Prep. Sampl.

		1.0	M=1					1.0	Vertical
			M=2						Hybrid
	Average rejection rate	0.2 0.4 0.6 0.8	M=3 M=4 M=5				Average rejection rate	0.2 0.4 0.6 0.8	Horizontal
		0.0						0.0
		0.0	0.1	0.2	0.3	0.4	0.5	10%	20%	30%	40%	50%
				Probability of network failure				Probability of node failure (with M=5)
				Fig. 1: Average number of rejections with failures

Table 4 :

 4 Characteristics and execution times for classes Person and Organisation 5.2 Outlier detection in distributed databases using pattern sampling This section aims at detecting outliers in knowledge bases of the Semantic Web by approximating Frequent Pattern Outlier Factor (FPOF) with sampled patterns. More precisely, we use this measure for identifying misclassified entities from two classes (Person and Organisation) described in DBpedia and Wikidata. In this context, each entity of a class C is a transaction distributed over DBpedia and Wikidata where each property p from a RDF triple (x, p, y) is an item. Table 4 provides some statistics about the two classes. Execution times are longer than that for UCI benchmarks because we use public SPARQL endpoints.

Long tail problem and norm constraints. The exact FPOF of a transaction can be approximated using pattern sampling

[START_REF] Giacometti | Anytime algorithm for frequent pattern outlier detection[END_REF]

. More precisely, given a database D and a sample S, the approximated FPOF of a transaction t ∈ D is defined by fpof S (t, D) = |{X∈S : X⊆t}| max u∈D

(|{X∈S :

  reports (S out , D) and (S in , D) w.r.t. the pattern budget (each measure is the arithmetic mean of

			DBpedia+Wikidata(Person)		DBpedia+Wikidata(Organization)
					400000
	Number of instances	200000 400000 600000 800000 1e+06	M=∞ M=10 M=5 M=4 M=3 M=2 M=1	Number of instances	50000 100000 150000 200000 250000 300000 350000	M=∞ M=10 M=5 M=4 M=3 M=2 M=1
		0			0
		[0 ,0 ]	]0 ,0 .1 ] ]0 .1 ,0 .2 ] ]0 .2 ,0 .3 ] ]0 .3 ,0 .4 ] ]0 .4 ,0 .5 ] ]0 .5 ,0 .6 ] ]0 .6 ,0 .7 ] ]0 .7 ,0 .8 ] ]0 .8 ,0 .9 ] ]0 .9 ,1 .0 ]		[0 ,0 ] ]0 ,0 .1 ] ]0 .1 ,0 .2 ] ]0 .2 ,0 .3 ] ]0 .3 ,0 .4 ] ]0 .4 ,0 .5 ] ]0 .5 ,0 .6 ] ]0 .6 ,0 .7 ] ]0 .7 ,0 .8 ] ]0 .8 ,0 .9 ] ]0 .9 ,1 .0 ]
			FPOF		FPOF

communication cost on several benchmarks datasets whatever the partitioning. We also illustrate the interest of the sampled patterns in RDF data for detecting abnormal entities among persons and organizations without centralizing all the data. In future work, we plan to replace the exact drawing of transactions with a stochastic method so that we do not have to centralize the lengths of all transactions for each fragment.


Aurore Lê

Didier Floner
email: didier.floner@univ-rennes1.fr

Thierry Roisnel

Olivier Cador

Léa Chancelier

Florence Geneste
email: florence.geneste@univ-rennes1.fr

Highly soluble Fe(III)-triethanolamine complex relevant for redox flow batteries

Keywords: all-Fe battery, Fe complex, organometallic electrolyte, triethanolamine, volumetric capacities

Fe-triethanolamine is a promising candidate as anolyte for redox flow batteries (RFBs), owning to its low potential, high solubility and low cost. We report here a new dinuclear structure of this complex at solid state when prepared with a stoichiometric amount of triethanolamine and iron in basic medium, whereas more than two equivalents of ligands are usually used to prepare Fe-triethanolamine for RFBs application. We achieve a calibration curve to estimate Fe(III) concentration in solution and coulometric experiments highlight a oneelectron reduction process per iron atom, corresponding to the reduction of the two Fe(III) atoms of the dinuclear complex into Fe(II). A solubility higher than 1.2 mol dm -3 can be reached for Fetriethanolamine with the new synthesis proposed in this work. All-Fe alkaline RFBs implemented with Fe-triethanolamine exhibit good performances in terms of coulombic, voltage and energy efficiencies, and is stable over a hundred of cycles. A power density around 80-120

Introduction

Triethanolamine (TEA) has been widely used as ligand owning to its tetradentate chelating character, low cost and commercial availability. Among the transition metals that have been tested for its complexation, iron has retained a lot of attention with applications ranging from the formation of biologically active metallated ionic liquids [START_REF] Ushakov | The NMR study of biologically active metallated alkanol ammonium ionic liquids[END_REF] and the production of pure iron oxide magnetite Fe 3 O 4 by electrochemical deposition [START_REF] Kulp | Electrodeposition of Epitaxial Magnetite Films and Ferrihydrite Nanoribbons on Single-Crystal Gold[END_REF][START_REF] Kothari | Electrochemical deposition and characterization of Fe3O4 films produced by the reduction of Fe(III)-triethanolamine[END_REF][START_REF] Goujon | Electrochemical Deposition of Thick Iron Oxide Films on Nickel Based Superalloy Substrates[END_REF][START_REF] Saadat | Coaxial Fe 3 O 4 /CuO hybrid nanowires as ultra fast charge/discharge lithium-ion battery anodes[END_REF] to photocatalytic production of hydrogen in aqueous medium [START_REF] Dong | Simple Nickel-Based Catalyst Systems Combined With Graphitic Carbon Nitride for Stable Photocatalytic Hydrogen Production in Water[END_REF]. In addition, Fe(III)-triethanolamine (Fe(III)-TEA) exhibits a reversible Fe(III/II) system at a very low potential (-1.15 V vs Ag/AgCl) and a high solubility in alkaline medium, making it a very efficient reducing agent. Thus it has been used as an environmentally friendly mediator for indirect electroreduction of vat dyes such as indigo for industrial dyeing processes [START_REF] Xu | Indirect Electrochemical Reduction of Indigo on Carbon Felt: Process Optimization and Reaction Mechanism[END_REF][START_REF] Kulandainathan | Potentiostatic studies on indirect electrochemical reduction of vat dyes[END_REF][START_REF] Bechtold | Electrochemical reduction processes in indigo dyeing[END_REF][START_REF] Bechtold | Multi-cathode cell with flow-through electrodes for the production of iron(II)-triethanolamine complexes[END_REF][START_REF] Bechtold | Alkali-stable iron complexes as mediators for the electrochemical reduction of dispersed organic dyestuffs[END_REF][START_REF] Bechtold | The redox-catalyzed reduction of dispersed organic compounds. Investigations on the electrochemical reduction of insoluble organic compounds in aqueous systems[END_REF]. Another particularly attractive application of this interesting redox behaviour of Fe(III)-TEA is redox flow battery (RFB) [START_REF] Arroyo-Curras | An Alkaline Flow Battery Based on the Coordination Chemistry of Iron and Cobalt[END_REF][START_REF] Gong | All-Soluble All-Iron Aqueous Redox-Flow Battery[END_REF][START_REF] Wen | A study of the Fe(III)/Fe(II)-triethanolamine complex redox couple for redox flow battery application[END_REF]. This large-scale energy storage technology is well-adapted for renewable energies owning to power/energy independent sizing, low cost and high-efficiency. Its performances in terms of voltage and stability strongly depend on the involved redox couples. Thus, Fe(III)-TEA/Fe(II)-TEA has proved its efficiency as negative redox couple in RFBs, leading to high cell voltage. Concentrations raising 0.6 and 0.8 mol dm -3 A c c e p t e d M a n u s c r i p t in Fe(III) salts have been reported [START_REF] Arroyo-Curras | An Alkaline Flow Battery Based on the Coordination Chemistry of Iron and Cobalt[END_REF]. To facilitate its implementation in RFBs, the Fe(III)-TEA complex has been synthesized in situ using more than 2 eq. of TEA per Fe atom [START_REF] Arroyo-Curras | An Alkaline Flow Battery Based on the Coordination Chemistry of Iron and Cobalt[END_REF][START_REF] Gong | All-Soluble All-Iron Aqueous Redox-Flow Battery[END_REF][START_REF] Wen | A study of the Fe(III)/Fe(II)-triethanolamine complex redox couple for redox flow battery application[END_REF].

Whereas its structure in solution is not known, the crystal structure of the complex corresponding to a protocol involving 2 eq. of TEA per Fe atom at solid state has been reported to be a mononuclear complex with an iron atom coordinated by two TEA ligands in a regular octahedral configuration [START_REF] Sen | Characterization and studies of some triethanolamine complexes of transition and representative metals[END_REF][START_REF] Naini | New complexes of triethanolamine (TEA): novel structural features of [Y(TEA) 2 ](ClO 4 ) 3 •3C 5 H 5 N and [Cd(TEA) 2 ](NO 3 ) 2[END_REF].

Herein, we report a new X-ray crystal structure of a Fe(III)-TEA complex prepared in basic medium at high concentrations using a stoichiometric amount of TEA. The oxidation state of iron atoms and the number of electrons transferred were determined. The performance of the complex in RFB was tested, underlining the interest of this high soluble complex to achieve high volumetric capacities.

Experimental

Materials

Iron(III) sulfate hydrate and triethanolamine (98%) were purchased from Alfa Aesar. Potassium sulfate was from Sigma Aldrich and sodium hydroxide from Acros Organics. All solutions were prepared with ultrapure water (18.2 MΩ, Millipore Simplicity).

Instrumentation

High resolution mass spectra (HRMS) were obtained with a Bruker Maxi 4G and a Thermo Fischer Scientific Q-exactive with an electrospray ionisation (ESI) at the Centre Régional de Mesures Physiques de l'Ouest (CRMPO) in Rennes. Elemental analysis was performed at the Centre Régional de Mesures Physiques de l'Ouest (CRMPO) in Rennes. Voltammetric A c c e p t e d M a n u s c r i p t experiments were carried out using a BioLogic SP150 potentiostat/galvanostat apparatus. A glassy carbon electrode (0.071 cm 2 ), a platinum wire auxiliary electrode, and a silver/silver chloride reference electrode (Ag/AgCl) were used in a standard three-electrode configuration for cyclic voltammetry experiments. Linear Sweep Voltammetry (LSV) at a 0.0314 cm² gold rotating disc electrode (Metrohm) was performed at 20 mV s -1 scan rate with a rotation rate of 1000 rpm, under a dinitrogen atmosphere.

Synthesis of Fe(III)-TEA complex.

100 g (< 0.25 mol) iron(III) sulfate hydrate (Fe 2 (SO 4 ) 3 , xH 2 O) were dissolved in 200 cm -3 water leading to a dark red solution. 78 g (0.52 mmol) triethanolamine were then added under stirring.

To this solution, 140 g (2.5 mol) potassium hydroxide dissolved in a minimum of water were added. The dark solution was stirred for 30 min and K 2 SO 4 formed was then filtered on Buchner funnel. The precipitate was rinsed with water and the filtrate was collected and added to the previously filtered solution. It was then filtered again on a fritted glass and completed to 500 mL with water in a volumetric flask. This synthesis protocol led to a concentration of 0.84 mol dm -3 of Fe-TEA, according to the calibration curve obtained by linear sweep voltammetry at a rotating disk electrode.

Crystals were grown by slow evaporation of water. They were rinsed with cold water and dried at air during several days (around 10 g of crystals for 100 cm -3 of solution).

The crystals were crushed and the resulting powder was dried at a vacuum pump at 70-80°C during three days, to dehydrate the substrate. Results in elemental analysis correspond to the µoxo bis(TEA) diiron complex with three molecules of water. Elemental

Magnetic susceptibility

The magnetic susceptibility measurements were performed on solid polycrystalline sample with a Quantum Design MPMS-XL SQUID (Superconducting QUantum Interference Device) magnetometer between 2 and 300 K in an applied magnetic field of 0.5 T for temperatures in the range 2-20 K and 1 T for temperatures between 20 and 300 K. These measurements were all corrected for the diamagnetic intrinsic contribution as calculated with Pascal's constants and for the diamagnetism of the sample holder measured separately. Dimer magnetic model [START_REF] Kahn | Molecular Magnetism[END_REF]:

= - ⋅ H J 1 2
S S (eq. 1)

β χ           + + + +                     =           + + + + +                     2

Coulometry

The reduction of Fe(III)-TEA (solution prepared as reported above) was performed under argon in a home-made flow cell (Fig. 1) [START_REF] Geneste | Electrocatalytic oxidation of alcohols by a [Ru(tpy)(phen)(OH 2 )] 2+modified electrode[END_REF].

A c c e p t e d M a n u s c r i p t obtained at the end of the electrolysis to be sure to have a proportionality between the diffusionlimited current and the concentration of Fe(III) atoms, using the following equation:

𝑛𝑛 𝑒𝑒-= 𝑄𝑄 𝑒𝑒𝑒𝑒𝑒𝑒 (𝐶𝐶 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 -𝐶𝐶 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 )𝑉𝑉𝑉𝑉
with Q exp the charge passed, V the solution volume, F the Faraday's constant, C initial the initial concentration = 0.84 mol dm -3 and C final the concentration of remaining Fe(III)-TEA estimated from the calibration curve.

RFB performance

RFB tests were performed using a home-made cell containing graphite felt electrodes (square cuboid 50 × 50 × 4.6 mm with 35% compression of the thickness leading to 95.5% porosity) with a geometrical surface area of 25 cm 2 , composite graphite current collectors and cationic ionexchange membrane. The positive and negative electrolytes (20 cm -3 ) were pumped at a flow rate of 150 cm 3 min -1 (linear flow velocity: 3.1 cm s -1 ) through KNF Pump (NF1-100KT-18S).

The battery was operated through a Bio-Logic SP150 potentiostat with a 20 A booster.

NMR

NMR spectra were recorded on a Bruker AH300 FT spectrometer. After 110 cycles, a sample of the positive electrolyte was taken, evaporated and dried under vacuum. The solid was dissolved in D 2 O until saturation was reached. The sample was filtered and then analyzed.

Results and discussion

Synthesis and characterization of the Fe-TEA complex

A c c e p t e d M a n u s c r i p t

The Fe-TEA complex was prepared by reaction of Fe 2 (SO 4 ) 3 (around 1 mol dm -3 ) with around 2 eq. of TEA (i.e. around 1 eq. per Fe(III) atom) in water before adding a solution of KOH (pH close to 14). It is interesting to note that the use of KOH instead of NaOH as reported before [START_REF] Arroyo-Curras | An Alkaline Flow Battery Based on the Coordination Chemistry of Iron and Cobalt[END_REF][START_REF] Gong | All-Soluble All-Iron Aqueous Redox-Flow Battery[END_REF][START_REF] Wen | A study of the Fe(III)/Fe(II)-triethanolamine complex redox couple for redox flow battery application[END_REF] allows reaching concentrations higher than 1.2 mol dm -3 in Fe(III) salts. Slow volume reduction afforded large dark brown crystals, which were suitable for X-ray crystallographic study (Fig. 2). The structure of the complex was also confirmed by mass spectrometry and elemental microanalysis.

Temperature variable magnetization measurements have been used to verify the oxidation state, and spin state of iron solid state. At room temperature, χ M T (χ M the molar magnetic susceptibility and T the temperature in Kelvin) is equal to 8.75 cm 3 K mol -1 which perfectly fits with the expected value for two uncoupled high spin Fe(III) (S=5/2) spins (8.75 cm 3 K mol -1 with g=2) (Fig. 3) [START_REF] Kahn | Molecular Magnetism[END_REF]. Fig. 3. Temperature variation of χ M T between 2 and 300 K (circles) with the best fitted curve (red line) with the model exposed in text [START_REF] Kahn | Molecular Magnetism[END_REF]. χ M T decreases monotonically on cooling to reach 0.22 cm 3 K mol -1 at 2 K. The experimental curve can be reproduced with a dimer model featuring superexchange interaction between the two Fe(III) spins (H=-J S 1 ⋅S 2 ) through water molecule [START_REF] Kahn | Molecular Magnetism[END_REF]. The best fit with the analytical expression exposed in the experimental part incorporating a very small amount of paramagnetic impurities (0.2 cm 3 K mol -1 , g=2.07 and J=-31 cm -1 ) is represented on Fig. 3.

Electrochemical characterization of Fe-TEA

The Fe-TEA solution, prepared according to our protocol was diluted in 1 mol dm -3 KOH solution and analyzed by cyclic voltammetry (Fig. 4). A reversible system at -1.15 V vs Ag/AgCl is obtained, with a difference between anodic and cathodic peaks around 70 mV, showing that the system corresponds to one electron exchange (Fe III/II ). The stability of the Fe-TEA solution prepared according to our protocol was also estimated by cyclic voltammetry after six-month storage at room temperature (Fig. S1 in supplementary material). No degradation of the Fe-TEA complex could be observed.

To estimate the real concentration of Fe-TEA complex in solution, a calibration curve was established by Linear Sweep Voltammetry (LSV) at a rotating disk electrode using a solution of

A c c e p t e d M a n u s c r i p t

Fe-TEA prepared from isolated crystals (Fig. S2 in supplementary material). The concentration of the initial solution in Fe(III) complexed by TEA was calculated by using the following equation:

𝐶𝐶 𝐹𝐹𝑒𝑒(𝐼𝐼𝐼𝐼𝐼𝐼) = 𝑚𝑚 2𝑀𝑀
With m, the mass of crystals (3.5 g) and M the molecular mass estimated from X-ray analysis (696.41 g mol -1 ).

The diffusion-limited current was proportional to the concentration of Fe-TEA in the range 0.01 to 0.1 mol dm -3 (Fig. 5), whereas the linearity was lost at higher concentrations (Fig. S3 in supplementary material). a diffusion-limited current of 25.8 mA cm -². According to the calibration curve depicted in Fig. 5, it corresponds to a concentration in complexed Fe(III) of 0.84 mol L -1 .

Another important point for RFB application is the number of electrons that can be exchanged by the Fe-TEA complex. Indeed, the formation of a mixed valence compound [(TEA)Fe(III)OFe(II)(TEA)] 3-has been previously mentioned for Fe-TEA synthesized with more than 2 eq. of TEA per Fe atom [START_REF] Arroyo-Curras | An Alkaline Flow Battery Based on the Coordination Chemistry of Iron and Cobalt[END_REF]. This dinuclear compound would be the oxidized form of [(µ-O)(Fe(TEA)) 2 ] 4-, resulting in a one electron process per dinuclear complex instead of two. To investigate the number of electrons exchanged during the reduction of the Fe-TEA complex prepared according to our procedure, bulk electrolyses were carried out in a flow electrochemical cell (Fig. 1). Coulometry was performed on a solution of 0.84 mol dm -3 Fe-TEA in a KOH solution (pH=14) saturated with K 2 SO 4 at constant current intensity ranging from 50 to 400 mA in order to maintain a potential at the working electrode higher than -1.2 V vs. SCE. The same Fe-TEA solution was reoxidized at air after reduction and used for coulometry experiments three times more. The reaction was followed by LSV at a rotating disk electrode to estimate the concentration of remaining Fe-TEA in the solution (Fig. S4 in supplementary material). In the experimental conditions used to reduce Fe(III) into Fe(II), the reduction of Fe(II) into Fe(0) was avoided as evidenced by the stability of the total current during the electrolysis (Fig. S5 in supplementary material). A value of 0.90 ± 0.08 electrons per Fe atom was found, corresponding to the reduction of the two Fe(III) atoms of the dinuclear complex into Fe(II).

Battery performance

The performance of the all-Fe alkaline RFB with 0.4 mol dm -3 (80:20) negative electrolyte was first evaluated by a charge-discharge test with a charging protocol of constant current density at 40 mA cm -2 , followed by a constant voltage (CV) step at 1.5 V (Fig. 6a). The initial coulombic efficiency (CE), voltage efficiency (VE) and energy efficiency (EE) found to be 92, 79 and 71% respectively, are in agreement with previous results

(93, 78, respectively) reported for this type of battery using 0.2 mol dm -3 of K 4 [Fe(CN) 6 ]

and Fe-TEA [START_REF] Gong | All-Soluble All-Iron Aqueous Redox-Flow Battery[END_REF]. Fig. 6b shows the cycling volumetric capacity and efficiencies curves over 110 cycles, corresponding to 57h, at a current density of 40 mA cm -2 . The efficiencies were stable with the CE ranging from 92 to 97.5%, the VE higher than 74% and the EE around 70%. These values were close to those previously reported for an all-Fe alkaline RFB, although the CE was significantly higher [START_REF] Gong | All-Soluble All-Iron Aqueous Redox-Flow Battery[END_REF]. Interestingly, the volumetric capacity (around 8.5 Ah dm -3 ) was the double of the value previously reported [START_REF] Gong | All-Soluble All-Iron Aqueous Redox-Flow Battery[END_REF] with a concentration The power density estimated during the cycling (Fig. 9) ranged between 101 and 117 mW cm -2

(for 50% state of charge). The initial volumetric capacity was around 14 Ah dm -3 , as expected for a K 4 [Fe(CN) 6 ] concentration of 0.775 mol dm -3 . This value slightly decreased during the first 20 cycles and stabilized around 12 Ah dm -3 . The decrease of the volumetric capacity could be due to different factors such as water transfer, species equilibrium or internal resistance but no significant cross-over of Fe-TEA was observed (Fig. S6 in supplementary material). Such volumetric capacity is high for redox flow batteries based on metallic complexes and/or organic redox species. For example, values around 4 Ah dm -3 have been reported for all-Fe alkaline batteries [START_REF] Hruska | Investigation of factors affecting performance of the iron-redox battery[END_REF], up to 1.9 Ah dm -3 for quinone based batteries and from 1.34 to 5.36 Ah dm -3 [START_REF] Cao | A highly reversible anthraquinone-based anolyte for alkaline aqueous redox flow batteries[END_REF] for batteries with flavin solubilized in hydrotropic agent [START_REF] Orita | A biomimetic redox flow battery based on flavin mononucleotide[END_REF]. KOH/NaOH (50:50) negative electrolyte.

Conclusions

In conclusion, the formation of a Fe-TEA complex at high concentrations (> 1.2 mol dm -3 ) was achieved thanks to a simple new synthetic protocol easily scaled up to a larger scale. The compound was characterized in terms of structure of the complex at solid state and oxidation degree of the Fe atom, showing a µ-oxo dimer containing two Fe(III) atoms. Electrochemical methods were applied to establish the exact concentration of iron after synthesis and the number of electrons exchanged during the reduction process, showing that the Fe-TEA complex exchange 1 electron per Fe atom. The RFB results are very promising since it exhibited good performances in terms of power density (80-120 mW cm -2 ) and stability with a volumetric capacity (around 12 Ah dm -3 ) that is high for redox flow batteries based on metallic complexes and/or organic redox species.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the flow electrochemical cell used for coulometry experiments. The

Fig. 2 .

 2 Fig. 2. Structure of the Fe-TEA complex showing 50% probability thermal ellipsoids and atomlabelling scheme. Hydrogen atoms, K + and H 2 O molecules are omitted for clarity. Selected

  A c c e p t e d M a n u s c r i p t a Fe-Fe distance of 3.6083(8) Å and a Fe-O-Fe angle of 169.28(17)°, close to the linearity. Right side of the complex is generated from asymmetric unit content and applying the 1-x,y,-z+1/2 symmetry operator.

Fig. 4 .

 4 Fig. 4. Cyclic voltammogram of Fe-TEA ( ___ ) prepared according to our protocol and diluted in 1 mol dm -3 KOH solution (10 -3 mol dm -3 of Fe) on a 0.071 cm² glassy carbon electrode. Blank in 1 mol dm -3 KOH solution ( ___ ). Scan rate 100 mV s -1 .

Fig. 5 .

 5 Fig. 5. Plots of current vs. concentration in Fe(III) constructed from LSV analysis at a gold rotating disk electrode of solutions of Fe(III) prepared by dilution of a 0.1 mol dm -3 Fe(III) solution obtained from isolated crystals of Fe(III)-TEA (3.5 g) dissolved in 1 mol dm -3 KOH solution saturated with K 2 SO 4 . Dilutions were performed with the same 1 mol dm -3 KOH

A c c e p t e d M a n u s c r i p t

K 4 [

 4 Fe(CN) 6 ] in 1 mol dm -3 KOH/NaOH (75:25) positive electrolyte and 0.5 mol dm -3 Fe-TEA in 1 mol dm -3 KOH/NaOHA c c e p t e d M a n u s c r i p t

Fig. 6 .

 6 Fig. 6. a) Cell voltage curve of a charge-discharge test of an all-Fe alkaline RFB with 0.4 mol dm -3 K 4 [Fe(CN) 6 ] in 1 mol dm -3 KOH/NaOH (75:25) positive electrolyte and 0.5 mol dm -3 Fe-TEA in 1 mol dm -3 KOH/NaOH (80:20) negative electrolyte (CC: 40 mA cm -2 ; CV 1.5 V) b)

A c c e

  p t e d M a n u s c r i p t of Fe-TEA and K 4 [Fe(CN) 6 ] of 0.2 mol dm -3 . This result was expected since the concentration of K 4 [Fe(CN) 6 ] is the double for our all-Fe alkaline battery. It clearly highlights the interest of this new synthesis of Fe-TEA, allowing an improvement of the solubility of the complex in the electrolyte medium. Polarization curves were plotted to evaluate the power density of the RFB during cycling (Fig.7).

Fig. 7 .

 7 Fig. 7. Maximum power density vs number of cycles obtained by polarization curves performed

A c c e p t e d M a n u s c r i p tFig. 8 .

 8 Fig. 8. a) Cell voltage curve of a charge-discharge test of an all-Fe alkaline RFB with 0.775 mol dm -3 K 4 [Fe(CN) 6 ] in 1 mol L -1 KOH/NaOH (50:50) positive electrolyte and 1.1 mol dm -3 Fe-TEA in 2 mol dm -3 KOH/NaOH (50:50) negative electrolyte at 40 mA cm -2 b)

A c c e p t e d M a n u s c r i p tFig. 9 .

 9 Fig. 9. Maximum power density vs number of cycles obtained by polarization curves performed at a 50% state of charge (SOC) for an all-Fe alkaline RFB with 0.775 mol dm -3 K 4 [Fe(CN) 6 ] in 1 mol dm -3 KOH/NaOH (50:50) positive electrolyte and 1.1 mol dm -3 Fe-TEA in 2 mol dm -3

A c c e p t e d M a n u s c r i p t