Olivier Pivert

Olfa Slama

Virginie Thion

Expression and efficient evaluation of fuzzy quantified structural queries to fuzzy graph databases

Keywords: Graph Databases, Fuzzy Quantified Queries 2010 MSC: 00-01, 99-00

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Even though the concept of a graph database is not exactly new, cf. [START_REF] Angles | Survey of graph database models[END_REF], it is only recently that the database community has started to show a strong interest in it, due in particular to the rise of linked data on the Web and the profusion of domains where networked objects have to be handled: social 5 networks, genomics, cartographic databases, etc.

Simultaneously, the need for flexible querying has been acknowledged by database researchers and many approaches to relational database preference queries have been proposed in the last decade, see e.g. [START_REF] Stefanidis | A survey on representation, composition and application of preferences in database systems[END_REF]. However, the pioneering work in this domain dates back to the 70's and is based on fuzzy set 10 theory [START_REF] Tahani | A conceptual framework for fuzzy query processing -A step toward very intelligent database systems[END_REF]. Since then, much effort has been made to come up with expressive and efficient flexible querying tools based on fuzzy logic, see e.g. [START_REF] Pivert | Fuzzy Preference Queries to Relational Databases[END_REF]. In particular, fuzzy quantified queries have proved useful in a relational database context for expressing different types of imprecise information needs, see e.g. [START_REF] Bosc | Quantified statements and database fuzzy querying[END_REF]. In a graph database context, such queries have an even higher potential since they can exploit the structure of the graph, beside the attribute values attached to the nodes or edges. Nevertheless, only one approach from the literature, described in [START_REF] Castelltort | Fuzzy queries over NoSQL graph databases: Perspectives for extending the cypher language[END_REF], considered fuzzy quantified queries so far and only in a limited way.

The present paper is an extended version of our work in [START_REF] Pivert | Fuzzy quantified structural queries to fuzzy graph databases[END_REF] in which we have integrated fuzzy quantified queries in a framework named fudge that we defined previously in [START_REF] Pivert | On a fuzzy algebra for querying graph databases[END_REF]. In this paper, we describe in more detail the syntactic form of the queries considered as well as a processing strategy aimed to efficiently evaluate them. Moreover, we provide some experimental results that show the tractability of the approach.

The remainder of the paper is organized as follows. Section 2 presents some background notions about graph databases, fuzzy graph theory, fuzzy graph databases, and the principles underlying the fudge query language introduced in [START_REF] Pivert | Expression and efficient processing of fuzzy queries in a graph database context[END_REF]. Section 3 is a refresher about fuzzy quantified statements. Section 4 discusses related work. In Section 5, we consider a specific type of fuzzy quantified structural query, we propose a syntactic format for expressing it in the fudge language and we describe its interpretation. Section 6 deals with query processing and discusses implementation issues. In Section 7, some experimental results showing the feasibility of the approach are presented. Finally, Section 8 recalls the main contributions and outlines research perspectives.

Background Notions

In this section, we recall important notions about graph databases, fuzzy graph theory, fuzzy graph databases, and the query language fudge.

Graph Databases

A graph database management system enables managing data for which the structure of the schema is modeled as a graph (nodes are entities and edges are relations between entities), and data is handled through graphoriented operations and type constructors. Different models of graph databases have been proposed in the literature (see [START_REF] Angles | Survey of graph database models[END_REF] for an overview), including the attributed graph (aka. property graph) aimed to model a network of entities with embedded data. In this model, nodes and edges may contain data in attributes (aka. properties).

Fuzzy Graphs

A graph G is a pair (V, R), where V is a set and R is a relation on V .

The elements of V (resp. R) correspond to the vertices (resp. edges) of the graph. Similarly, any fuzzy relation ρ on a set V can be regarded as defining a weighted graph, or fuzzy graph, see [START_REF] Rosenfeld | Fuzzy graphs[END_REF], where the edge (x, y) ∈ V × V has weight or strength ρ(x, y) ∈ [0, 1]. Having no edge between x and y is equivalent to ρ(x, y) = 0.

A fuzzy data graph may contain both fuzzy edges and crisp edges as a fuzzy edge with a degree of 0 or 1 can be considered as crisp. Along the same line, a crisp data graph is simply a special case of fuzzy data graph (where ρ : V × V → {0, 1} is Boolean). We then only deal with fuzzy edges and data graph in the following.

An important operation on fuzzy relations is composition. Assume ρ 1 and ρ 2 are two fuzzy relations on V . Thus, composition ρ = ρ 1 • ρ 2 is also a fuzzy relation on V s.t. ρ(x, z) = max y min(ρ 1 (x, y), ρ 2 (y, z)). The composition operation can be shown to be associative:

(ρ 1 • ρ 2 ) • ρ 3 = ρ 1 • (ρ 2 • ρ 3 ).
The associativity property allows us to use the notation ρ k = ρ • ρ • . . . • ρ for the composition of ρ with itself k -1 times. In addition, following [START_REF] Yager | Social network database querying based on computing with words[END_REF], we define ρ 0 to be s. t. ρ 0 (x, y) = 0, ∀(x, y).

Useful notions related to fuzzy graphs are those of strength and length of a path. Their definitions, drawn from [START_REF] Rosenfeld | Fuzzy graphs[END_REF], are given hereafter.

Strength of a path. -A path p in G is a sequence x 0 → x 1 → . . . → x n (n > 0) where the x i 's are nodes from V , such that ρ(x i-1 , x i ) > 0, 1 ≤ i ≤ n and where n is the number of links in the path. The strength of the path is defined as

ST (p) = min i=1..n ρ(x i-1 , x i ). (1)
In other words, the strength of a path is defined to be the weight of the weakest edge of the path. Two nodes for which there exists a path p with ST (p) > 0 between them are called connected. We call p a cycle if n ≥ 2 and x 0 = x n . It is possible to show that ρ k (x, y) is the strength of the strongest path from x to y containing at most k links. Thus, the strength of the strongest path joining any two vertices x and y (using any number of links) may be denoted by ρ ∞ (x, y).

Length and distance. -The length of a path p = x 0 → x 1 → . . . → x n in the sense of ρ is defined as follows:

Length(p) = n i=1 1 ρ(x i-1 , x i ) . (2)
Clearly Length(p) ≥ n (it is equal to n if ρ is Boolean, i.e., if G is a nonfuzzy graph). We can then define the distance between two nodes x and y in G as

Distance(x, y) = min

all paths p f rom x to y

Length(p).

It is the length of the shortest path from x to y.

Fuzzy Graph Databases

We are interested in fuzzy graph databases where nodes and edges can carry data (e.g. key-value pairs in attributed graphs). So, we consider an extension of the notion of a fuzzy graph: the fuzzy data graph as defined in [START_REF] Pivert | On a fuzzy algebra for querying graph databases[END_REF].

Definition 1 (Fuzzy data graph). Let E be a set of labels. A fuzzy data graph G is a quadruple (V, R, κ, ζ), where V is a finite set of nodes (each node n is identified by n.id) and R = e∈E {ρ e : V × V → [0, 1]} is a set of labeled fuzzy edges between nodes of V . Data may be attached to nodes (resp. edges), in the form of a set of key-value pairs for each node (resp. edge), modeled through the κ (resp. ζ) function.

In the following, a graph database is meant to be a fuzzy data graph. Figure 1 is an example of a fuzzy data graph in which the degree associated with A -contributor-> B is the proportion of journal papers co-written by A and B, over the total number of journal papers written by B. The degree associated with J -domain-> D (denoted by ρ domain (J, D) in Definition 1) is the extent to which the journal J belongs to the research domain D.

The nodes are assumed to be typed. If n is a node of V , then T ype(n) denotes its type. In Figure 1, the nodes IJWS12, IJAR14, IJIS16, IJIS10 and IJUFK15 are of type journal, the nodes IJWS12-p, IJAR14-p, IJIS16-p, IJIS10-p, IJIS10-p1 and IJUFK15-p of type paper, and the nodes Andreas, Peter, Maria, Claudio, Michel, Bazil and Susan are of type author ; the nodes named Database are of type domain and the other nodes are of type impact factor. For nodes of type journal, paper, author and domain, a property, called name, contains the identifier of the node and for nodes of type impact factor, a property, called value, contains the value of the node. In Figure 1, the value of the property name or value for a node appears inside the node. Data are attached to nodes only in this example. For instance here, κ(IJWS12) = {volume: 14, when: July 2012 }.

The fudge Query Language

115 fudge, based on the algebra described in [START_REF] Pivert | Expression and efficient processing of fuzzy queries in a graph database context[END_REF], is an extension of the Cypher language (see [12]), initially used in the Neo4j graph DBMS (see [START_REF]Neo4j web site[END_REF]) and now in other systems too, thanks to the OpenCypher initiative. These languages are based on graph pattern matching, meaning that a query Q over a fuzzy data graph DB defines a graph pattern and answers to Q 120 are its isomorphic subgraphs that can be found in DB. More concretely, a pattern has the form of a subgraph where variables can occur. An answer maps the variables in elements of DB.

A fuzzy graph pattern expressed à la Cypher consists of a set of expressions (n1:Type1)-[exp]->(n2:Type2) or (n1:Type1)-[e:label]->(n2:Type2) where n1 and n2 are node variables, e is an edge variable, label is a label of E, exp is a fuzzy regular expression and Type1 and Type2 are node types. Such an expression denotes a path satisfying a fuzzy regular expression exp (that is simple in the second form e) going from a node of type Type1 to a node of type Type2. All its arguments are optional, so the simplest form of an expression is ()-[]->() denoting a path made of two nodes connected by any edge. Conditions on attributes are expressed on nodes and edges variables in a where clause.

Example 1. We denote by P the graph pattern: This pattern "models" information concerning authors (au2) such that there exists a related -in the sense of a contributor path -author (au1) who published a paper (ar1) in IJWS12 and also published a paper (ar2) in another journal (j2).

Let us now illustrate how a fuzzy query can be expressed in the extension of Cypher named fudge, that embarks fuzzy preferences over the data and the structure specified in the graph pattern. Given a graph database DB, a fudge query is composed of:

1. A list of define clauses for fuzzy term declarations. We choose to use linear membership functions. If a fuzzy term fterm has a trapezoidal function defined by the quadruple (A', A, B, B') -meaning that its support is ]A', B'[ and its core [A, B] -, then the clause has the form define fterm as (A', A, B, B'). If fterm is a decreasing function, then the clause has the form defineDesc fterm as (δ,γ) meaning that the support of the term is [0, γ[ and its core [0, δ]. For an increasing function, the clause has the form defineAsc fterm as (δ,γ) meaning that the support of the term is ]δ, +∞[ and its core [γ, +∞[.

2.

A match clause, which has the form match pattern where conditions that expresses the fuzzy graph pattern.

Example 2. Listing 2 is an example of a fudge query. This pattern specifies information concerning authors (au2) such that there exists a closely related author -i.e., connected by a short path made of contributor edges -(au1) who published a paper (ar1) in IJWS12 and also published a paper (ar2) in another journal (j2) which has a high impact factor (i.value is high). The fuzzy terms short and high are defined on line 1.

Refresher on Fuzzy Quantified Statements

In this section, we recall important notions about fuzzy quantifiers, then, we present two approaches that have been proposed in the literature for interpreting fuzzy quantified statements.

Fuzzy Quantifiers

Fuzzy logic extends the notion of quantifier from Boolean logic and makes it possible to model quantifiers from the natural language (most of, at least half, few, around a dozen, etc). [START_REF] Zadeh | A computational approach to fuzzy quantifiers in natural languages[END_REF] distinguishes between absolute and relative fuzzy quantifiers. Absolute quantifiers refer to a number while relative ones refer to a proportion. An absolute quantifier Q is represented by a function µ Q from an integer range to [0, 1] whereas a relative quantifier is a mapping

µ Q from [0, 1] to [0, 1].
Quantifiers may also be increasing, as "at least half", or decreasing, as "at most three". According to [START_REF] Yager | On ordered weighted averaging aggregation operators in multicriteria decisionmaking[END_REF], when an increasing (proportional) quantifier is used, if all of the x i 's fully satisfy A, then the statement "Q X are A" is entirely true and if all of the x i 's do not satisfy A at all, then the statement "Q X are A" is entirely false. Moreover, the transition between those two extremes is continuous and monotonous. Therefore, when Q is increasing (e.g., "most", "at least a half"), function µ Q is increasing. Similarly, decreasing quantifiers (e.g., "at most two", "at most a half") are defined by decreasing functions.

Figure 2 gives two examples of monotonous decreasing and increasing fuzzy quantifiers respectively.

Calculating the truth degree of the statement "Q X are A" raises the problem of determining the cardinality of the set of elements from X which satisfy A. If A is a Boolean predicate, this cardinality is a precise integer (k) and then, the truth value of "Q X are A" is µ Q (k). If A is a fuzzy predicate, this cardinality cannot be established precisely and then, computing the quantification corresponds to establishing the value of function µ Q for an imprecise argument.

Interpretation of Fuzzy Quantified Statements

We now consider two proposals from the literature for interpreting quantified statements of the type "Q B X are A" (which generalizes the case "Q X are A" by considering that the set to which the quantifier applies is itself fuzzy) where X is a (crisp) referential and A and B are fuzzy predicates. Examples of such fuzzy statements are: "most of the close contributors of Mr. Brown are young", "more than two thirds of the papers authored by Mr. Smith have been published in journals with a high impact factor", etc.

Zadeh's Interpretation

Let X be the usual (crisp) set {x 1 , x 2 , . . ., x n } and n the cardinality of X. [START_REF] Zadeh | A computational approach to fuzzy quantifiers in natural languages[END_REF] defines the cardinality of the set of elements of X which satisfy A, as: Σcount(A) = n i=1 µ A (x i ). The truth degree of the statement "Q B X are A" (with Q relative) is then given by:

µ(Q B X are A) = µ Q Σcount(A ∩ B) Σcount(B) = µ Q x∈X (µ A (x), µ B (x)) x∈X µ B (x) (4)
where denotes a triangular norm (e.g., the minimum).

One may notice, however, that a large number of elements with a small degree µ A (x) has a same effect as a small number of elements with a high degree µ A (x), due to the definition of Σcount. This is a drawback inasmuch as one cannot distinguish between situations that are quite different.

Interpretation Based on the OWA Operator

In [START_REF] Yager | On ordered weighted averaging aggregation operators in multicriteria decisionmaking[END_REF], Yager suggests to compute the truth degree of statements of the form "Q B X are A" by an OWA aggregation of the implication values

µ B (x) → KD µ A (x) where → KD denotes Kleene-Dienes implication (a → KD b = max(1 -a, b)). Let X = {x 1 , . . . , x n } such that µ B (x 1 ) ≤ µ B (x 2 ) ≤ . . . ≤ µ B (x n ) and n i=1 µ B (x i ) = d.
The weights of the OWA operator are defined by: Finally:

w i = µ Q (S i )-µ Q (S i-1 ), with S i = i
µ(Q B X are A) = n i=1 w i × c i . (5)
It has been shown in [START_REF] Bosc | Quantified statements and database fuzzy querying[END_REF] that the OWA-based interpretation of a fuzzy quantified statement corresponds to using a Choquet integral. This interpretation, inasmuch as it does not use the Sigma-count, obviates the drawback mentioned at the end of the previous subsection.

Related Work

Fuzzy quantified queries have been studied by several authors in a relational database context. Kacprzyk and Zadrozny [START_REF] Kacprzyk | FQUERY III +:a "humanconsistent" database querying system based on fuzzy logic with linguistic quantifiers[END_REF] propose to use fuzzy quantifiers as connectives for combining atomic conditions, which makes it possible for instance to retrieve those items that satisfy most of the conditions among {c 1 , . . . , c n }. Bosc et al. [START_REF] Bosc | Quantified statements and database fuzzy querying[END_REF] suggest to use fuzzy quantifiers in a having clause in order to express a condition on the cardinality of a fuzzy subset of a group, as in "find the departments where most of the employees are well-paid ". Let us also mention that fuzzy quantified statements have also been used to model data summaries, see for instance [START_REF] Rasmussen | Summary SQL -A fuzzy tool for data mining[END_REF][START_REF] Kacprzyk | Linguistic database summaries and their protoforms: towards natural language based knowledge discovery tools[END_REF].

In a graph database context, beside data values, a new dimension can be exploited that concerns the structure of the graph. In [START_REF] Yager | Social network database querying based on computing with words[END_REF], Yager briefly mentions the possibility of using fuzzy quantified queries in a social network database context, such as the question of whether "most of the people residing in western countries have strong connections with each other" and suggests to interpret it using an OWA operator (cf. Subsection 3.2). However, the author does not propose any formal language for expressing such queries.

A first attempt to extend Cypher with fuzzy quantified queries -in the context of a regular (crisp) graph database -is described in [START_REF] Castelltort | Fuzzy queries over NoSQL graph databases: Perspectives for extending the cypher language[END_REF][START_REF] Castelltort | Extracting fuzzy summaries from NoSQL graph databases[END_REF]. The authors take as an example a graph database representing hotels and their customers and consider the following fuzzy quantified query:

1 match (c1:customer)-[:knows**almost3]->(c2:customer) return c1,c2
looking for pairs of customers linked through almost 3 hops. The syntax ** is used for indicating what the authors call a fuzzy linker. However, the interpretation of such queries is not formally given. The authors give a second example that involves the fuzzy concept popular applied to hotels. They assume that a hotel is popular if a large proportion of customers visited it. First, they consider a crisp interpretation of this concept (large being seen as equivalent to at least n) and recall how the corresponding query can be expressed in Cypher: where popular(cpt) > 0 return h which returns a fuzzy set of hotels.

In [START_REF] Castelltort | Extracting fuzzy summaries from NoSQL graph databases[END_REF], Castelltort and Laurent propose an approach aimed to summarize a (crisp) graph database by means of fuzzy quantified statements of the form Q X are A, in the same spirit as what Rasmussen and Yager [START_REF] Rasmussen | Summary SQL -A fuzzy tool for data mining[END_REF] did for relational databases. Again, they consider that the degree of truth of such a statement is obtained by a sigma-count (according to Zadeh's interpretation) and show how the corresponding queries can be expressed in Cypher. More precisely, given a graph database G and a summary S = a-[r ]->b, Q, the authors consider two degrees of truth of S in G defined by

truth 1 (S) = µ Q (count(distinct S)/count(distinct a)) and truth 2 (S) = µ Q (count(distinct S)/count(distinct a-[r ]->(?)))
where ? denotes an edge type variable and distinct(S) gives the number of times a relation of type r appears in the graph between a node of type a and a node of type b. They illustrate these notions using a database representing students who rent or own a house or an apartment. The degree of truth (in the sense of the second formula above) of the summary "S = student-[rent]->apartment, most" -meaning "most of the students rent an apartment" (as opposed to a house) -is given by the membership degree to the fuzzy quantifier most of the ratio: (number of times a relationship of type rents appears between a student and an apartment) over (number of relations of type rents starting from a student node). The corresponding Cypher query is: A limitation of this approach is that only the quantifier is fuzzy (whereas in general, in a fuzzy quantified statement of the form "Q B X are A", the predicates A and B may be fuzzy too).

Fuzzy Quantified Statements in FUDGE

In this section, we show how fuzzy quantified statements may be expressed in the fudge query language. We first propose a syntactic format for these queries, then we show how they can be efficiently evaluated in practice.

Syntax of a Fuzzy Quantified Query

In the following, we consider fuzzy quantified queries involving fuzzy predicates (beside the quantifier) over fuzzy graph databases. The fuzzy quanti-fied statements considered are of the form"Q B X are A", where the quantifier Q is represented by a fuzzy set and denotes either an increasing/decreasing relative quantifier (e.g., most) or an increasing/decreasing absolute one (e.g., at least three), where B is the fuzzy condition "to be connected (according to a given pattern) to a node x", X is the set of nodes in the graph and A is the fuzzy (possibly compound) condition.

An example of such a statement is: "most of the recent papers of which x is a main author, have been published in a renowned database journal".

The general syntactic form of a fuzzy quantified query of the form"Q B X are A" in the fudge language is given in Listing 3. This query contains a list of define clauses for the fuzzy quantifiers and the fuzzy terms declarations, a match clause for fuzzy graph pattern selection, a having clause for the fuzzy quantified statement definition and a return clause for specifying which elements should be returned in the result set. B(res, x) denotes the fuzzy graph pattern involving the nodes res and x and expressing the (possibly fuzzy) conditions in B. B(res, x) takes the form of a fuzzy graph pattern expressed à la Cypher by P B where C B , where P B is the "structural" part of B and C B its "value-based" part (made of conditions on node and edge variables, see Section 2.4). A(x) denotes the fuzzy graph pattern involving the node x and expressing the (possibly fuzzy) conditions in A. A(x) takes the form of a fuzzy graph pattern expressed à la Cypher by P A where C A .

Example 3. The query, denoted by Q mostAuthors , that consists in "retrieving every author (a) such that most of the recent papers (p) of which he/she is a main author, have been published in a renowned database journal (j)" may be expressed in fudge as follows (it is assumed that renowned means having a high impact factor): respectively. In this query, a corresponds to res, p corresponds to x, lines 3 and 4 correspond to B and lines 6 to 8 correspond to A.

According to the general syntax introduced in Listing 3, the variable a instantiates res and the variable p instantiates x.

Evaluation of a Fuzzy Quantified Query

From a conceptual point of view, the interpretation of such a query can be based on one of the formulas ( 4) and ( 5). Its evaluation involves three stages :

1. the compiling of the fuzzy quantified query Q into a crisp query denoted by Q derivedBoolean , 2. the interpretation of the crisp query Q derivedBoolean , 3. the calculation of the satisfaction degrees associated with the answers to Q derivedBoolean .

Compiling. The compiling stage translates the fuzzy quantified query Q into a crisp query denoted by Q derivedBoolean . This compilation involves two translation steps. First, Q is transcripted into a derived query Q derived that allows to interpret the fuzzy quantifier embedded in Q. The query Q derived , whose general form 1 is given in Listing 5, aims to retrieve the elements of the B part of the initial query, matching the variables res and x, for which we will then need to calculate the final satisfaction degree. It is obtained by removing the with and having clauses from the initial query and adding the optional match clause before the fuzzy graph pattern in condition A. The evaluation of Q derived is based on the derivation principle introduced by [START_REF] Pivert | Fuzzy Preference Queries to Relational Databases[END_REF] in the context of relational databases: Q derived is derived into another query denoted by Q derivedBoolean . The derivation translates the fuzzy query into a crisp one by transforming its fuzzy conditions into Boolean ones that select the support of the fuzzy statements. For instance, the fuzzy condition p.year IS recent where the fuzzy term recent is defined as defineAsc recent as (2013,2016) is derived into the Boolean condition p.year > 2013. Basically, the derivation principle relies on the following property:

x somewhat satisfies c 1 ∧ . . . ∧ c n ⇔ (x.A 1 ∈ support(c 1 ) ∧ . . . ∧ x.A n ∈ support(c n ))
where every c i denotes a fuzzy atomic condition and A i denotes the attribute concerned by c i . Listing 7 of Example 4 below is an illustration of the derivation of the query Q derived . The way the derivation principle can be applied to fudge queries is detailed in [START_REF] Pivert | Expression and efficient processing of fuzzy queries in a graph database context[END_REF].

1 Hereafter, the define clauses are omitted for the sake of simplicity.

Crisp interpretation. The previous compiling stage translates the fuzzy quantified query Q embedding fuzzy quantifiers and fuzzy conditions into a crisp query Q derivedBoolean , whose interpretation is the classical crisp one.

For the sake of simplicity, we consider in the following that the result of Q derived , denoted by Q derived , is made of the quadruples (res i , x i , µ Bi , µ Ai ) matching the query.

Final result calculation. The last stage of the evaluation calculates the satisfaction degrees µ B and µ A according to I B and I A . If the optional part does not match a given answer, then µ A = 0. The answers of the initial fuzzy quantified query Q (involving the fuzzy quantifier Q) are answers of the query Q derived derived from Q and the final satisfaction degree associated with each element e can be calculated according to the two different interpretations mentioned earlier in section 3. Hereafter, we illustrate this using [START_REF] Zadeh | A computational approach to fuzzy quantifiers in natural languages[END_REF] and [START_REF] Yager | On ordered weighted averaging aggregation operators in multicriteria decisionmaking[END_REF]'s approaches (which are the most commonly used). Following Zadeh's Sigma-count-based approach (cf. Subsection 3.2.1) we have:

µ(e) = µ Q {(res i , x i , µ B i , µ A i )∈ Q derived |res i =e} min(µ A i , µ B i ) {(res i , x i , µ B i , µ A i )∈ Q derived |res i =e} µ B i (6)
In this formula, µ A i and µ B i denote the satisfaction degrees of element res i = e with respect to conditions A and B respectively.

In the case of a fuzzy absolute quantified query, the final satisfaction degree associated with each element e is simply

µ(e) = µ Q   {(res i , x i , µ B i , µ A i )∈ Q derived |res i =e} µ A i   .
Example 4. Let us consider the fuzzy quantified query Q mostAuthors of Listing 4. We evaluate this query according to the fuzzy data graph DB of Figure 1. In order to interpret Q mostAuthors , we first derive the following query Q derived from Q mostAuthors , that retrieves "the authors (a) who are main authors of at least one recent paper (p) (corresponds to B(a,p) in lines 1 and 2) possibly (optional) published in a renowned database journal (corresponds to A(p) in lines 3 to 5)". This query returns a list of author (a) with their papers (p), satisfying the conditions of query Q derived , along with their respective satisfaction degrees. Here, µ B is min(µ strong (ρ author (a, p)), µ recent (p.year)) and µ A is µ high (i.value).

Let us consider Q derived (a) the set of answers of the query Q derived for a given author a. The set Q derived (a) provides a list of papers with their respective satisfaction degrees. This result set is of the form Q derived (a) = {((µ B , µ A )/p 1 ), ..., ((µ B , µ A )/p n )}.

For the running example, Q derived returns the four answers {Peter, Maria, Claudio, Michel}. The authors Andreas, Susan and Bazil do not belong to the result of Q mostAuthors because Susan has not written a journal paper yet and Andreas and Bazil do not have a recent paper.

For the running example, we then have Q derived (P eter) = {((0.2, 1)/IJAR14_p)}, Q derived (M aria) = {((0.33, 1)/IJAR14_p), ((0.6, 0.33)/IJIS16_p)}, Q derived (Claudio) = {((0.33, 1)/IJAR14_p), ((0.3, 0.07)/IJUFK15_p)} and Q derived (M ichel) = {((0.3, 0.07)/IJUFK15_p)}.

Finally, assuming for the sake of simplicity that µ most (x) = x, the final result of the query Q mostAuthors evaluated on DB using Formula 6 is {µ(Peter) = µ most ( 0.2 0.2 ) = 1, µ(Maria) = µ most ( 0.66 0.93 ) = 0.71, µ(Claudio) = µ most ( 0.4 0.63 ) = 0.63, µ(Michel) = µ most ( 0.07 0.3 ) = 0.23}. Using Yager's OWA-based approach (cf. subsection 3.2.2), for each element e returned by Q derived we calculate Then, with µ most (x) = x, we get µ Q (S IJAR14 ) = 0.35 and µ Q (S IJIS16 ) = 1. Therefore, the weights of the OWA operator are:

µ(e) = {(res i ,x i ,µ Bi ,µ Ai )∈ Q derived |res i =e} w i × c i . (7
W 1 = µ Q (S IJAR14 )-µ Q (S 0 ) = 0.35 and W 2 = µ Q (S IJIS16 )-µ Q (S IJAR14 ) = 0.65.
The implication values are: Lastly, the final result of the query Q mostAuthors evaluated on DB, given 440 by Formula 7, is: {µ(Peter) = 1, µ(Claudio) = 0.84, µ(Michel) = 0.7, µ(Maria) = 0.61}.

About Query Processing

For the implementation of these quantified queries, we updated the sugar software of [START_REF] Pivert | On a fuzzy algebra for querying graph databases[END_REF][START_REF] Pivert | SUGAR: A Graph Database Fuzzy Querying System[END_REF], which is a software add-on layer that implements the 445 fudge language over the Neo4j graph DBMS. This software efficiently evaluates fudge queries that contain fuzzy preferences, but its initial version did not support fuzzy quantified statements.

The current work is an extended version in which we also consider Yager's OWA-based interpretation (whereas the initial version only used Zadeh's interpretation) and a single fuzzy query is derived for the evaluation of the quantified query instead of three queries as initially proposed in [START_REF] Pivert | Fuzzy quantified structural queries to fuzzy graph databases[END_REF] where a fuzzy quantified query Q was indeed derived into:

• a first query Q 1 aimed to retrieve the elements matching the variable

x, for which one needs to calculate a satisfaction degree. Query Q 1 was obtained by removing the with and having clauses from the initial query;

• a second (resp. third) derived query, denoted by Q 2 (e) (resp. Q 3 (e)) aimed to compute the degree corresponding to the denominator (resp. numerator) of Equation 6for every element e returned by Q 1 .

The sugar software basically consists of two modules, which implement the Compiling and Final result calculation stages defined in Section 5.2. These modules interact with a Neo4j crisp engine, which implements the Crisp implementation stage defined in Section 5.2. Figure 4 illustrates this architecture.

In a pre-processing step, the Query compiler module produces (i) the query-dependent functions that allow us to compute µ B , µ A and µ, for each returned answer, according to the chosen interpretation, and (ii) the (crisp) Cypher query Q derivedBoolean , which is then sent to the crisp Neo4j engine for retrieving the information needed to calculate µ B and µ A .

In a post-processing step, the Score calculator module performs a grouping (according to the with clause of the initial query) of the elements, then calculates µ B , µ A and µ for each returned answer and finally ranks the answers.

Remark 1. Since repeating the definition of the fuzzy terms in every query may be tedious for the user, an improvement would be to store the definitions of the fuzzy terms in user profiles that could be accessed by the query compiler.

For quantified queries of the type introduced in the previous sections (i.e. using relative quantifiers), the principle is to first evaluate the fuzzy query Q derivedBoolean derived from the original query. For each element x ∈ Q derivedBoolean , we return the satisfaction degrees related to conditions A and B, denoted respectively by µ A and µ B . The final satisfaction degree µ can be calculated according to Formulas (4) or (5) using the values of µ B and µ A . [START_REF] Zadeh | A computational approach to fuzzy quantifiers in natural languages[END_REF]'s approach and [START_REF] Yager | On ordered weighted averaging aggregation operators in multicriteria decisionmaking[END_REF]'s OWA-based approach have been implemented and the choice of the interpretation to be used is made through the system configuration tool. Finally, a set of answers ranked in decreasing order of their satisfaction degree is returned.

As a proof-of-concept of the proposed approach, the fudge prototype is available at www-shaman.irisa.fr/fudge-prototype. A screenshot of this prototype is shown in Figure 5 which contains the final result of the evaluation of the query Q mostAuthors of Example 3. The GUI is composed of two frames i) a central frame for visualizing the graph and the results of a query and ii) an input field frame (placed under the central one), for entering and running a fudge query.

Experimental Results

In order to confirm the effectiveness and efficiency of the approach, we carried out some experiments on a computer running on Windows 7 (64 bits) with 8 Gb of RAM. We considered four queries (based on the topology of [START_REF] Angles | A comparison of current graph database models[END_REF]) with various forms of condition A. The third query Q 3 , where A is a reachability pattern involving regular simple path, aims to find the authors a such that most of the recent papers of which they are main authors, have been co-authored by another author b who is a contributor (not necessarily direct) of Claudio. The fourth query Q 4 is the query Q mostAuthors of Listing 4, where A is a pattern matching.

Our experiments have been performed on a graph database inspired from DBLP containing crisp (e.g., published) and fuzzy edges (e.g., contributor). Four database sizes have been considered, see Table 1.

The results of the processing of these queries over the datasets from Table 1 are depicted in Figure 6 where Figure 6.(a) (resp. Figure 6.(b)) presents the execution time in milliseconds using Zadeh's interpretation (resp. Yager's OWA-based interpretation).

A crucial result is that the processing time taken by the compiling step and the score calculation step, which are directly related to the introduction of flexibility into the query language, are very strongly dominated by the time taken by the crisp Cypher evaluator.

Moreover, the fudge compiling step takes so little time compared to the other two steps that it cannot even be seen in Figure 6. This time remains almost constant and is independent on the size of the dataset while slightly increasing in the presence of complex patterns or fuzzy conditions. As to the score calculation step, it represents around 9% of the time needed for evaluating a crisp Cypher query. The time used for calculating the final satisfaction degree is of course dependent on the size of the result set and the nature of the patterns.

Finally, these results show that introducing fuzzy quantified statements into a fudge query entails a reasonable increase of the overall processing time in the case of selection graph pattern queries. It represents, in the worst case, around 11% of the time needed for evaluating a fuzzy quantified fudge query. Of course, as can be seen in Figures 6.(a) and Figure 6.(b), some queries are more costly than others, depending on the complexity of the graph pattern involved, as in the Boolean case.

Conclusion and Perspectives

In this paper, we have dealt with a specific type of fuzzy quantified queries, addressed to fuzzy graph databases. We have defined the syntax and semantics of an extension of the query language Cypher that makes it possible to express and interpret such queries. A query processing strategy based on the derivation of non-quantified fuzzy queries has also been proposed. We performed some experiments in order to study its performances. The results of these experiments show that the cost of dealing with fuzzy quantification in a query is reasonable w.r.t. the cost of the overall evaluation.

As a future work, we first plan to study other types of fuzzy quantified queries. An example of a fuzzy quantified statement that is out of the scope of the present approach is "find the authors x that had papers published in most of the renowned database journals". More generally, it would be interesting to study fuzzy quantified queries that aim to find the nodes x such that x is connected (by a path) to Q nodes reachable by a given pattern and satisfying a given condition C.

Figure 1 :

 1 Figure 1: Fuzzy data graph DB

1 match 2 ( 3 ( 4 (

 1234 au2:author)-[:contributor+]->(au1:author), au1)-[:author_of]->(ar1:paper), (ar1)-[:published]->(j1), au1)-[:author_of]->(ar2:paper), (ar2)-[:published]->(j2) 5 where j1.name="IJWS12" and j1.name <> j2.name Listing 1: Pattern expressed à la Cypher

1 defineDesc 2 match 3 ( 4 ( 5 ( 6 (

 123456 short as[START_REF] Tahani | A conceptual framework for fuzzy query processing -A step toward very intelligent database systems[END_REF][START_REF] Bosc | Quantified statements and database fuzzy querying[END_REF], defineAsc high as (0.5,2) in au2:author)-[(contributor+)|Length is short]->(au1:author), au1)-[:author_of]->(ar1:paper), (ar1)-[:published]->(j1), au1)-[:author_of]->(ar2:paper), (ar2)-[:published]->(j2), j2)-[:impact_factor]->(i)7where j1.name="IJWS12" and j1.name <> j2.name and i.value is high Listing 2: A fudge query

Figure 2 :

 2 Figure 2: Quantifiers "at most 2" (left) and "at least 3" (right)

  j=1 µ B (x j ) d and S 0 = 0. The implication values are denoted by c i = max(1 -µ B(x) , µ A(x) ) and ordered decreasingly such that c 1 ≥ c 2 ≥ . . . ≥ c n .

1 match

 1 (c:customer)-[:visit]->(h:hotel) with h, count(*) as cpt 2 where cpt > n -1 return h Then, the authors switch to a fuzzy interpretation of the term popular and propose the expression: 1 match (c:customer)-[:visit]->(h:hotel) with h, count(*) as cpt2

1 MATCH 2 WITH 5 RETURN

 125 (s:student)-[rents]->(a:apartment) toFloat(count(*)) AS countS 3 MATCH (s1:student)-[rents]->(m) 4 WITH toFloat(count(*)) AS count2 MuMost(countS/count2)

1

  define... in 2 match B(res, x) 3 with res having Q(x) are A(x) 4 return res Listing 3: Syntax of a fuzzy quantified query

1 defineQrelativeAsc 2 defineAsc

 12 most as (0.3,0.8), defineAsc recent as (2013,2016), strong as (0,1), defineAsc high as (0.5,2) in 3 match (a:author)-[author_of|ST IS strong]->(p:paper) Membership function of most (d) Membership function of strong

Figure 3 : 7 ( 9 return a Listing 4 :

 3794 Figure 3: Membership functions

1

  match B(res, x) 2 optional match A(x) 3 return res x I A I B Listing 5: Derived query Q derived Such a query retrieves the pairs {res, x} that belong to the graph and all the information needed for the calculation of µ B and µ A , i.e., the combination of fuzzy degrees associated with relationships and node attribute values involved in B(res,x) and in A(x), respectively denoted by I B and I A . Listing 6 of Example 4 below presents the derived query associated with the query Q mostAuthors .

1

  match (a:author)-[author_of|ST IS strong]->(p:paper) 2 where p.year is recent 3 optional match (p)-[:published]->(j:journal), 4 (j)-[:impact_factor]->(i:impact_factor), (j)-[:domain]->(d:dom) 5 where i.value is high and d.name="database" 6 return a p µ A µ B Listing 6: Query Q derived derived from Q mostAuthors Then, we evaluate the Cypher query Q derivedBoolean given in Listing 7, derived from the fudge non-quantified query Q derived of Listing 6.

1

  MATCH fudge_p0 = (a:author)-[:author_of]->(p:paper) 2 WITH REDUCE(min=1.0, edge IN relationships(fudge_p0)| 3 case when edge.fdegree<min 4 then edge.fdegree else min end) AS fudge_p0,a AS a,p AS p 5 WHERE fudge_p0>0.0 AND p.year>2013 6 OPTIONAL MATCH (p)-[:published]->(j:journal), 7 (j)-[:impact_factor]->(i:impact_factor), (j)-[:domain]->(d:domain) 8 WHERE i.value>0.5 AND d.name='database' 9 return a p µ A µ B Listing 7: Query Q derivedBoolean derived from Q derived

c

  IJAR14 = max(1 -0.33, 1) = 1, and c IJIS16 = max(1 -0.6, 0.33) = 0.4. Thus, c 1 = 1 and c 2 = 0.4. Finally, we get: µ(Maria) = 0.35 × 1 + 0.65 × 0.4 = 0.35 + 0.26 = 0.61.

  calculator (computation of µ and ranking) Answers to Q ( Q ) sugar add-on pre-processing post-processing Classical Cypher querying

Figure 4 :

 4 Figure 4: sugar software architecture

1 match 2 where p.year is recent 3 with a 4 having 7 return a Listing 10 :

 1234710 (a:author)-[author_of|ST is strong]->(p:paper) most(p) are ( (b:author)-[:author_of]->(p), 5 (b)-[:contributor*]->(c:author) 6 where c.name="Claudio" ) Fuzzy quantified with reachability query Q 3

  Figure 6: Experimental results

)

  Example 5. In order to calculate µ(Maria) from Q derived , let us consider B (resp. A) the set of satisfaction degrees corresponding to condition B (resp. A) of element Maria as follows B={0.33/IJAR14, 0.6/IJIS16} and A={1/IJAR14, 0.33/IJIS16}. We have d = 0.93 and:

	S IJAR14 =	0.33 0.93	= 0.35, and S IJIS16 =	0.33 + 0.6 0.93	= 1.

Table 1 :

 1 Fuzzy graph datasets

		Dataset	Size
		DB 1	700 nodes & 1447 edges
		DB 2	2100 nodes & 4545 edges
		DB 3	3500 nodes & 7571 edges
		DB 4	4900 nodes & 10494 edges
	6	return a
		Listing 9: Fuzzy quantified query with reachability pattern Q 2

Acknowledgement: This work has been partially funded by the French DGE (Direction Générale des Entreprises) under the project ODIN.

The first query Q 1 , where A is an adjacency pattern, aims to find the authors such that most of the recent papers of which they are main authors, have been published in a journal. In the following, we omit the define clause of the queries in order to focus on the most interesting part. The second query Q 2 , where A is a reachability pattern involving fixed length path, aims to find the authors such that most of the recent papers of which they are main authors, have been published in a ranked journal.


Camille Lepaulard
email: lepaulard@cerege.fr

Jérôme Gattacceca

Minoru Uehara

Pierre Rochette

Yoann Quesnel

Robert J Macke

S J Walter Kiefer

A survey of the natural remanent magnetization and magnetic susceptibility of Apollo whole rocks

Keywords: Apollo samples, remanence, paleomagnetism, magnetic susceptibility

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The existence of past or present magnetic fields on the Moon was a matter of debate before lunar exploration started, as it was one of the key factors in distinguishing between a "hot Moon" versus a "cold Moon" model. Early orbital magnetic measurements by Luna 10 [START_REF] Dolginov | Measurements in the vicinity of the Moon by the artificial satellite Luna 10[END_REF] and Explorer 35 [START_REF] Sonett | The geomagnetic tail, topology, reconnection and interaction with the Moon[END_REF] satellites were only able to determine upper limits of the magnetic fields in the vicinity of the Moon, indicating that any lunar fields would be much weaker than that of the Earth, and suggesting a non-magnetic body. However, the Apollo orbital and surface magnetometers revealed patches of significant field due to crustal remanence. The paleomagnetic study of Apollo samples suggested the record of past strong (i.e., Earth-like) magnetic fields at the surface of the Moon (e.g., [START_REF] Fuller | Lunar paleomagnetism[END_REF], besides the particular difficulties linked with retrieving reliable paleointensities in metal-bearing rocks (e.g. [START_REF] Lawrence | Lunar paleointensity measurements: Implications for lunar magnetic evolution[END_REF].

Analysis of Apollo samples establishes constraints on the temporal evolution of the lunar magnetic field. This evolution reveals two main magnetic epochs: a high-field epoch with paleointensities of several tens of µT between about 4. 25 and3.56 Ga (Garrick-Bethell et al., 2009, 2017;Cournède et al., 2012;[START_REF] Shea | A long-lived lunar core dynamo[END_REF]Tikoo et al., 2012a;[START_REF] Suavet | Persistence and origin of the lunar core dynamo[END_REF]; followed by a weak-field epoch with paleointensities below 10 µT [START_REF] Tikoo | Decline of the lunar core dynamo[END_REF] and down to < 4 µT after 3.19 Ga [START_REF] Tikoo | Decline of the lunar core dynamo[END_REF][START_REF] Tikoo | A twobillion-year history for the lunar dynamo[END_REF][START_REF] Weiss | The lunar dynamo[END_REF][START_REF] Buz | Magnetism of a very young lunar glass[END_REF].

In part because the age distribution of Mare basalts peaks between 3.8 and 3.3 Ga [START_REF] Hiesinger | Ages and stratigraphy of lunar mare basalts: A synthesis[END_REF], the high-field epoch is rather well constrained, with around 60 studied samples (Cisowski et al., 1983;[START_REF] Weiss | The lunar dynamo[END_REF]. However, paleointensities up to 100 µT are at odds with dynamo models that predict much lower surface fields [START_REF] Dwyer | A long-lived lunar dynamo driven by continuous mechanical stirring[END_REF][START_REF] Bars | An impact-driven dynamo for the early Moon[END_REF][START_REF] Laneuville | A long-lived lunar dynamo powered by core crystallization[END_REF][START_REF] Evans | The case against an early lunar dynamo powered by core convection[END_REF]. Also, some samples may indicate paleointensities < 10 µT during this high-field epoch [START_REF] Fuller | Lunar paleomagnetism[END_REF], raising the possibility of a highly variable or intermittent dynamo [START_REF] Evans | The case against an early lunar dynamo powered by core convection[END_REF]. Further data from this high-field epoch is thus needed to strengthen the robustness of the high paleofields claims, and test the possibility of an intermittent or highly variable dynamo.

The weak-field epoch is defined as the period of decline of the field intensity, down to the present day situation with no internally generated field [START_REF] Fuller | Lunar paleomagnetism[END_REF][START_REF] Tikoo | Decline of the lunar core dynamo[END_REF]. This epoch is constrained by 30 paleointensity estimates (Cisowski et al., 1983;[START_REF] Weiss | The lunar dynamo[END_REF][START_REF] Tikoo | A twobillion-year history for the lunar dynamo[END_REF][START_REF] Wang | Lifetime of the lunar dynamo constrained by young Apollo returned breccias 15015 and 15465[END_REF]. Recent analyses of young samples reveal surface field intensities from 5 ± 2 µT (~2.5-1 Ga, [START_REF] Tikoo | A twobillion-year history for the lunar dynamo[END_REF] to <1 µT (~1 Ga, [START_REF] Wang | Lifetime of the lunar dynamo constrained by young Apollo returned breccias 15015 and 15465[END_REF]. These results could be explained by the persistence of a weak dynamo powered by core crystallization and mantle precession from 2.5 to 1 Ga [START_REF] Tikoo | A twobillion-year history for the lunar dynamo[END_REF].

Numerical modeling studies show that core convection could indeed generate surface fields of 2 µT until up to 200 Ma [START_REF] Evans | The case against an early lunar dynamo powered by core convection[END_REF]. The field evolution over the weak-field epoch is rather poorly defined, especially between 3 and 2 Ga, a time interval where both basalts and regolith breccias are rare [START_REF] Hiesinger | Ages and stratigraphy of lunar mare basalts: A synthesis[END_REF][START_REF] Fagan | Ages of globally distributed Lunar paleoregoliths and soils from 3.9 Ga to the present[END_REF].

So far, fewer than 70 small (usually well below 1 g) samples of Apollo rocks have been studied for paleomagnetism, compared to a total of 1402 rock samples available in the Apollo collection (this latter number is a personal communication by Ryan Zeigler). The aim of this study is to obtain a rough estimate of the paleomagnetic potential of a larger fraction of the Apollo rock collection (fraction that span the entire age range that the lunar dynamo has been previously investigated) using non-destructive methods. We measured the natural remanent magnetization (NRM) and the magnetic susceptibility (χ) of whole Apollo samples. These data increase the database of lunar rocks magnetic properties. The measurement of these two properties on the same sample also allows a qualitative assessment of the paleomagnetic potential of these samples. These methods, in addition to being nondestructive, are fast, easy to implement and allow us to measure whole samples. The NRM intensity is a crude indication of whether a rock can be targeted for paleomagnetic study. χ is a proxy for the metal content. The ratio between NRM and χ can be viewed as a crude proxy for the intensity of the paleofield recorded by a rock, enabling the selection of samples for detailed laboratory demagnetization follow-up studies that address specific questions (timing of the dynamo, maximum field recognized in the weak-field epoch, etc). A high NRM/χ may also be an indication of suitable paleomagnetic intrinsic properties for laboratory follow-up studies. Finally, these analyses increase the database of lunar rock magnetic properties for the purposes of modeling the lunar crustal magnetic field (e.g., [START_REF] Oliveira | Iron abundances in lunar impact basin melt sheets from orbital magnetic field data[END_REF].

Samples and methods

Samples

We studied 161 Apollo samples with a focus on large samples (mean mass 497 g, median mass 232 g, mass range 21 to 4688 g, see Table S1). This is much larger than the typical size of specimens used in most previous studies, which were well below 1g (see section S2 in Supplementary material). Of the 161 samples in this study, only 28 had been studied previously for remanent magnetization (samples in literature were measured using a squid magnetometer and with a mean mass of 1 g), and only 37 for magnetic properties. We measured the natural remanent magnetization of 123 samples, the magnetic susceptibility of 154 samples, and 119 samples for which both properties were measured. All specimens in this survey were single coherent bulk samples. We studied basalts, anorthosites, norites, and different types of breccias, spanning the Apollo 11, 12, 14, 15, 16 and 17 landing sites. In the following, a simplified lithological classification of lunar rocks is used [START_REF] Heiken | Lunar source book, a user's guide to the moon[END_REF]. Breccias are classified into three categories: impact melt breccia (IMB) with clasts set in a melt matrix, dimict breccia (DB) with two distinct lithologies (usually anorthosite and melt rock) and regolith breccia that represent lithified regolith. The other lithologies are impact melt rocks (IMR), basalts, and feldspathic rocks from the lunar highlands (norites, troctolites, anorthosites).

Methods

All measurements were performed directly in the Apollo storage vault at the Lunar Sample Laboratory Facility (LSLF) at Johnson Space Center (JSC), NASA, Houston. A specific susceptibility meter and a spinner magnetometer were used to be assembled in the LSLF and which satisfy curatorial constraints.

The samples were left in their original packaging for this study. Sample packaging varied from sample to sample. Samples kept in steel containers could not be studied. We measured those stored in Teflon bags, aluminum containers, or a combination of both. Prior to measurements, the samples were placed at the center of a hollow cubic sample holder made of acrylic plastic. They were maintained in place using padding material made of acrylic and Teflon.

The magnetometer and associated data processing methods are described in [START_REF] Uehara | A spinner magnetometer for large Apollo lunar samples[END_REF]. The errors associated with the NRM measurements provided in section S1, Table S1, are based on the departure of the measured signal from that of the signal from an ideal dipole [START_REF] Uehara | A spinner magnetometer for large Apollo lunar samples[END_REF]. We validated only samples that fit a dipole source to better than 20% (see discussion in [START_REF] Uehara | A spinner magnetometer for large Apollo lunar samples[END_REF]. We did not obtain a satisfactory measurement for 24 samples, either because the magnetization was too low or because the measured signal departed too much from the signal generated by a centered dipole (samples noted with † in the Table S1). These 24 samples range in mass from 24.5 to 1444 g and do not correlate with lithology or size. We did not measure NRMs for 14 samples that consisted of multiple small pieces in a single container (samples notedin Table S1 in Supplementary material).

The susceptibility meter is described in section S3 in Supplementary material. It consists of an acrylic cylindrical body with two inducing coils and a Hall probe to measure the sum of the alternating inducing and induced fields (Figures S2 and S3 in Supplementary materials).

The peak inducing field was 30 µT p-p , with absolutely no effect on the paleomagnetic signal, and its frequency was at 5 Hz. This low frequency limits eddy currents in metal and thus enables measurements of samples wrapped in aluminum foils or encapsulated in aluminum containers. This susceptibility meter can accommodate samples as large as 8000 cm 3 , corresponding to 20 x 20 x 20 cm sample capsule (which could correspond to a 4.5 kg mass sample). Measurements are based on the assumption that samples are perfectly centered in the instrument. The method of the calculation of the uncertainty on our magnetic susceptibility results is detailed in section S3 in Supplementary material. To limit errors from sample geometric offset and/or anisometric sample shape, we measured the susceptibility in two to three orthogonal positions and used the average result. Magnetic anisotropy of lunar rocks is at most a few percent (e.g., [START_REF] Rochette | Magnetic properties of lunar materials: Meteorites, Luna and Apollo returned samples[END_REF]Cournède et al. 2012) and is neglected in this study. The magnetic susceptibility for 41 samples was also measured using a commercial susceptibility meter MS2 from Bartington equipped with a MS2C coil. This coil, with internal diameter of 12 cm, is designed for measurement of cores and was calibrated for discrete samples against a KLY2 large volume kappabridge. The calibration and procedure is provided in section S4 in Supplementary material. Like during the calibration procedure, the measurements were performed along three orthogonal positions and averaged, not so much to account for the weak magnetic anisotropy of lunar rocks, but to account for the effect of sample anisometry on the measurements. For samples measured with both instruments (MS2 and our susceptibility meter), the results compare well, with of coefficient of correlation R 2 = 0.98 and a slope of 1.03 (see section S5 in Supplementary material).

Results and discussion

Magnetic properties and lithology

The NRM intensity spreads over two orders of magnitude (S1, Table S1) and shows a strong lithological dependence, in ascending order of NRM: feldspathic rocks, basalts, breccias and regolith breccia (Figure 1a, Table 1). This corroborates previous results, although on a much larger dataset. Magnetic susceptibility is also lithology-dependent, with impact-processed rocks being more magnetic (i.e., higher NRM and χ) (Figure 1b). This confirms that lunar impact processes are the main factors controlling the amount of ferromagnetic minerals in lunar rocks, with a magnetic enhancement due to incorporation of meteoritic metal (e.g., [START_REF] Rochette | Magnetic properties of lunar materials: Meteorites, Luna and Apollo returned samples[END_REF].

3.2.Comparison with other measurements

The NRM and the magnetic susceptibility values averaged by lithology are given in Table 1, both for this study and for literature data. The literature dataset consists of 109 NRM estimates (from 61 different Apollo rocks), and 105 susceptibility estimates (from 73 different Apollo rocks). A comparison of both datasets shows that the NRM intensities are larger by an average factor of ~2.7 in small sample studies (Figure 2a). This discrepancy (slope of 2.74 and R 2 =0.84) is evidence of the NRM directional heterogeneity within some lunar rocks and/or of possible magnetic contamination of small samples during preparation and handling.

The two datasets provide similar values for the susceptibility (R 2 = 0.90) with a slope of 0.66 (Figure 2b). Standard deviations in the literature data are generally higher than in ours, except for regolith breccias, indicating small-scale heterogeneities in metal content.

For the subset of samples that were previously studied in the literature (section S6, Table S2 in Supplementary material), we compare our NRM measurements on large samples with the laboratory measurements performed on small subsamples of the same rocks (Figure 3).

One sample, 15015 is not used due to anomalous result in literature (Table S2). The results of our bulk sample study show a R 2 = 0.52. It is noteworthy that when several measurements are available in the literature, there is a large scatter between measurements (with a relative variation from 6% to 163%, 56% in average). This scatter is not due to analytical uncertainties that are small compared to magnetic moments, both in this study and in previous studies. Indeed, basalts have a better correlation (R 2 =0.91) than impact melt breccias (R 2 =0.18). This is rather further evidence of small-scale heterogeneity of NRM, attributed to lithological heterogeneities (illustrated in section S2), with heterogeneous distribution of metallic iron, especially in breccias. It can indicate too the presence, in the same rock, of lithologies that have recorded contrasting paleofields. Overall, the NRM is higher for small samples than large samples (see also Table 1), indicating possible magnetic contamination during small samples preparation or handing, or heterogeneous directions of NRM within a large sample.

Our susceptibility data for the large samples and the small subsamples in the literature data (Figure 4) are also correlated (R 2 = 0.69). Impact melt breccia and regolith breccia, shows the best correlation with R 2 of 0.74 and 0.87, respectively. Regolith breccia 60255 and two impact melt breccia (72275 and 77135) are not used due to anomalous result in literature but these results are in Table S2. Feldspathic rocks show the largest discrepancy, our values being up to of 2 orders of magnitude larger than literature data. It is noteworthy that there is a large scatter in literature data (S5, in Table S2) for multiple measurements of subsamples from the same rock, especially for impact melt breccia and impact melt rocks with coefficient of variation of 43% and 59% respectively. Again, small-scale lithological heterogeneities can account for these variations, especially in smaller samples. For feldspathic rocks, the larger whole rock susceptibility may be due to the fact that these rocks often have a coating of melted or regolithic material and that this coating, enriched in metal, was avoided in the subsampling. As an example, samples 65315 and 60215 had diamagnetic χ values of -1.4 x 10 -9 m 3 .kg -1 and -5.4 × 10 -9 m 3 .kg -1 , respectively, in the small sample study of Cournède et al. (2012) and 20 x 10 -9 m 3 .kg -1 and 228 × 10 -9 m 3 .kg -1 , respectively, in this study.

In Figure 5, we compare our susceptibility results with measurements performed at JSC on another set of large samples (mean mass 19.8 g) from the returned sample collection [START_REF] Macke | New lunar sample density and magnetic susceptibility measurements[END_REF]. These measurements were performed using a SM30 instrument calibrated following [START_REF] Gattacceca | Calibration of in situ magnetic susceptibility measurements[END_REF] and are provided in section S7, Table S3. The coefficient of correlation of 0.70, together with a slope of 0.998 between the two independent dataset confirm that the scatter observed between our measurements and the small sample laboratory measurements is due to the intrinsic properties of lunar rocks (mostly small scale heterogeneities) rather than instrumental issues. Still returned samples yield on average lower values than the whole rocks, again suggesting the possible effect of regolith contamination of the surface of whole rocks. It is also possible that measurements using the SM30 contact probe are underestimated because they are sensitive to the surface roughness of the samples and were calibrated for smooth pebble shaped samples [START_REF] Gattacceca | Calibration of in situ magnetic susceptibility measurements[END_REF].

Paleomagnetic potential of this dataset

The ratio of NRM to magnetic susceptibility, noted ß hereafter, may convey information about lunar paleomagnetism. Indeed, high ß values may point to more single domain like behavior (for a given paleofield) or to higher paleofields (for a given magnetic mineralogy).

The paleointensity of 23 selected samples measured in this study have already been determined in previous laboratory studies (part S8, Table S4). ß appears broadly correlated with the paleointensity (Figure 6). However, a trend line does not intercept at the origin (B,ß)=(0,0). Indeed probably because of secondary magnetization components, hardly any samples have ß below 1. Qualitatively, we estimate that ß values below 4 have no real paleointensity significance. Selecting samples with ß ≥4, and forcing the x-intercept of the linear fitting to ß=4, we have a relation B = 9.7 * (ß-4) (R 2 =0.63) with B in µT and ß in A/m. This relation indicates that the overall ß dataset may be informative about the overall evolution of lunar paleointensities and allows using ß as a crude paleointensity proxy.

However, there are various reasons why ß may not be a good proxy for the paleointensity. First, the minerals that carry the magnetic susceptibility signal are not necessarily the same ones that carry the NRM. This may introduce a mineralogical bias in the interpretation. Unfortunately, the formation ages of the various lithologies are clustered in discrete and mostly non overlapping periods so that this lithological bias is difficult to circumvent. Second, it would be better to normalize NRM by ferromagnetic susceptibility only, since the paramagnetic minerals contribute to the susceptibility but not to the remanence. As we do not know the paramagnetic susceptiblity of most Apollo rocks, however, we chose not to apply a blanket correction to the susceptibility that may introduce an additional source of uncertainty. Third, we measured the total NRM of the samples without any magnetic cleaning. Therefore these rocks may be affected by secondary magnetizations (i.e., not acquired on the Moon) like viscous remanent magnetization (VRM) acquired during the ~50 years stay of the Apollo rocks in the geomagnetic field, but also by all magnetic contaminations that may have occurred at differents stages of the Apollo samples' history (during sampling on the Moon, in the return spacecraft, during cutting and processing at JSC).

Regarding VRM, it is noteworthy that the studied samples were kept in a mumetal container in ambient field < 200 nT for at least 24 hours before their NRM was measured, allowing for relaxation of about 50% of a 50 year VRM assuming a similar acquisition and decay rate for VRM. As an example we measured in the laboratory (CEREGE) the NRM and the magnetic susceptibility of a small piece of sample 14169 (73 mg). The NRM (6.45 x 10 -5 Am 2 kg -1 ) was measured using a SQUID cryogenic magnetometer and the susceptibility (2.34 x 10 -6 m 3 kg -1 ) using a MFK1 instrument. The NRM was then stepwise demagnetized using alternating fields (AF). This particular sample has a large secondary component (possibly a VRM), removed between 2 mT and 6 mT AF (Figure 7a). As a consequence ß decreases rapidly during AF demagnetization and reaches much lower values than that obtained on the whole rock at LSLF (Figure 7b). This ß value of 16.5, measured on a 72.4 g sample, is clearly dominated by the secondary component of the sample. It is therefore hardly related to the high coercivity component of magnetization which would relate to the lunar surface field during the formation of this rock on the Moon. This effect can enhance ß for some samples with no relation to the paleofield intensity. Finally the possibility that the large block magnetic susceptibility is biased toward higher value by surface contamination and coating by regolith material may add another bias since this contamination will increase the magnetic susceptibility but not necessarily the bulk NRM.

A plot of ß as a function of time reveals different behaviors over the high-field epoch (between 4.25 and 3.56 Ga in literature) and weak-field epoch (after ~3.5 Ga) discussed above (Figure 8). There is more scatter in the high-field epoch with 7/40 samples with high ß value (ß > 10 A/m, including samples 14053 and 14305 with ß > 20 A/m) between 4 and 3 Ga (Figure 8). However, 23 samples, accounting for 64% of samples in this period, have low ß values (ß < 5 A/m). A majority of samples have ß < 5 A/m in the weak-field epoch (18/48 samples higher than 5 A/m) and only 1 sample has ß > 20 A/m (sample 12010 with ß = 25.2 A/m). A median value computed on a sliding window of 15 ß values (grey line in Figure 8) agrees qualitatively with the existence of the high-field and weak-field epochs. It is noteworthy that a significant fraction of samples from the high-field epoch have low ß values (23/40 with ß < 5 A/m). They may correspond to samples with low paleointensities, suggesting that the average field in the high field epoch may have been overestimated in previous studies because of the focus on highly magnetized samples. Alternatively they could correspond to samples whose original NRM has been partially or totally erased by a later process. Shock in particular is able to remagnetize rocks and in view of the low efficiency of shock magnetization compared to TRM (e.g., [START_REF] Tikoo | Preservation and detectability of shock-induced magnetization[END_REF], the result of shock remagnetization would be a drastic decrease of ß regardless of the intensity of the lunar surface field at the time of impact. Other explanation for low ß could be post NRM acquisition "cold" brecciation (without resetting of radiometric ages but clast-scale randomization of the NRM) and recent contamination by metal rich regolith.

With all these caveats in mind, the dataset presented in Figure 8 allows selecting samples for laboratory studies addressing specific questions about the lunar surface field evolution.

Conclusions

Using two portable instruments specifically designed for this project, we obtained magnetic susceptibility and NRM measurements on 154 and 123 Apollo whole samples, respectively. This dataset represents a significant increase in number (+52% for NRM and +64% for susceptibility) and in mass (80 kg against 20.5 g for the literature dataset) of the existing database of lunar magnetic properties.

Our lithology-averaged NRM values may be more representative of the large-scale magnetic parameters than the values measured on small samples usually well below 1 gram that appears overestimated by an average factor of ~2.7. The NRM values obtained in this work are therefore more suitable to model and interpret the lunar magnetic field anomalies. Conversely, our magnetic susceptibility estimates may be overestimated for some weakly magnetic lithologies because of coating by glass or regolith material on bulk samples. Overall, magnetic susceptibility is strongly controlled by lithology and is clearly enhanced by impact processing and associated incorporation of meteoritic metal.

The ratio of natural remanence and magnetic susceptibility may be used as a crude paleointensity proxy. Its overall evolution with time is coherent with the existence of the high field and weak field epochs defined in the literature. However our data raise the possibility that the paleointensity during the high field epoch may have been more variable and lower on average than previously suggested. Although this ratio cannot provide robust paleointensities, our dataset represents a valid tool to select samples of interest for detailed paleomagnetic laboratory study. S2. S2, with all literature data. S2. S4. The right axis indicates the approximate paleointensity for ß > 4 (broken line, see section 3.3 and fig. 6). The dark line corresponds to the literature limit (at 3.5 Ga) between the high and the weak field epoch.

Figures and table captions

Figure 1

 1 Figure 1: a) Distribution of the NRM intensities of the large Apollo samples measured in this study, grouped by lithology. b) Distribution of the magnetic susceptibility of the large Apollo samples measured in this study, grouped by lithology. Numbers of analysed samples are indicated for each lithology (number for NRM, number for susceptibility). Note the non-linear scales for both NRM and susceptibility.

Figure 2 :

 2 Figure 2: a) NRM average values of small samples (literature data, with standard deviations) versus NRM average values of large samples (this study, with standard deviations) by lithology. b) Magnetic susceptibility average values of small samples (literature data, with standard deviations) versus magnetic susceptibility average values of large samples (this study, with standard deviations) by lithology. In both figures numbers in parentheses refer to the number of samples in our dataset and the literature dataset, respectively, and the light broken line is a 1:1 line. Literature data are listed in TableS2.

Figure 3 :

 3 Figure 3: NRM intensities measured on small samples (literature data) versus NRM values of large intensities measured on large samples of the same rocks (this study). Box symbols connected by vertical lines are for rocks with several small sub-samples in the literature dataset. The error bars are smaller than the symbol size. Sample 15015 is not plotted due to anomalous results in the literature, but it is listed in TableS2, with all literature data.

Figure 4 :

 4 Figure 4: Magnetic susceptibility of small samples (literature data) versus magnetic susceptibility values of large samples (this study, with error bars). Box symbols connected by vertical lines are for rocks with several sub-samples in the literature dataset. Samples 77135, 72275 and 60255 are not plotted due to anomalous results in the literature, but all literature data are listed in TableS2.

Figure 5 :

 5 Figure 5: Magnetic susceptibility data measured with our custom made susceptibility meter (average sample mass 1130 g for this subset of samples) versus magnetic susceptibility measured using a SM30 susceptibility meter on different samples (with average mass 19 g) of the same Apollo rocks. The dotted line is a linear fit between the two dataset.

Figure 6 :

 6 Figure 6: Paleointensity estimates (literature data, solid circles: paleointensity based on AF methods, open circles: paleointensity based on thermal methods) as a function of ß (with error bars). When several estimates are available the average paleointensity is used. When both AFand thermal-based results are available, the AF-based result is used, except for recent result on 15015 that was obtained using oxygen fugacity controm (Wang et al,. 2017). The single pTRM result of Brechet 1974 for 74275 was not used. The shaded area corresponds to ß <4 (see section 3.3). Sample 14305 was not included in this plot because of its peculiar magnetic mineralogy (NRM being carried by cohenite; e.g Cournède et al., 2012) that sets it apart from all other Apollo samples. Paleointensities results from literature are listed TableS4.

Figure 7 :

 7 Figure 7: Laboratory NRM measurements for Apollo rock 14169. a) Orthogonal projection plot for a 73 mg sub-sample. b) ß (A.m -1 ) as a function of alternating field. The dashed line is the ß value for the 72.4 g block measured at JSC.

Figure 8 :

 8 Figure 8: ß (with error bars) as a function of the age of studied rocks. Color-coding refers to lithology. The grey line corresponds to a sliding median, calculated on 15 successive ß values.

Table 1 :

 1 NRM and magnetic susceptibility sorted by lithology. N: number of reliable measurements (see text); No: number of samples studied in this work; S.D: standard deviation. University, CNRS, IRD, Coll. France, INRA, CEREGE, Aix-en-Provence, France 2 Vatican Observatory, Vatican City-State 3 Lunar and Planetary Institute, Houston, Texas, USA Corresponding author: Camille Lepaulard (lepaulard@cerege.fr) Highlights  We measured the natural remanence and magnetic susceptibility of 161 large (mean mass 500g) Apollo lunar samples in the LSLF (Houston)  Magnetic properties are primarily controlled by impact processing  The ratio of remanence to susceptibility can be used for sample selection for laboratory study, and is a crude proxy to paleointensity

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Acknowledgements

This work was funded by the Agence Nationale de la Recherche (project Maglune, ANR-14-CE33-0012) and the Programme National de Planétologie (INSU/CNES).

Physics of the Earth and Planetary Interiors

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