J Fliegans

O Tosoni

N M Dempsey

G Delette
email: gerard.delette@cea.fr

Modelling of demagnetization processes in permanent magnets measured in closed-circuit geometry

The hysteresis loops of nucleation-type magnets made of exchange-decoupled grains (i.e. sintered Nd-Fe-B magnets) reflect the discrete character of magnetization switching in such materials. Due to this discrete character, the experimental determination of coercivity depends on the measurement protocol. Finite element modelling allows to investigate how the pattern of reversed grains develops during sample demagnetization performed under closed-circuit conditions, provided that the basic features of the hysteresigraph are known. Numerical modelling provides a quantitative understanding of the collective effects which are very pronounced in the closed-circuit configuration and shows how they affect both the slope of the demagnetizing curve and the sample coercivity. With a grain coercive field standard deviation adjusted to 0.1 T, it is numerically found that the difference in coercivity between closed-and open-circuit configurations is 40 kA/m, in good agreement with previous experimental data.

Introduction

Demagnetization of hard ferromagnets is intricately dependent on their microstructure.

In high-performance sintered Nd-Fe-B magnets, grains are mostly exchange-decoupled and magnetization reversal spreads over the whole polycrystalline material by switching of individual grains. Basically, the polarization of a given grain is reversed when the local magnetic field exceeds the grain coercivity [START_REF] Soderžnik | Magnetization reversal of exchange-coupled and exchange-decoupled Nd-Fe-B magnets observed by magneto-optical Kerr effect microscopy[END_REF][START_REF] Dobrynin | Revisiting magnetization processes in granular hard magnetic materials[END_REF][START_REF] Dobrynin | Influence of dipolar collective effects on coercivity and demagnetizing factors in hard magnetic materials[END_REF][START_REF] Dobrynin | Experimental determination of the magnetization dependent part of the demagnetizing field in hard magnetic materials[END_REF][START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF]. As a consequence, a pattern of reversed and non-reversed grains that are magnetostatically coupled, develops as long as demagnetization proceeds. It has been proposed in a previous work [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF] that the discrete pattern should be governed by the minimization of the magnetostatic energy. Such effects are of importance since they could lead to some misinterpretation of the demagnetization curves usually measured on magnets for system design purposes. Differences in the coercivity values measured in open-circuit and closed-circuit geometries were attributed to protocol-dependent collective effects [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF]. In this work, a numerical simulation of the demagnetization of hard grains in 3D assemblies has been performed in order to analyze features of the reversal pattern and to assess the previous formalism. Open-circuit demagnetization has previously been simulated [START_REF] Dobrynin | Revisiting magnetization processes in granular hard magnetic materials[END_REF][START_REF] Dobrynin | Influence of dipolar collective effects on coercivity and demagnetizing factors in hard magnetic materials[END_REF][START_REF] Dobrynin | Experimental determination of the magnetization dependent part of the demagnetizing field in hard magnetic materials[END_REF][START_REF] Navau | Demagnetizing effects in granular hard magnetic bodies[END_REF]. The analytical expression of the field produced by a rectangular prism in free space is used in [START_REF] Dobrynin | Revisiting magnetization processes in granular hard magnetic materials[END_REF][START_REF] Dobrynin | Experimental determination of the magnetization dependent part of the demagnetizing field in hard magnetic materials[END_REF] to take into account the contribution of each grain, while the dipolar approximation is employed in [START_REF] Navau | Demagnetizing effects in granular hard magnetic bodies[END_REF]. A finite element computation of the magnetic field has been implemented here to extend the simulation to the closed-circuit geometry. Besides allowing a comparison of open-and closed-circuit loops, multi-grain simulation enables investigation of magnetostatic coupling effects in heterogeneous magnets with property gradients.

Numerical method and results

The polycrystalline magnet material is depicted in the model as a regular array of identical cubic grains of side length 10 µm. Each grain is assumed to be homogeneously magnetized along its easy axis and able to switch between the initial positive value +Ms to the negative value -Ms. Reversal occurs in a given grain when the projection of the local magnetic field H along the z axis, averaged over the grain volume, exceeds the coercive field assigned to the grain Hc. A Gaussian distribution of the coercivity is ascribed to the individual grains of the array. At the beginning of the simulation, all the grains are magnetized along the +z direction, yielding the remanent state reached after saturation. An external magnetic field is progressively applied on the magnet in the opposite direction by small increments H.

After each field increment, the reversal condition is tested on all grains. The grains for which the condition for reversal is fulfilled are identified and at the end of the screening, the magnetization of these grains is switched simultaneously, leading to a new grain reversal pattern. Before further increasing the field, several equilibrium sub-steps are performed: since each grain reversal changes the overall magnetostatic field, the screening of grains is repeated after each reversal pattern update until the establishment of a stable magnetization pattern. Then, the next step, with a new field increment, is performed considering the stabilized grain magnetization pattern.

The magnetostatic problem is solved by a Finite Element (FE) commercial software (Flux 3D, Altair) monitored by a specific Python script. Each grain is meshed with (444) quadratic cubic elements with size refinement near the edges and corners. It has been verified that this spatial discretization of the cubic grain leads to an accurate description of the demagnetizing field: the FE simulation of an open-circuit (oc) configuration provided results close to those previously obtained with an equivalent numerical simulation developed in [START_REF] Dobrynin | Revisiting magnetization processes in granular hard magnetic materials[END_REF]. The demagnetizing field was computed differently in [START_REF] Dobrynin | Revisiting magnetization processes in granular hard magnetic materials[END_REF] by summing the individual contribution of each grain given by an analytical expression derived by Aharoni [START_REF] Aharoni | Introduction to the Theory of Ferromagnetism[END_REF] and valid for isolated bodies. In [START_REF] Dobrynin | Revisiting magnetization processes in granular hard magnetic materials[END_REF] the summation of individual fields allows the demagnetizing field in arrays of saturated cubic grains to be accurately depicted. Considering the field in free space around each grain corresponds to the open circuit case but for closed-circuit (cc) configurations, the flux lines are canalized around the sample by the yoke of the hysteresigraph and then by the pole pieces that are made of a non-linear (saturable) magnetic material. The demagnetizing field in a given grain depends intricately on the neighboring grains and the surrounding magnetic circuit.

Accordingly, in this work FE computation replaces the analytical expression for calculating the magnetostatic field in closed-circuit. The 3D FE simulation takes into account the following features of the hysteresigraph: (i) the external field is created by a coil excited by an electric current and wound around a ferromagnetic armature, (ii) the flux is canalized towards the sample thanks to vertical arms and planar poles having a large surface area compared to the sample and (iii) the magnetic field is always measured in the mid-plane of the sample using a measuring coil. Although simple, this geometrical model gives a sufficient description of the hysteresigraph frame. In order to reduce the size of the model, three planes of symmetry (z=0, x=0 and y=0) are considered allowing only 1/8 of the total volume to be actually simulated. In the following, the simulations are labelled with the number of grains used in each direction of the reduced model, i.e. nx  ny  nz.

The closed-and open-circuit demagnetizing curves are plotted in Fig. 1 for an array of 10105 grains. This corresponds to the aspect ratio of Nd-Fe-B samples measured in our previous study [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF]. The mean value <Hc> and the standard deviation Hc are determined in such a way that the M(H) curves obtained after simulation of demagnetization in closed-circuit (red solid line) fit the experimental closed-circuit demagnetization curves (red dots). The closed-circuit demagnetizing curve is found to be more rectangular than the coercivity distribution, which is explained by the expansion scheme of reversal resulting from magnetostatic coupling. 3D grain reversal patterns plotted in Fig. 2 show that demagnetization starts from the least coercive grains and tends to form clusters of reversed grains. Such clusters expand by aggregating more and more grains as the demagnetizing field exerted by the clusters on the neighboring grains is stronger.

The open-circuit simulation exhibits different characteristics as seen in Fig. 2. In this case, reversal begins in the grains where the cumulative effect of a low coercivity and a high local demagnetization field is critical, i.e. where the quantity [Hc/Ms -Nz] is the lowest.

The last term Nz represents the local demagnetizing factor calculated at the first step when Hext = 0. Nz depends on the shape ratio of the magnet but it is not uniform in the magnet, as previously reported for rectangular samples [START_REF] Chen | Demagnetizing factors of rectangular prisms and ellipsoids[END_REF]. The open-circuit demagnetization curve evolves almost quasi-linearly with few expansion phases. The simulated pattern of grain reversal is notably different compared to closed-circuit and exhibits more diffuse structures and quasi-homogeneous distribution of reversed grains (cf. Fig. 2 for M/Ms = 0). It has to be pointed out that once the grain coercivity parameters were adjusted to account for the closed-circuit experimental curve, the simulated M(H) open-circuit curve followed the experimental one without any parameter adjustment. Interestingly, the difference in coercivity obtained numerically between open-and closed-circuit configurations (~40 kA/m) is close to the experimental value, but larger when the latter is corrected for viscosity effects (~25 kA/m from [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF]). Finally, it has been found that this difference increases (from ~40 kA/m to ~60 kA/m) when the standard deviation of the grain coercivity distribution rises (from 0.1 T to 0.2 T). All these characteristics will be discussed in the following section, by considering magnetostatic interactions between hard grains.

A third configuration has also been investigated to shed light on magnetostatic coupling effects. It consists of a "duplex" magnet made of a low coercivity Nd-Fe-B magnet (Hc = 1050 kA/m, 10 mm diameter and 5 mm height) sandwiched between two symmetrical Nd-Fe-B magnets with higher coercivity (Hc = 1500 kA/m, 10 mm diameter and 10 mm height). Both regions have the same remanence (1.3 T).

This stack is simulated in the closed-circuit testing condition using arrays with the same shape ratio as the individual magnets. Experimental and simulated demagnetization curves plotted in Fig. 3 show three main stages: (i) a plateau that extends up to the Hk value taken at the first 'knee' of the M(H) curve, resulting from a linear combination of the curves of the individual magnets, (ii) a quasi-linear decrease in M and (iii) an abrupt loss of magnetization ending at the coercivity of the "duplex" magnet. The simulated curve obtained with values of M and H extracted at the mid-plane, which corresponds to the position of the measuring coil, is in good agreement with the experimental one, contrary to the case where the simulated curve is plotted with the average values of M and H.

Grain reversal patterns reported in Fig. 3 indicate that the second stage (ii) can in turn be decomposed into two steps. In a first step (A-D), reversal begins in the less coercive central magnet, its evolution shows few rapid expansion steps and the demagnetizing curve is found to be broader than the experimental curves of the individual magnets.

Then, in a second step (E-F), reversal enters into the high coercivity region where grain switching occurs for field values lower than expected from the coercivity distribution. At the end of the second stage, expansion of clusters occurs in the more coercive magnets (F) and finally, the overall coercivity of the sample is lower than the value that would be obtained by a simple linear combination of the demagnetizing curves of all three magnets.

Discussion

A formalism for the demagnetizing field has been established [START_REF] Kronmüller | Theory of nucleation fields in inhomogeneous ferromagnets[END_REF] for polycrystalline hard magnets considering the contributions of: (i) the self-demagnetizing field of the magnet, (ii) the so-called cavity field and (iii) the grain self-demagnetizing field. The two first contributions depend on the demagnetized state of the magnet; the first one depends only on the average polarization <M> taken over the sample whereas the second one is intricately related to the magnetization pattern existing in the material. Since the grains are considered to be exchange-decoupled and made of a hard ferromagnetic phase, their polarization remains constant before reversal, as does the grain self-demagnetizing field (=-NgMs where Ng stands for the grain demagnetizing factor).

The cavity field reflects magnetostatic interactions between a given grain and the surrounding grains. In the previous study [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF], the cavity field was expressed as a linear combination of a "homogeneous" term, proportional to the average polarization M, and a "collective" term proportional to Ms, that accounts for the effect of the adjacent and nonreversed grains.

𝐻 𝐷 𝑐𝑎𝑣 = (1 -𝛼)𝛽 ℎ𝑜𝑚 𝑁 𝑔 𝑀 + 𝛼𝛽 𝑐𝑜𝑙 𝑁 𝑔 𝑀 𝑠 (1)
The parameter  gives the weight of the collective effects in the cavity field. Note that when the parameter is equal to 0 the formalism leads to the well-known "hard demag" correction. When tends to 1 the M(H) curve should become more rectangular in its shape since the cavity field is not dependent on M. The simulation in the closed-circuit condition clearly shows that when reversed grains form large clusters, this condition for  is fulfilled. This is a strong argument for a formalism involving collective effects.

Basically, the collective character means that the cavity field experienced by a nonreversed grain is a fraction of + NgMs.

The parameter  hom was inferred to be close to unity while the last term  col was assumed to be equal to 0.5 considering that, before its reversal, a grain is close to a boundary between two regions with opposite magnetization directions [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF]. This is again consistent with grain reversal patterns exhibiting large clusters oriented along the z-axis.

It was also previously demonstrated [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF] that collective effects have an influence on the slope of the demagnetizing curve M(H) (after the knee) and also on the coercive field: both effects have been experimentally observed. Basically, when the parameter increases, the slope of M(H) tends to increase, as reflected by the first right-hand term of (1).

Moreover, the second term, independent of M, which acts on the grain as a "magnetizing" field, increases at the same time. As a result, an increase in coercivity is expected. This last term is larger for the closed-circuit configuration, for which collective effects are maximal ( cc = 1), than for the open-circuit case, for which these effects were found to be weaker ( op = 0.82).

It has to be pointed out that this unique fitting parameter  op accounts for both the slope of the M(H) curve in open-circuit and the coercivity difference between open-and closedcircuit measurements, given by:

Δ𝐻 𝑐 = (𝛼 𝑜𝑝 -1)𝛽 𝑐𝑜𝑙 𝑁 𝑔 𝑀 𝑠 (2)
While this parameter value was initially fitted to experimental results from [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF], an identical value ( op = 0.82) has been obtained in this study without making any assumptions other than that concerning a dispersion in coercivity field values (Hc = 0.1 T). With this low standard deviation value,  op is found to be close to 1, which corresponds to a reversed grain structure that doesn't form a fully homogeneous pattern.

Actually, reversed grains tend to align into axially oriented one-grain-wide columns. This is explained by magnetostatic interactions between hard grains acting as follows: since the coercive field of grains are rather close in value, once a low coercivity grain reverses it increases locally the demagnetizing field felt by the upper and lower grains, by an amount that is sufficient to trigger reversal. It is worth noting that when the standard deviation is raised from 0.1 T to 0.2 T, the parameter  op is found to be lower (= 0.6), meaning a reduction of the weight of collective effects. Correspondingly, the simulation shows a more homogeneous pattern of grain reversal.

In the duplex magnet case simulated here in the closed-circuit configuration, it has been found that, during the first step (A-D), demagnetization spreads into the central magnet differently than it would for the collective scheme, which exhibits large clusters of grains. This is due to volume charges that accumulate at the interface between magnets where magnetization is discontinuous, which tend to promote multi-domain patterns.

Moreover, the magnetostatic field induced by the higher coercivity grains delays demagnetization of the central magnet. However, before expansion of reversal into the central region, the demagnetizing field felt by the higher coercivity grains located at the interface becomes strong enough to trigger reversal in the upper magnet, which in turn rapidly becomes dominated by collective effects.

Conclusions

The numerical simulation of hard ferromagnetic magnets composed of exchangedecoupled grains has been implemented in this study to provide a quantitative understanding of demagnetization curves. The findings apply mainly to sintered Nd2Fe14B-type materials which are the most common high-performance magnets used in industry. The granular nature of this material has been explicitly taken into account to describe the behavior of an assembly of about 1000 grains. The simulation is in very good agreement with the demagnetizing field formalism proposed previously [START_REF] Fliegans | Closed-Circuit Versus Open-Circuit Characterization of Hard Magnets[END_REF], in which the cavity field comprises a well-known homogeneous contribution and a second part reflecting collective effects. The latter is justified here by the development of large clusters of reversed grains, with such patterns being more prevalent in closed-circuit

Figure captions Fig. 1 :

 captions1 Figure captions

Fig. 2 :Fig. 3 :

 23 Fig. 2: D grain reversal patterns at selected demagnetization values simulated in closed-(up) and open-circuit (down) configurations (10x10x5 grains). The filled cubes refer to reversed grains. The 3 visible faces of the cubic boundary correspond to the 3 planes of symmetry.

Fig. 1 :

 1 Fig. 1: Demagnetizing curves: experimental (lines) and simulated (dots) in closed-(red) and open-circuit (black) configurations (10x10x5 grains).

Fig. 2 :

 2 Fig. 2: 3D grain reversal patterns at selected demagnetization values simulated in closed-(up) and open-circuit (down) configurations (10x10x5 grains). The filled cubes refer to reversed grains. The 3 visible faces of the cubic boundary correspond to the 3 planes of symmetry.

Fig. 3 :

 3 Fig. 3: Experimental demagnetizing curve (solid, red) measured by the hysteresigraph device (closed-circuit) and simulated demagnetizing curves for the duplex magnet (dotted points, red open circles = values at mid plane, black open squares = averaged values). The dotted lines refer to the experimental curves of individual magnets (green for high coercivity, blue for low coercivity) and the fine dotted line (black) represents the linear combination of the measured demagnetizing curves of individual magnets. Grain reversal patterns for specified field values labelled D-E-F in mid-plane with symmetries duplicated.

measurements than in open-circuit ones. Besides offering a quantitative explanation of the coercivity differences observed between open-and closed-circuit measurements, the model helps in understanding demagnetization processes in heterogeneous magnets.

Magnetostatic interactions between low and high coercivity zones, that exist for instance in large Dy-diffused magnets, may lead to a loss of squareness of M(H) curves measured in closed-circuit geometry. More generally, this study emphasizes the fact that the performance of a given magnet designed for an application should be evaluated by taking into account the real distribution of demagnetizing fields.


Lucie Cadorel
email: lucie.cadorel@inria.fr

Andrea G B Tettamanzi
email: andrea.tettamanzi@univ-cotedazur.fr

Mining RDF Data of COVID-19 Scientific Literature for Interesting Association Rules

In the context of the global effort to study, understand, and fight the new Coronavirus, prompted by the publication of a rich, reusable linked data containing named entities mentioned in the COVID-19 Open Research Dataset, a large corpus of scientific articles related to coronaviruses, we propose a method to discover interesting association rules from an RDF knowledge graph, by combining clustering, community detection, and dimensionality reduction, as well as criteria for filtering the discovered association rules in order to keep only the most interesting rules. Our results demonstrate the effectiveness and scalability of the proposed method and suggest several possible uses of the discovered rules, including (i) curating the knowledge graph by detecting errors, (ii) finding relevant and coherent collections of scientific articles, and (iii) suggesting novel hypotheses to biomedical researchers for further investigation.

Introduction

The COVID-19 pandemics has brought about an immediate and unprecedented response by the whole scientific community, beyond the disciplines of Biology and Medicine. Researchers from all over the world and from all disciplines have mobilized to contribute to a common effort to study, understand, and fight the new Coronavirus and its associated diseases and pandemics.

One of the major challenges in this context is to make sense of the huge scientific literature that has been (and is being) produced about this subject. The number of articles published about COVID-19 is so large that it would be impossible for any researcher to be aware of (let alone read) them all. This is one problem where knowledge representation and knowledge discovery have the potential to play an important role to support the effort of the scientific community.

The Wimmics team, 1 a joint research group of Inria and the I3S Laboratory, decided to join the global effort by starting the Covid-on-the-Web Project, 2 to publish, as thoroughly and quickly as possible, a maximum of rich and reusable linked data about the coronaviruses [START_REF] Michel | Covid-on-the-Web: Knowledge graph and services to advance COVID-19 research[END_REF].

1. https://team.inria.fr/wimmics/ 2. https://www.inria.fr/en/covid-web

The Covid-on-the-Web dataset 3 is based on the "COVID-19 Open Research Dataset" (CORD- [START_REF] De Padua | Preprocessing data sets for association rules using community detection and clustering: a comparative study[END_REF], which gathers more than 50,000 full-text scientific articles related to the coronaviruses. It consists of two main modules, namely the CORD-19 Named Entities Knowledge Graph, containing named entities mentioned in the CORD-19 corpus and linked to DBpedia, Wikidata, and other BioPortal vocabularies, and the CORD-19 Argumentative Knowledge Graph, a graph of arguments, conceptual structures linking claims to evidence supporting or attacking them. A third, crosssectional module describes in RDF the metadata and content of the CORD-19 articles.

While publishing a comprehensive knowledge graph of named entities mentioned in the scientific literature and linked to Linked Open Data vocabularies is already an invaluable tool to support the search of the literature and semantic information retrieval, we thought it would be even more interesting to use data mining methods to discover interesting knowledge hidden in this dataset. Such knowledge, in the form of multi-relational association rules, might further enrich the dataset and suggest novel hypotheses to biomedical researchers for further investigation.

Our main contributions may be summarized as follows:

• we propose a method for discovering association rules from an RDF dataset combining clustering, community detection, and dimensionality reduction;

• we propose a method for filtering the discovered association rules in order to keep the most "interesting" ones;

• we apply the proposed method on the CORD-19 Named Entities Knowledge Graph, showing its effectiveness as a technique to discover interesting hidden patterns that can be used to further enrich the dataset and suggest novel hypotheses for biomedical research.

The rest of the paper is organized as follows: Section 2 positions our contribution with respect to the literature; Section 3 provides a detailed description of the method we propose to mine the CORD-19 Named Entities Knowledge Graph for interesting association rules; Section 4 reports 3. https://github.com/Wimmics/CovidOnTheWeb the results of the experimental validation of the proposed approach and Section 5 draws some conclusions and outlines directions for further research.

Related Work

The growth of the semantic Web and of its most prominent implementation, the Linked Open Data (LOD), has made a huge number of interconnected RDF (Resource Definition Framework) triples, of the form (subject, predicate, object), which represent knowledge in a machineunderstandable format, freely available for sharing and reuse [START_REF] Bizer | Linked data -the story so far[END_REF]. LOD have thus become a giant real-world data resource that can be exploited for mining implicit knowledge, this opening up exciting new opportunities for data mining research [START_REF] Barati | Mining semantic association rules from RDF data[END_REF].

Finding association rules, also known as association rule mining, is an important data mining task. Since the seminal work on the Apriori algorithm [START_REF] Agrawal | Fast algorithms for mining association rules in large databases[END_REF], association rule mining has been a thriving field of research, which has contributed many effective techniques for detecting frequent patterns [START_REF] Han | Frequent pattern mining: current status and future directions[END_REF].

Several methods have been proposed in the literature for mining association rules from large RDF knowledge graphs. These methods can be divided into two broad categories: on the one hand, we find methods inspired by inductive logic programming [START_REF] Muggleton | ILP turns 20 -biography and future challenges[END_REF] or statistical relational learning [START_REF] Getoor | Introduction to Statistical Relational Learning[END_REF]; on the other hand, methods that follow the mainstream of association rule mining are adapted and applied to RDF graphs.

From the former category, we may mention AMIE [START_REF] Galárraga | AMIE: association rule mining under incomplete evidence in ontological knowledge bases[END_REF], AMIE+ [START_REF] Galárraga | Fast rule mining in ontological knowledge bases with AMIE+[END_REF], and their descendants or DL-Foil and DL-Focl [START_REF] Rizzo | Class expression induction as concept space exploration: From DL-Foil to DL-Focl[END_REF]. Somehow atypical, but very promising, are methods using evolutionary metaheuristics to drive the exploration of the hypothesis space [START_REF] Tran | An evolutionary algorithm for discovering multi-relational association rules in the semantic web[END_REF].

Example of the latter category are [START_REF] Völker | Statistical schema induction[END_REF], aimed specifically at ontology learning, [START_REF] Abedjan | Improving RDF data through association rule mining[END_REF], using association rule mining for RDF base enrichment, or [START_REF] Nebot | Finding association rules in semantic web data[END_REF], for mining RDFbased medical data. It is typical of these methods to generate transactions from an RDF graph using SPARQL queries and then use frequent pattern analysis algorithms to extract rules from those transactions. In addition, the idea is to exploit the knowledge embedded in the RDF graph at the schema level in order to add more semantics to the rules [START_REF] Barati | Mining semantic association rules from RDF data[END_REF]. Our contribution can be viewed as being part of this stream of research.

One of the open issues of association rule mining is that of measuring the interest of the discovered patterns, beyond straightforward interestingness measures like confidence or lift. As a matter of fact, obvious and trivial rules tend to have high support and confidence, but they do not teach us anything novel; to be subjectively "interesting" for an expert, a rule should be somehow surprising, unexpected, improbable. Some proposals to attack this problem are surveyed in [START_REF] Kontonasios | Knowledge discovery interestingness measures based on unexpectedness[END_REF].

An alternative approach that has been proposed in the literature is to look for associations by restricting one's attention to specific subsets of a dataset. Often, the aggregate behavior of a dataset may be very different from localized segments. In such cases, it is desirable to design algorithms which are effective at discovering localized associations, because they expose a customer pattern which is more specific than the aggregate behavior [START_REF] Aggarwal | Finding localized associations in market basket data[END_REF].

One way in which relevant localized segments can be automatically detected is by clustering the dataset using community detection algorithms [START_REF] Yang | A comparative analysis of community detection algorithms on artificial networks[END_REF]. To be sure, detecting communities in very large graphs (in our specific case, in a graph of named entity co-occurrences) requires scalable and efficient algorithms, like [START_REF] Pons | Computing communities in large networks using random walks[END_REF].

Discovering Interesting Association Rules

Association Rule Mining is often based on finding frequent patterns in a large transactional database by calculating support and then selecting them based on some interestingness measure, traditionally confidence. Support and confidence are defined as follows :

Supp(X → Y ) =
No. of transactions with both X and Y Tot. No. of transactions

= P(X ∩ Y ); (1)
Conf (X → Y ) =
No. of transactions with both X and Y No. of transaction with

X = P(Y | X). (2)
The limit of this approach is that the interesting rules with a low support are not discovered and the itemsets with a high support are over-represented. In the literature, one of the solutions proposed to overcome this issue is preprocessing the database. Accordingly, we tried different preprocessing methods and then we combined them.

Clustering and Community Detection

The first pre-processing idea consists of creating subgroups of transactions and/or itemsets and separately finding rules inside each group. Two methods have been tried: Clustering and Community Detection [START_REF] De Padua | Preprocessing data sets for association rules using community detection and clustering: a comparative study[END_REF].

3.1.1. Clustering. In a transactional database, two different types cluster analysis can be performed. The first one is clustering the transactions, whereas the second one is clustering the itemsets. We chose to try both approaches. In this section, we describe the method of clustering the transactions, which follows essentially the same approach used for clustering itemsets, except for the input data.

We used hierarchial agglomerative (i.e., bottom-up) clustering (HAC) to cluster the transactions. As we had more features about transactions, we added them in the input data and computed their one-hot encoding matrix. In our approach, it is required to have enough transactions in each clusters in order to obtain more local rules. To determine the optimal number of clusters, we applied the elbow method. The steps are described below:

1) Compute Similarity: similarity can be viewed as the inverse of a distance metric. It is used to combine similar elements or clusters. 2) HAC with Complete Linkage: the algorithm starts creating a cluster for each element and then it combines them into larger clusters (linkage) until all elements are in the same cluster. 3) Elbow Method: it consists of calculating the second derivative of the distance between clusters and taking the number of clusters where such second derivative is highest. As we wanted a large enough number of clusters, we added a further condition, which is that of having at least three clusters with a predefined minimum number of transactions in each cluster. Otherwise we take the next highest value of the second derivative, until the condition is satisfied. The clustering approach of the itemsets is similar, except that we do not have more features than the transactions in the one-hot encoding matrix.

Community Detection.

The second approach is Community Detection. Community Detection algorithms seek to gather nodes in a large network. The main idea that distinguishes community detection from clustering, is that the network structure is used to split the database. Thus, a co-occurrence matrix is used instead of a transactional matrix to represent the links.

Both Clustering and Community Detection suffer from the curse of dimensionality, that is why our next preprocessing step is dimensionality reduction, before splitting the database into sub-groups.

Dimensionality Reduction

The second pre-processing step we applied is dimensionality reduction, both for Clustering and Community Detection. The most well-known methods are linear transformations such as Principal Component Analysis [START_REF] Jolliffe | Principal Component Analysis[END_REF]. However, data do not always lie on a linear subspace, that is why methods like auto-encoders [START_REF] Kramer | Nonlinear principal component analysis using autoassociative neural networks[END_REF] are more powerful. We decided to implement this approach using neural networks.

The idea of Auto-encoders is to learn a latent-space representation thanks to the encoder part and reconstruct the data with the decoder. The results of the encoder part is our new input data.

We applied Auto-encoders on both one-hot encoding matrix and co-occurrences matrix. Then, we repeated the Clustering methods on both results of the encoder in order to get our subgroups.

Combining Methods

The final data pre-processing step is to combine different proposed methods before applying association rule mining algorithms. The idea is to mix them in different ways in order to find different rules.

For example, Algorithm 1 presents a model combining Community Detection and Clustering. First, we gather itemsets into communities. Then, inside each community, we create a new one-hot encoding of transactions and thus we cluster the transactions. Dimensionality reduction with auto-encoders is applied to both Community Detection and Clustering. Eventually, we find rules for each cluster in each community.

The other way round, clustering transactions first and then finding communities might be another possibility. Data: X co-occurrences matrix, Y one-hot encoding matrix

Initialisation : list R rules, CD communities, C clusters ← [ ] X ← Auto_encoders(X) CD ← Community_Detection( X) foreach community in CD do Select itemsets in community Y ← Y [. . . , itemsets] Ŷ ← Auto_encoders(Y ) C ← Clustering( Ŷ ) foreach cluster in C do Select transactions in cluster R ← Rules(Y [transactions, itemsets]) end end return the list of rules R Algorithm 1: Example of aggregated methods

Filtering Rules

While we focused on pre-processing methods, the postprocessing is also important to filter rules and keep the most interesting ones. We identified three filters that can help us keep the best rules.

1) Minimum Confidence: confidence can be viewed as the reliability of a rule. We chose to keep rules that have a confidence higher than a given threshold. 2) Interestingness Measure: we wanted to penalize rules whose antecedents and/or consequents are very frequent. In [START_REF] Spyropoulou | Interesting pattern mining in multi-relational data[END_REF], the authors define a measure (see Equation 3) based on the support of a rule. We kept the rules with a value of this measure higher than a given threshold. 3) Redundant Rules: we deleted all rules that comply with the following definition of redundancy:

A, B, C → D is redundant if Conf (A, B → D) ≥ Conf (A, B, C → D). Interestingness(X → Y ) = ( Supp(X → Y ) Supp(X) ) × ( Supp(X → Y ) Supp(Y ) ) × (1 - Supp(X → Y ) Tot. No. of transactions ) (3)

Experiments and Results

RDF Dataset and Evaluation Metrics

The Covid-on-the-Web Dataset [START_REF]Covid-on-the-web dataset[END_REF] As our work is focused on discovering association rules between named entities, we queried the dataset's SPARQL endpoint to get articles and named entities but also other metadata related to the articles such as publication, date, or authors. We used a sample of 27,636 articles and 6,398 named entities. We split the dataset into a training and a test set, with 22,000 articles (80%) and 5,636 articles (20%), respectively.

The named entities have been first identified in the abstracts of the articles and then in their bodies. We decided to use the named entities in the articles' abstracts to discover rules because of the smaller number of NEs. Also, abstracts summarize all the important ideas of the article and, thus, are expected to mention the most important entities. Nevertheless, the named entities in the bodies remain interesting in order to evaluate our model. We tried to predict NEs in the bodies from association rules found in the abstracts. Then we compared the predicted named entities with the named entities extracted in the body by Entity-fishing. We finally computed the precision of our predictions, defined as

P recision =
Number of predicted NEs truly in the body Total number of predicted NE .

(4) Algorithm 2 describes the method we followed to evaluate our model.

Data Pre-Processing

Before applying our methodology, we prepared the data to be more informative. As NEs are textual data, we first chose to apply lowercasing, removed punctuation and NEs which only consisted of numbers. Also, we decided to remove NEs according to their frequency. Thus, we kept NEs occurring in more than 5 articles and removed the 15 most frequent NEs (Figure 2). The second step was to create new data used as input for Clustering and Community Detection algorithms. We generated a one-hot encoding matrix and a co-occurrences matrix. From the co-occurrences matrix we implemented a graph structure.

Finally, we chose to aggregate different algorithms and, after some preliminary experiments, we kept the following five models:

• AE + C is a clustering on the articles with an auto-encoder to reduce dimensionality. Indeed, applying clustering without dimensionality reduction was computationally too slow.

• WT is a Community Detection algorithm called Walk Trap [START_REF] Pons | Computing communities in large networks using random walks[END_REF]. It is a vertices based algorithm using a hierarchical clustering. The idea is to apply a random walk to compute a probability to go from node i to node j in t steps. Afterwards, a distance is calculated thanks to the probabilities in order to apply HAC.

• AE + CD is also a Community Detection algorithm. In this approach, we used an auto-encoder on the co-occurrences matrix to get latent variables. We applied HAC to this new variables to find NEs communities.

• WT + (AE + C) is a combination between Community Detection and Clustering. First, we applied Walktrap algorithm to find communities. Then, for each community, we took their NEs and applied auto-encoder and clustering on the articles having those NEs.

• AE + CD + (AE + C) is the same as WT + (AE + C), except that it uses the AE + CD Community Detection approach.

We compared the above five models to a baseline consisting of applying the association rule mining algorithm directly to the data, without any pre-processing.

Finally, we set up hyperparameters for the different approaches and the association rules algorithm.

Cosine similarity, computed as

Cosine(X, Y ) = a (a + b)(a + c) 2 , ( 5
)
where

• a = (X = 1 ∩ Y = 1), • b = (X = 1 ∩ Y = 0), • c = (X = 0 ∩ Y = 1),
measures the cosine angle between two vectors projected in a multi-dimensional space. Since we created a multidimensional latent space with auto-encoders, this metric is more advantageous than other metrics such as Euclidean distance.

The hyperbolic tangent activation function, defined as

tanh(X) = 2 1 -e -2x -1, (6)
is a non-linear function whose shape is sigmoidal. The advantage of this function is its output range between [-1, 1]. Thus, the negative inputs will be mapped to strongly negative outputs and positive inputs will be mapped to strongly positive outputs, which is useful to differentiate data.

Results

To determine the most effective model, we analyzed three metrics which determine interest, novelty and predictive power of the rules.

The comparison of the number of rules before and after filtering, presented in Table 1, highlights how many rules are not considered as interesting. For example, our baseline found 14,233 rules before filtering, which is the largest number among all approaches. However, those rules are mainly redundant or uninteresting according to our criteria. Thus, only 1.7% of those rules have been kept after filtering. On the contrary, methods AE + C found fewer rules (6,409), which however turned out to be less redundant, since 17% of them have been kept. All our combining methods get more interesting rules compared to the baseline. The second evaluation consists of a two-by-two comparison of the number of rules that are in both models as presented in Table 2. Thanks to this metric, we analyzed if a model is able to create novelty by providing rules which are not found in another model. We noticed that the Community Detection approaches (AE + CD and WT) yield rules very similar to those found by the baseline. The AE + CD method even has 100% of its rules that have been already found with the baseline. Thus, this method does not bring out novelty. On the other hand, we noticed that the rules of the Clustering method AE + C yield more novelty since they are totally different from the other models and more particularly from the baseline (only 2,6% of common rules). We finally noticed that the last two approaches, and more precisely AE + CD + (AE + C), are similar neither to the baseline (10,1%), nor to AE + C (22%). Therefore, we can conclude that the combination of Community Detection and Clustering is able to generate new interesting rules that we have not found with the other models.

Finally, we evaluated the different models by predicting NEs that may be in the body according to the extracted NEs and our association rules. For instance, if the NE fever has been extracted in the body and is also found as an antecedent in our rules, then the consequent NEs of fever, such as cough, are predicted as being also NEs in the body. The precision helps us to determine how many NEs have been correctly predicted. The first precision, presented in Table 3, represents the percentage of predicted NEs that have been actually extracted in the body. The second one removes the predicted NEs that are already in the abstract. Indeed, most NEs in the abstract are obviously in the body. However, predicting NEs truly in the body that are not in the abstract could be a hallmark of interesting and non-trivial rules. Table 3 shows that AE + CD method outperforms the baseline and Community Detection approaches. The combinations of Community Detection algorithms and Clustering give also a better precision than our baseline. If we combine all the rules generated by all the model, we obviously reach a higher precision mostly due to AE + C and AE + CD + (AE + C) models. To conclude this evaluation, the Clustering model AE + C gives the best results in terms of number of interesting rules, novelty and precision. Nevertheless, the other rules generated by the combination of Community Detection and Clustering also bring some novelty, which could be useful to find surprising rules for the users.

Discussion of Results

The proposed methodology applied to the CORD-19 Named Entities Knowledge Graph allowed us to enrich the dataset. In this section, we give some details on three applications of the discovered rules.

First, the discovered rules allowed us to detect some errors in the extraction of named entities by Entity-Fishing. Most of the errors are due to wrongly-associated acronyms.

For example, we found nokia n95 associated to personal protective equipment in a rule. Since the topics of all the papers in the CORD-19 base are related to biomedical research, the correct named entity is obviouslsy mask n95, instead of nokia n95. Table 4 presents some of the errors discovered this way in the dataset.

The second application could be finding relevant and coherent collections of articles in the database. Indeed, our clustering methods yielded groups of articles that turned out to deal with the same topic. For example, several clusters gather NEs related to one disease or a specific organ. This is the case of clusters number 4, 5, and 7, which mention NEs about COVID-19, Ebola, and the central nervous system, respectively. We also detected more biological clusters about proteins or enzymes such as cluster number 15.

We could also gather named entities, since we have computed communities. Finally, Table 6 presents some rules generated by our methodology. The rules give information about disease symptoms as well as biology or Web actors. For example, we found the COVID-19 symptoms such as fever, dyspnea and cough but also the hot topics such as surgical mask and respirator or cruise ship and Diamond Princess. Then, we discovered associations between geographic locations and diseases. The NE People's Republic of China associated with pneumonia has a high probability of co-occurring with Wuhan. On the other hand, African countries associated with Ebola virus are often linked with other African countries. In terms of biology, some associations refer to proteins or enzymes whereas others link diseases together. Finally, we found associations between Web companies such as Facebook and Twitter or two biological databases PubMed and Embase. The rules are diverse and provide information not only on the medical or biological aspect but also on the social or economic one.

Conclusion

In this paper, we have proposed a method combining clustering, community detection, and dimensionality reduction for discovering association rules from an RDF knowledge graph. We have also proposed a post-processing method to filter the association rules according to their interestingness. We have presented five different models applied to the CORD-19 Named Entities Knowledge Graph.

The experimental results provide evidence supporting the effectiveness of the proposed method. We have demonstrated that clustering and community detection help to reduce the number of rules and to detect weaker (i.e., lowersupport) but more interesting hidden patterns. The evaluation method showed that the clustering method combined with auto-encoders gives the best results in terms of precision.

Nevertheless, adding community detection to clustering generates novelty in the rules.

The application of the discovered rules allowed us to clean the database by finding errors among extracted named entities. As a useful byproduct of our method, we have also found relevant clusters and communities that might ease the navigation of the knowledge graph. We finally discovered rules covering various topics that give us insight to add to the database and might suggest novel hypotheses to biomedical researchers for further investigation.

Even though this first association rules analysis has provided interesting results, the proposed methodology could be improved. We have not yet applied the method on the other named entities linked to DBpedia and Bioportal: adding these and other resources will certainly boost the number and quality of the discovered rules. The validation of the rules by a medical expert is also an important test to assess the reliability of our rules. As future work, the visualisation of our rules may play a key role to make the search of information in scientific literature more interactive and leverage it.

Figure 1 .

 1 Figure 1. Overall methodology

  Figure 1 presents all the possible combinations.

Figure 2 .

 2 Figure 2. A histogram showing the distribution of the 15 most frequent named entities.

• 20 •

 20 AE + C -Number of hidden layers : 4 -Activation function : tanh -Number of output variables : 128 -Similarity of HAC : Cosine -Maximum number of clusters : 30 • AE + CD -Number of hidden layers : 4 -Activation function: tanh -Number of output variables: 128 -Similarity of HAC : Cosine -Maximum number of communities : 20 • WT + (AE + C) -Number of hidden layers : 2 -Activation function : tanh -Number of output variables : 32 -Similarity : Cosine -Maximum number of clusters : 20 • AE + CD + (AE + C) -Number of hidden layers : 4 (AE + CD) and 2 (AE + C) -Activation function : tanh -Number of output variables : 128 (AE + CD) and 32 (AE + C) -Similarity : Cosine -Maximum number of clusters : Association Rule Mining -Algorithm : Frequent Pattern Growth Algorithm -Maximum length : 3 -Minimum support : the tuples must be at least in 5 articles -Minimum confidence : 0.7 -Minimum interestingness : 0.3

  is an RDF dataset composed of several distinct graphs, including the CORD-19 Named Entities Knowledge Graph. This graph gathers Data: T test set, R discovered rules A ← Extract_NEs_from_abstracts(T ) B ← Extract_NEs_from_bodies(T ) foreach article in T do list P ← [ ] (the list of predicted NEs) foreach antecedent in R do if antecedent in A[article, :] then NEs) identified in scientific articles and linked to DBpedia, Wikidata, and other BioPortal vocabularies. We chose to use named entities identified and disambiguated by Entity-Fishing and linked to Wikidata.

	P .add(consequent)
	end
	Num_NE ← sum_same_NE(P , B[article, :])
	precision ← N um_N E length(P )
	end
	end
	avg_precision ← Average(precision)
	return avg_precision, the average precision
	Algorithm 2: Model Evaluation
	named entities (

TABLE 2 .

 2 COMPARISON OF NUMBER OF COMMON RULES.

		Baseline	AE + C	WT	AE + CD	WT + Clust	AE + CD + (AE + C)
	Baseline
	AE + C	28 (2,6%)
	WT	193 (49,7%)	46 (11,9%)
	AE + CD	127 (100%)	16 (12,6%)	111 (87,4%)
	WT + (AE +C)	25 (7%)	123 (34%)	45(12,5%)	19 (5,3%)
	AE + CD + (AE + C)	41 (10,1%)	89 (22%)	47 (11,6%)	41 (10,1%)	66 (16,3%)

TABLE 1 .

 1 NUMBER OF RULES BY MODEL.

		Number of rules	Number of rules
	Model	before filtering	after filtering
	Baseline	14,233	251 (1.7%)
	AE + C	6409	1091 (17%)
	WT	10,050	388 (3.8%)
	AE + CD	2264	127 (5.6%)
	WT + (AE +C)	3419	361 (10.5%)
	AE + CD + (AE + C)	3380	405 (12%)

TABLE 3 .

 3 PRECISION EVALUATION

	Model	Body Precision	Body \ Abstract Precision
	Baseline	4%	1%
	AE + C	34%	20%
	WT	7%	2%
	AE + CD	3%	1%
	WT + (AE + C)	24%	14%
	AE + CD + (AE + C)	26%	16%
	All	39%	24%

TABLE 4 .

 4 DETECTED ERRORS IN THE DATASET

	Error	Acronym	Associated Named Entities	Correct Named Entity
	nokia n95	n95	personal protective equipment	mask n95
	íþróttabandalag vestmannaeyja	IBV	avian infectious bronchitis, respiratory tract,	Infectious bronchitis virus
			chicken
	a59 road	a59	glycoprotein	Mouse hepatitis virus A59
	international federation of basque pelota	FIPV	feline infectious peritonitis,	Feline Infectious Peritonitis Virus
			coronavirus, transmissible gastroenteritis virus
	new international version	NIV	henipavirus, vaccine, malaysia	Nipah Virus

TABLE 5 .

 5 EXAMPLE OF CLUSTERS

	Cluster	Named Entities
	Cluster #4	hubei, wuhan, cough, fever, pneumonia,
	COVID-19	people's republic of china, elisa,
		reverse genetics
	Cluster #5	guinea, liberia, sierra leone,
	Ebola	ebola virus disease, crab-eating macaque,
		filoviridae
	Cluster #7	myelin, multiple sclerosis, spinal cord,
	Central nervous system	white blood cell, microglia, CXCL8
	Cluster #15	eif2ak3, atf6, eif2ak2, kinase, ern1,
	Proteins	genotype, serotype

TABLE 6 .

 6 EXAMPLE OF RULES

	Antecedents	Consequents
	fever, dyspnea	cough
	runny nose	cough
	anxiety	mental depression
	surgical mask	respirator
	cruise ship	diamond princess
	exponential growth	basic reproduction
		number
	liberia, western african ebola virus	guinea
	epidemic
	people's republic of china, pneumonia	wuhan
	camelus, middle east respiratory	arabian peninsula
	syndrome coronavirus
	poultry, people's	influenza a virus
	republic of china	subtype h7n9
	tnf, cytokine	il10
	eif2ak3, eif2ak2	atf6
	p38 mitogen-activated protein	sb203580
	kinases, pyrazolanthrone
	methyl, cholesterol	cyclodextrin
	etiology, vasculitis	kawasaki disease
	steroid, magnetic-resonance imaging	osteonecrosis
	hepatitis, liver cirrhosis	hepatocellular carcinoma
	pubmed	embase
	facebook	twitter

Acknowledgments

This work has been supported by the French government, through the 3IA Côte d'Azur "Investments in the Future" project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.