Kazuki Ashihara

Cheikh Brahim

El Vaigh

Chenhui Chu
email: chu@ids.osaka-u.ac.jp

Benjamin Renoust
email: renoust@ids.osaka-u.ac.jp

Noriko Okubo

Noriko Takemura
email: takemura@ids.osaka-u.ac.jp

Yuta Nakashima

Hajime Nagahara
email: nagahara@ids.osaka-u.ac.jp

Improving topic modeling through homophily for legal documents

Keywords: homophily network, topic modeling, legal documents

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

RESEARCH

Improving Topic Modeling through Homophily for Legal Documents

Kazuki Ashihara 1 , Cheikh Brahim El Vaigh 2 , Chenhui Chu 3* , Benjamin Renoust 3 , Noriko Okubo 4 , Noriko Takemura 3 , Yuta Nakashima 3 and Hajime Nagahara 3 1 Introduction

Computational law applies quantitative computational methods to the study of laws, intersecting various research fields such as natural language processing, network science, machine learning, statistical methods, and sociophysics [START_REF] Katz | What is Computation Legal Studies? University of Houston[END_REF]. In the last decade, the interests of studying laws from the perspective of computational social science have significantly increased [START_REF] Lettieri | Computational social science and its potential impact upon law[END_REF]. It has been proven that no matter the country an efficient approach for searching patterns in citation analysis of legal documents will always become a complex network analysis study [START_REF] Fowler | Network analysis and the law: Measuring the legal importance of precedents at the us supreme court[END_REF][START_REF] Kim | The emergent network structure of the multilateral environmental agreement system[END_REF][START_REF] Pelc | The politics of precedent in international law: A social network application[END_REF][START_REF] Koniaris | Network analysis in the legal domain: A complex model for european union legal sources[END_REF][START_REF] Khanam | Application of network analysis for finding relatedness among legal documents by using case citation data[END_REF][START_REF] Lettieri | Network, visualization, analytics. a tool allowing legal scholars to experimentally investigate eu case law[END_REF][START_REF] Lee | Network structure reveals patterns of legal complexity in human society: The case of the constitutional legal network[END_REF].

In computational law, topic modeling is one key task for many higher-level goal such as law document retrieval, law document clustering for exploration, or the comparison of legal documents [START_REF] Katz | Legal n-grams? a simple approach to track the 'evolution'of legal language[END_REF][START_REF] Lu | Legal document clustering with built-in topic segmentation[END_REF][START_REF] O'neill | An analysis of topic modelling for legislative texts[END_REF][START_REF] Wang | Topic model based text similarity measure for chinese judgment document[END_REF][START_REF] Yoshioka | overview of japanese statute law retrieval and entailment task at coliee-2018[END_REF]. To date, LDA is still the favored model for topic modelling, but does not take into account the larger context of laws and cited cases. However, ambiguity may still rise in the topics because law is a specific domain: different real-world contexts can lead to the breach of a same law, while a same real-world context can breach different law cases -meaning that the relevance of the modeled topics strongly depends on the legal context they are used in [START_REF] Kanapala | Text summarization from legal documents: a survey[END_REF]. In the case of judgement cases, the citations to preexisting cases and laws are as important as the content to render a decision -similarly to scientific work.

Academic evaluation has frequently used citation analysis [START_REF] Hirsch | An index to quantify an individual's scientific research output[END_REF], despite the controversies that have been shown even recently [START_REF] Renoust | Multiplex flows in citation networks[END_REF]. Using scientific citation networks have also improved the quality of topic modeling [START_REF] Chang | Relational Topic Models for Document Networks[END_REF]. In comparison to academic citation networks, the structure of legal cases often follows a directed acyclic network (or DAG). The structure of these DAG may however differ. While academic citation networks are often deep with many authors citing one another, and even self citations, the structure of legal citations may look more shallow. This results in a flat organization with little cross references between cases but a clear emphasis in precedent cases making jurisprudence [START_REF] Pelc | The politics of precedent in international law: A social network application[END_REF]. As a consequence, the very shape of the networks built from legal citations shows a different topology to those built from academic citations.

In addition, the content of the cited documents is not always available, network analysis then relies on the investigation of the co-citation patterns [START_REF] Kim | The emergent network structure of the multilateral environmental agreement system[END_REF][START_REF] Khanam | Application of network analysis for finding relatedness among legal documents by using case citation data[END_REF]. Sometimes, this is referred to as homophily [START_REF] Mcpherson | Birds of a feather: Homophily in social networks[END_REF][START_REF] Borgatti | Network analysis in the social sciences[END_REF], also known as the property of entities to agglomerate when being similar and the implied similarity of two entities. Homophily is always corresponding to a bipartite structure, which can be projected into a single type network. Topic modeling can be one approach to use homophily in the projections [START_REF] Renoust | Entanglement in multiplex networks: understanding group cohesion in homophily networks[END_REF]. In the context of computational law, retrieval and topic modeling give rise to open challenges and publicly datasets such as the COLIEE data [START_REF] Yoshioka | overview of japanese statute law retrieval and entailment task at coliee-2018[END_REF]. The COLIEE dataset provides a testbed for legal information extraction and entailment. It provided over 6k cases from the Canadian Federal Court for about 40 years, with very rich annotations including among a lot of different entities, citations to past cases, rulings, and laws.

Our work contributes in a methodology for building topic modeling for legal documents, when the content of cited documents is not available. We propose in the current work to automatically build networks from cases of the Canadian court. The collection presents thousands of cases, and details two major types of citations. The first one refers to prior cases, while the other one to statute laws. We further propose to use these two types of citations to explore the similarity of cases, by constructing homophily relationships between cases. Furthermore, we use case homophily to improve topic modeling for legal cases. In particular, we work on the relational topic model (RTM) [START_REF] Chang | Relational Topic Models for Document Networks[END_REF] that uses the links between documents during topic modeling. We analyze a publicly available dataset part of the COLIEE challenge. We then construct a homophily network consisting of nodes for legal cases and edges with weights for the references. We compare different strategies of using the edge weights in the homophily network as link information for RTM. This work is an invited extension of the original presentation [START_REF] Ashihara | Legal information as a complex network: Improving topic modeling through homophily[END_REF]. In this paper, we extend our previous work as follows:

• We improve the strategy of using homophily relationships for topic modeling.

Previously, we only set a threshold based on edge weights in the homophily network to decide whether a link should be used for RTM or not, which we call the cutting model. In this paper, we also propose to use the weights to weight the link probability in RTM, which we call the weighting model. Experiments show that our weighting model significantly outperforms our previous cutting model. • In addition to the product or the sum of prior case and status information, we propose a list of weighting methods with fuzzy logic aggregation [START_REF] Detyniecki | Mathematical aggregation operators and their application to video querying[END_REF] for the weighting model, showing similar coherence score to the simple weighting model and better coherence scores than previous cutting model. • We also investigate the use of a kernel such as Node2vec [START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF] to embed homophily network in low-dimension space. Experiments show that our Node2vec model, also significantly outperforms our previous cutting model.

• We analyze the topic words in detail for the best performing topic model, and verify the effectiveness of the proposed models for legal case topic modeling. The remaining of the paper is as follows: Section 2 presents the related work; Section 3 introduces the COLIEE dataset [START_REF] Yoshioka | overview of japanese statute law retrieval and entailment task at coliee-2018[END_REF] along with its characteristics; Section 4 presents homophily network modeling for this dataset. Section 5 describes topic modeling for legal cases and our proposed models for improving RTM using homophily networks; We report experiments and results in Sections 6 and conclude the paper in Section 7.

Related Work

In the current section, we present the related work to both legislation networks and topic modeling.

Legislation Networks

The interest in network analysis for legal documents has been significantly increasing recently. Many previous studies have shown that analyzing legal networks is closely related to complex networks [START_REF] Fowler | Network analysis and the law: Measuring the legal importance of precedents at the us supreme court[END_REF][START_REF] Kim | The emergent network structure of the multilateral environmental agreement system[END_REF][START_REF] Pelc | The politics of precedent in international law: A social network application[END_REF][START_REF] Koniaris | Network analysis in the legal domain: A complex model for european union legal sources[END_REF][START_REF] Khanam | Application of network analysis for finding relatedness among legal documents by using case citation data[END_REF][START_REF] Lettieri | Network, visualization, analytics. a tool allowing legal scholars to experimentally investigate eu case law[END_REF][START_REF] Lee | Network structure reveals patterns of legal complexity in human society: The case of the constitutional legal network[END_REF]. Fowler et al. [START_REF] Fowler | Network analysis and the law: Measuring the legal importance of precedents at the us supreme court[END_REF] developed a centrality measure based on Authorities and Hubs [START_REF] Kleinberg | Authoritative sources in a hyperlinked environment[END_REF], which is dedicated to citations of cases in the US Supreme Court consisting of 26k+ cases in a citation network. In order to find complex network properties and homophily behavior in a treaty network, Kim [START_REF] Kim | The emergent network structure of the multilateral environmental agreement system[END_REF] explored a structure consisting of 1k citations for 747 treaties. Pelc [START_REF] Pelc | The politics of precedent in international law: A social network application[END_REF] investigated the fundamental precedent concept, i.e., previous deliberations being cited in cases, in the international commercial cases. They also did a centrality study of Authorities and Hubs, which confirms that the network structure is relevant to predicting case output. From the Official Journal of the European Union, Koniaris [START_REF] Koniaris | Network analysis in the legal domain: A complex model for european union legal sources[END_REF] built a law reference network. They showed that it has the property of multilayer complex networks. With betweenness centrality, Khanam et al. [START_REF] Khanam | Application of network analysis for finding relatedness among legal documents by using case citation data[END_REF] proposed an analysis on citation for judgements in Indian courts. The relevance of the EUCaseNet project [START_REF] Lettieri | Network, visualization, analytics. a tool allowing legal scholars to experimentally investigate eu case law[END_REF] should also be underlined. It combines network analysis and centrality-based visualization to explore the entire EU case law corpus. Lee et al. [START_REF] Lee | Network structure reveals patterns of legal complexity in human society: The case of the constitutional legal network[END_REF] explored the court decision vs. constitution article patterns in Korea, which conducts topic analysis on the main clusters. Because in this paper we investigate the the Federal Court of Canada case law network, our target is close to these studies. We investigate the homophily in our analysis, which has been illustrated by all of the studies above. Different from previous studies, we improve topic modeling with the case co-citation structure, and feed back to homophily of documents in ways of topic proximity.

Topic Modeling

Latent Dirichlet allocation (LDA) is the first topic model introduced by Blei et al. [START_REF] Blei | Latent Dirichlet Allocation[END_REF]. As a graphical model, LDA can learn from observed documents to infer hidden word and document-topic distributions. In Section 5, we give a description of LDA in detail. The correlated topic model, proposed by Blei and Lafferty [START_REF] Blei | A correlated topic model of science[END_REF], models topic occurrences using the logistic normal for LDA. The dynamic topic model proposed by Blei et al. [START_REF] Blei | Dynamic topic models[END_REF] models temporal information in sequence data. Most topic models are unsupervised, but supervised topic modeling has been studied too. The supervised LDA proposed by Blei et al. [START_REF] Blei | Supervised topic models[END_REF] can model topics of responses and documents. Supervised LDA is suitable for data such as product reviews, which has both evaluation scores and corresponding descriptions of products. Ideal point topic models, proposed by Nguyen et al. [START_REF] Nguyen | Tea party in the house: A hierarchical ideal point topic model and its application to republican legislators in the 112th congress[END_REF], assume that the responses are also hidden. RTM models the topic of a document pair, which shares links, e.g. references, between a document pair [START_REF] Chang | Relational Topic Models for Document Networks[END_REF]. We describe RTM in detail in Section 5. Collaborative topic models proposed by Wang and Blei [START_REF] Wang | Collaborative topic modeling for recommending scientific articles[END_REF], can make recommendation for user preferences using user data.

Recent studies have also tried to bridge topic models to text representation methods based on word embeddings. E.g., Das et al. [START_REF] Das | Gaussian LDA for topic models with word embeddings[END_REF] modeled topics with distributions on word embeddings instead of word types and showed that the proposed model is more robust for handling out-of-vocabulary words; Similarly, Dieng et al. [START_REF] Dieng | Topic modeling in embedding spaces[END_REF] developed an embedded topic model that models words with categorical distributions on word embeddings and topic embeddings. We think that this can be one interesting direction for our future work.

In the context of joint network and topic modeling, Liu et al. [START_REF] Liu | Topic-link lda: Joint models of topic and author community[END_REF] proposed a framework to perform LDA based topic modeling and author community discovery simultaneously; Zhu et al. [START_REF] Zhu | Scalable text and link analysis with mixed-topic link models[END_REF] proposed a mixed-topic link model for joint topic modeling and link prediction; Brochier et al. [START_REF] Brochier | Inductive Document Network Embedding with Topic-Word Attention[END_REF] proposed a topic-word attention mechanism to generate document network embeddings via the interaction between topic and word embeddings. Different from previous studies, in this paper, we apply RTM for legal case analysis and improve it via co-citation homophily networks.

Data

We collect our data from the Competition on Legal Information Extraction/Entailment (COLIEE 2018) [START_REF] Yoshioka | overview of japanese statute law retrieval and entailment task at coliee-2018[END_REF]. [1] For our task we study the Case Law Competition Data Corpus, which has also been used in the Task 1 and 2 in COLIEE 2018. The data consists of 6,154 cases from the Federal Court of Canada for about 40 years, ranging between 1974 and 2016. Note that most cases in the corpus are the ones with a date after 1986. This data corpus is very rich. Each case is a textual document [1] https://sites.ualberta.ca/ ~miyoung2/COLIEE2018/ containing multiple parts, including a summary of the court, case content, references to relevant past cases and statutes, rulings, counsels, legal topics of interest, solicitors, miscellaneous information, and important facts.

In this paper, we only focus on the prior case and statutes noticed to form our networks. From the text input, they are divided by paragraph titles as follows:

• Cases Noticed, they are corresponding to the past trials which are relevant to this trial. • Statutes Noticed, they are corresponding laws referred to give the verdict of the trial. Each consecutive line has one reference. Recall that as they are Canadian cases, they may be written in both English and French. We found that there are only 5,576 cases that refer to prior cases and statutes noticed among all the cases.

The reference destination is always in detail, and references can be separated into paragraphs or chapters. If we directly use these as basic units for analyzing network modeling, only a small number of references are redundant across the cases making the network very sparse. Therefore, we treat the references to full articles of cases or statutes. The identification of each case or statute can be made based on a year, a title, and references. The parsing is conducted based on looking for the year structure, and make titles at a high granularity as nodes (Table 1). The cases without information about the year are discarded. In total, these correspond to 39 cases being cited. We also save the year information along with the nodes.

Legal Network

In this section, we first describe the used network model in Sec. 4.1. Second we provide the details about the construction of the underlying homophily network in Sec. 4.2. Finally, in Sec. 4.3 we lay out on the use of Node2vec to embed the homophily network in a low dimensional space.

Network Structure

In our data, each case refers to a set of prior cases and statutes noticed. In our network, G = (V, E), every case is represented as a node v 1 ∈ V , and a prior case or statute noticed is represented as another node v 2 ∈ V . We treat each citation as a link (v 1 , v 2 ) = e ∈ E. Figure 1 shows an overview of our network modeling.

Our initial set contains |C| = 5, 539 cases. Each case can refer to multiple prior cases and statutes noticed. Each case c ∈ C may refer to several prior cases p ∈ P , where |P | = 25, 112 in total. They also can have reference to a statute s ∈ S, where |S| = 1, 288. The citations to cases can be from the 18th century. Note that it cannot be guaranteed that the year information is the reliable for cases before the 18th century, concerning a small number of 78 cases.

!"#$"%&'()( *'()( +,',-,)( ! . ! / ! 0 ! 1 ! 2 * . * / * 0 * 1 * 2 + . + / + 0 + 1 + 2
Figure 1 An overview of the structure of the legal case data in the COLIEE dataset. With the above network modeling, our network includes 31,976 nodes, and 53,554 links. Note that reliable year information is only available for 29,319 nodes, where year information is unavailable for 1,224 status nodes. We can separate the network into two sub-networks. The former one G P is with only the cases and their cited prior cases, consisting of 29,952 nodes with 53,554 edges. The later one G S only considers the cases and their cited statutes, consisting of 4,441 nodes with 6,453 edges.

These two networks present one main connected component G consisting of 30,456 nodes with 52,453 links, which covers most nodes and edges. That number for G P is 27,353/44,871 , and 4,125/6,150 for G S . We further investigate the possibility of looking for case communities from these networks via Louvain clustering [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] and modularity [START_REF] Newman | Modularity and community structure in networks[END_REF]. The main components of G, G P , and G S show a modularity Q G = 0.739 with 34 communities, a modularity Q G P = 0.762 for 45 communities, and a modularity Q G S = 0.747 for 27 communities, respectively. This is illustrated in Figure 2.

If the communities were extremely imbalanced -consider an extremely large community surrounded by very small others -there would be more chances to be unsuccessful looking for homophily because most documents would have higher chances to share the same few common characteristics. Finding community structures within the citation network confirms that we can leverage on these structures by using homophily. In other words, there are groups of documents that share latent characteristics, and that may be differentiated enough to other groups.

Homophily Network

Prior case and statute citations indicate double bipartite structures in our network model, i.e., from G P are case-prior case, and from G S are case-statute relation-ships. Bipartite projections into one-mode networks indicate a complex network structure [START_REF] Guillaume | Bipartite graphs as models of complex networks[END_REF]. Thus, we further derive three one-mode networks: G P = (C, E P ), G S = (C, E S ), and G = (C, E ), in order to analyze homophily.

Homophily is the property for entities that are linked together to share some existing characteristics, that may also be shared among a group or a cluster. Although this may be captured through an entity-characteristics association, naturally forming bipartite networks, other models, including multilayer networks and hypergraphs could also be investigated [START_REF] Renoust | Analysis and visualisation of edge entanglement in multiplex networks[END_REF]. In the context of homophily, multilayer networks and bipartite graphs have equivalence [START_REF] Renoust | Voisinage et intrication dans les réseaux multiplexes[END_REF], and there exist the incidence/Levi graph for hypergraphs [START_REF] Levi | Finite Geometrical Systems: Six Public Lectues Delivered in February[END_REF]. All may be projected into 1-mode networks connecting the entities which may be linked by homophily. This single mode projection is the artifact which enables us to alleviate the limitation of not having access to the cited content but still embedding the network structure within our topic modeling.

To this end, we project the other two bipartite relationships onto case-case relationships. That is, let u ∈ C, v ∈ C be two initial cases that we are investigating. We can assign a set of references R u = {r 1 , r 2 , r 3 , . . .} to each of these cases, where r x represents either a prior case or a statute noticed. In a projected network G , the original cases {u, v} ⊆ C are the nodes, if and only if the intersection of their respective reference sets is non empty, there is a link (u, v) = e ∈ E : R u ∩ R v = ∅. Each reference r x ∈ P is a prior case in the network induced by prior cases G P . Each reference r x ∈ S is a statute law in the network induced by statutes G S . A reference r x ∈ S ∪ P can be of either a case or a statute noticed in the general projected network G .

We can obtain the weight of each link with the following two methods. The first one w n is the number of shared citations between two cases. The second one w j is the Jaccard index of these cases [START_REF] Jaccard | Étude comparative de la distribution florale dans une portion des alpes et des jura[END_REF].

w n = |R u ∩ R v | and w j = w n |R u ∪ R v | (1)
We find that the resulting networks are very dense, where the numbers of nodes and edges are 4,803/286,435 for G P , 3,138/379,447 for G S , and 5,576/643,729 for G , respectively. We can see that there is only a little overlap between links induced by prior cases and statutes. After investigating the main components of these networks, we get a size of 4,244/286,403, 3,033/379,426, and 4,870/643,725 nodes and edges for G P , G S and G , respectively. G P has modularity Q G P = 0.428 for 14 communities, G S has modularity Q G S = 0.542 for 13 communities, and G has modularity Q G = 0.502 for 7 communities. Figure 3 visualizes these networks with their communities.

Embedding of Homophily Network with Node2vec

Homophily networks can be embedded in a low dimensional space using kernels such as Node2vec [START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF]. Node2vec aims at embedding networks in low dimensional representations while preserving their properties such as node neighborhood, roles, or communities based on homophily. Node2vec is based on the model skip-grams [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF] used for words embedding leveraging their contexts. To compute node embedding, node2vec operates the model skip-grams over random walks in the KG, allowing it to represent the neighbourhood and the overall position of each node in the graph. Our aims is to embed the homophily networks and exploit the semantic of their embeddings, as provided by Node2vec. Thus we can obtain the weight of each link by computing the cosine similarity between the two nodes of the link under-consideration. For a link l = (n i , n j ), one can compute the weight

w s = sim(v ni , v nj ), (2)
where v x is the embedding of the node x provided by the Node2vec embedding, [2] and sim(•, •) is the cosine similarity. Using cosine similarity to weight links in the network is more general and robust, as this weight incorporates the similarity of two nodes and also the similarity of their neighborhood. The final Node2vec embedding of the case and status homophily networks are shown in Figure 4 using UMAP (Uniform Manifold Approximation and Projection for Dimension Reduction) [START_REF] Mcinnes | Umap: Uniform manifold approximation and projection for dimension reduction[END_REF] a dimension reduction technique that can be used for visualisation similarly to t-SNE. Node2vec embeddings preserve the property of homophily. As shown in Figure 4, one can see the different communities based, on prior case for G P , status for G S , and both prior case and status for G . Node2vec can force words in similar documents to fall in the same topics because the nodes are cases/status and Node2vec leverage only random walks to clusters nodes. [2] We used the best hyper-parameters for Homophily following [START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF] to get vx.

' ! " # ",% & ",% + , ! " # "$% & "$% ' ( ) * "$% + , ! " - # " -% & " -%

Relational Topic Model with Complex Network

In this section, the classical topic model of LDA [START_REF] Blei | Latent Dirichlet Allocation[END_REF] is first introduced, and second the RTM [START_REF] Chang | Relational Topic Models for Document Networks[END_REF] is described (as illustrated in the left and right parts of Figure 5, respectively). We then integrate the weights obtained by the homophily relationships to RTM for legal document analysis. LDA [START_REF] Blei | Latent Dirichlet Allocation[END_REF] is a generative model. In LDA, documents are represented as a mixture of latent topics, and each topic is characterized by a distribution over the vocabulary in the documents. α and β are parameters in corpus-level, which are sampled when generating the corpus. θ d are document-level variables, which are sampled for each document. We assume that words in documents are generated by the topic probability. From α and θ d , the topic appearance probability is generated in document d. Then, the word topic probability z d,n is generated in document d. At last, the word x d,n for the n-th word in the d-th document is generated from the occurrence probabilities of the vocabulary in the topic k of β k and z d,n . The LDA model only focuses on documents themselves, without considering the relationships among documents.

The link relationship between documents is considered in the RTM [START_REF] Chang | Relational Topic Models for Document Networks[END_REF]. Same as LDA, firstly documents are generated from topic distributions in RTM. Next, documents links are modeled as binary variables, with one link for one pair of documents. Given a pair of documents d, and d, an indicator of binary link is drawn as:

y|z d , z d ∼ ψ(•|z d , z d),
where ψ is the probability function of distributed link between the document pair. Sigmoid or exponential have been used for ψ. We adopt the exponential, because it performs better in the experiments reported in [START_REF] Chang | Relational Topic Models for Document Networks[END_REF]. ψ depends on the topic assignments z d and z d, which generate their words. It is calculated as:

ψ = exp (η T (z d • z d) + ν), (3)
where

z d = 1 N d n z d,n ,
• denotes element-wise product, and coefficients η and intercept ν are parameters, which can be estimated as:

ν = log(1 -1 T Π) -log(ρ K -1 K + 1 -1 T Π) η = log(Π) -log(Π + ρ K 2 1) -1ν
In the above two equations, 1 is the vector whose elements are all 1, ρ is a scalar to control the frequency of the negative observations of no links, and Π is given by

Π = (d, d) E[z d ] • E[z d]
where the summation is computed over all possible documents pairs. We use the homophily network structure presented in Section 4.2 to improve on RTM. We propose two models, called the cutting model and weighting model, in order to strengthen the effect of co-citation patterns in the networks. Prior cases and statutes can affect differently in the judicial processes, and we do not know which ones are more important for the decision making. The motivation behind the cutting/weighting models is to take the best of both prior cases and statutes. Besides using the networks of prior cases only or the networks of statutes only, we also propose to use both of them by aggregating their weights.

Cutting Model

In this model, we use either w n or w j in Equation ( 1) to cut inefficient links, only keeping the links with edge weights higher than a threshold. The links above will be used in RTM as document links. We tried different thresholds of edge weights in the three homophily networks, G P , G S , and G , respectively, in our experiments.

Weighting Model

In this model, instead of cutting the links with edge weights, we keep all the links but use edge weights to weight the link probability function, i.e., Equation [START_REF] Fowler | Network analysis and the law: Measuring the legal importance of precedents at the us supreme court[END_REF]. Note that we only use w j for weighting, because of the probability characteristic of Equation (3). Formally, the weighting is performed as follows:

ψ = w×ψ,
we compare different types of weighting methods:

• w = w P j : only use the edge weights w j in G P . • w = w S j : only use the edge weights w j in G S . • w = w P s : only use the edge weights w s in G P . • w = w S s : only use the edge weights w s in G S . • w = w P j + w S j : use the sum of the edge weights w j in G P and G S . • w = w P j ×w S j : use the product of the edge weights w j in G P and G S . • w = w P s + w S s : use the sum of the edge weights w s in G P and G S . • w = w P s ×w S s : use the product of the edge weights w s in G P and G S . Fuzzy logic aggregation, extensively studied in Detyniecki's thesis [START_REF] Detyniecki | Mathematical aggregation operators and their application to video querying[END_REF], can help balancing between the different types of weights. Fuzzy logic, especially triangular norm fuzzy logic (t-norm) which guarantees triangular inequality in probabilistic spaces, generalizes intersection in a lattice and conjunction in logic, offering many aggregation operators to define conjunction for values within [0, 1] [START_REF] Schweizer | Probabilistic metric spaces[END_REF]. Each tnorm operator is associated with an s-norm (t-conorm) with respect to De Morgan's law [START_REF] Hurley | A concise introduction to logic[END_REF]: S(x, y) = 1 -T (1 -x, 1 -y). The t-norm is the standard semantics for conjunction in fuzzy logic and thus the couple t-norm/s-norm acts as AN D/OR operators on real values in [0, 1]. We experiment with several fuzzy operators: beside the classical sum/product, we also consider the family of Hamacher t-norms (Hamacher product [START_REF] Hamacher | On logical connectives of fuzzy statements[END_REF]) defined for λ ≥ 0 as

T H,λ (x, y) = xy λ + (1 -λ)(x + y -xy) , (4)
the family of Yager t-norms [START_REF] Yager | On a general class of fuzzy connectives[END_REF] defined for λ > 0 as

T Y,λ (x, y) = max 0 1 -λ (1 -x) λ + (1 -y) λ
and the Einstein summation [50]

T E (x, y) = xy 1 + (1 -x)(1 -y)
.

recall that T H,0 is the classical product/sum. Fuzzy logic aggregators can also be used as weighting methods. We thus consider in addition to the methods presented above, the following weighting strategies:

• w = T H,λ (w P j , w S j ): the aggregation of w j in G P and G S , using Hamacher t-norm.

• w = T Y,λ (w P j , w S j ): the aggregation of w j in G P and G S , using Yager t-norm. • w = T E (w P j , w S j ): the aggregation of w j in G P and G S , using Einstein sum. The same weighting strategies, are also used to aggregate the weights obtained with Node2vec embedding, where w s is used instead of w j .

Experiments

We used the Canadian law corpus (Section 3), to conduct experiments for topic modeling.

Settings

For preprocessing, we performed the follows. Words were lemmatized and lowercased using NLTK [START_REF] Loper | NLTK: The Natural Language Toolkit[END_REF]. We also discarded word tokens containing non-alphabetic characters. In addition, we excluded stop words by using the English stop word list in NLTK. [3] We compared the homophily networks G P , G S , and G . Using the edge weights obtained from Equations ( 1) and (2), we either judged which edge to use according to a threshold in the cutting model (Section 5.1) or weight the edges using the methods in the weighting model (Section 5.2). We used a publicly available RTM implementation for all our experiments. [4] The parameters of RTM were trained [3] https://gist.github.com/sebleier/554280 [4] https://github.com/dongwookim-ml/python-topic-model/blob/master/notebook/RelationalTopicModel_ example.ipynb using the obtained network information and documents. [5] The number of topics and max iterator were set to 200, and 10, respectively.

The Node2vec kernel is used with 50 dimensions, an inout hyper-parameter set to 0.5 to preserve homophily property, and the remaining parameters are kept by default.

As a baseline, we compared with LDA [START_REF] Blei | Latent Dirichlet Allocation[END_REF], [6] which does not use link information. Furthermore, we compared to another baseline RTM (w → ∞), which is an RTM that does not use any link information. For all the experiments, we used the default values for the corpus-level parameters α and β (see Figure 5), which were 0.1 and 0.01, respectively.

Evaluation Metrics

To evaluate the output topics of the different models, we essentially used the coherence score [START_REF] Newman | Automatic Evaluation of Topic Coherence[END_REF]. Coherence measures the similarity among the output topic words, which is commonly used for evaluating topic modeling performance. Coherence can be computed as:

coherence = N k=2 k-1 l=1 sim(word k , word l ),
where word k and word l are the k-th and l-th topic words output by a topic model , and N is the number of output topic words. sim(•, •) is the cosine similarity of two words. The cosine similarity is calculated by representing the two word with word embeddings by GloVe840B [START_REF] Pennington | GloVe: Global Vectors for Word Representation[END_REF]. [7] Note that we used the top ten words output by topic models to calculate coherence.

In addition, we also reported the C V [START_REF] Röder | Exploring the space of topic coherence measures[END_REF], the C U M ass [START_REF] Mimno | Optimizing semantic coherence in topic models[END_REF], and the C U CI [START_REF] Newman | Automatic Evaluation of Topic Coherence[END_REF] scores to confirm the performance consistency among different evaluation metrics. The C V metric is reported as the measures with strongest correlations with human ratings. C V is based on a sliding window, a one-set segmentation of the top words and an indirect confirmation measure that uses normalized pointwise mutual information (NPMI) and the cosine similarity.

C U M ass is an intrinsic, asymmetrical confirmation measure between top word pairs that accounts for the ordering among the top words of a topic. C U M ass is computed as:

C U M ass = 2 N × (N -1) N k=2 k-1 l=1 log P (w k , w l ) + P (w l ) ,
where P (w x , w y ) donates the probability that the co-occurrence of the words w x and w y while P (w y ) donate the frequency of the word w y in the corpus. Word probabilities are estimated based on document frequencies of the original documents used for learning the topics. [5] https://github.com/dongwookim-ml/python-topic-model [6] https://radimrehurek.com/gensim/models/ldamodel.html [7] https://nlp.stanford.edu/projects/glove/ Unlike C U M ass , C U CI is an extrinsic measure based on pointwise mutual information (PMI) using an external resource such as Wikipedia to estimate the pointwise mutual information. C U CI coherence is calculated by:

C U CI = 2 N × (N -1) N k=2 k-1 l=1 log P (w k , w l ) + P (w k ) × P (w l ) ,
C U CI is similar to coherence where the sim(., .) is replaced with the pointwise mutual information.

Results

The coherence scores are shown in Table 2, comparing case G P with statute G S networks, for the baselines, and both the cutting and weighting models. In the table, w x with x ∈ {n, j, s} denotes the node weight as in Equations 1 and 2. In the cutting model, an edge is considered only when its weight w x ≥ w e ; an edge between two nodes is created in RTM and trained with RTM. w x ≥ 0 means that all edges are considered without considering their weights, w x → ∞ means none of the links is used for the model, which is the RTM baseline described in Section 6.1. In the weighting model, all edges between two nodes are created and trained using RTM, but they are weighted by the different weighting methods described in Section 5.2. In all cases, we can see that our RTM model given link information shows higher performance than the baselines, i.e., LDA and RTM with w x → ∞. In addition, the cutting model that uses a weight threshold to cut links improves compared to the all link inclusion w x ≥ 0 without weighting. This means that there is noise information in some links and give negative effect when we treat them equally. As for the settings of creating nodes either from prior cases G P or statute laws G S , we do not observe significant difference. This indicates that both citations are good sources for improving topic modeling. There is also no big difference comparing between w n and w j , where w n is the common citation number between two cases, and w j treats homophily as similarity, respectively. In the cutting model, we only use prior cases citations, w n ≥ 5 performs the best in G P ; when we only use statutes, w n ≥ 10 performs the best in G S . Also, w j ≥ 0.50 and w j ≥ 0.25 performs the best for G P and G S , respectively, in the cutting model. Comparing the performance of the cutting model to that of the weighting model, we can see that the weighting model outperforms using citations from either prior cases G P or statute laws G S . This indicates the importance of giving weights to distinguish good and noise links for RTM. Comparing w j and w s in the weighting model, we can see that w j performs slightly better than w s . We also evaluated the best balance between G P and G S in the cutting model for the combined links of G . For the cutting model, we use the best thresholds in Table 2 in order to balance the weights w P x and w S x . Table 3 shows the results. We also report the C V , C U M ass , and C U CI scores to confirm the consistency among different evaluation metrics. As shown in Table 3, we can see in general they are very consistent, and thus we discuss the results based on the coherence scores only. Although the performance is better than the no link model w → ∞, it is similar to when using them separately. It indicates that in the cutting model, the coherence score is not necessarily improved with both types of links, though the best parameters are used as each citation information. For the weighting model, the results shown in Table 3 indicate a trend similar to the cutting model that the coherence score is not necessarily improved by using both types of links no matter w j or w s are used or which weighting methods are used. However, the weighting model still outperforms the cutting model. The best weighting methods are the simple product and Hamacher product (smooth product) (see Equation ( 4)), which means simple weighting strategies work well if there are both prior cases and statutes (an AND operator). Using cosine similarity based on Node2vec is very consistent, as the different weighting models have close coherence scores. The results obtained with Node2vec similarity w s can be interpreted by the fact that the kernel generalizes well the homophily networks. Meanwhile, Node2vec does not improve over simple and fuzzy weighting strategies. The best weighting strategies for Node2vec is the sum, and the interpretation is: a prior cases or the cases are enough to generate coherence topic models (an OR operator).

In addition, we investigated the impact given different number of topics for our topic models. Table 4 on cases, we can see that as the topic number decreases coherence tends to increase. Table 5 shows the coherence for the weighting model using different topic numbers |T | with different weighting methods. [8] We observe the same trend as the cutting model, where the small topic number at |T | = 10 shows the best coherence score. In addition, the weighting model still outperforms the cutting model when changing |T |, and w = w P j +w S j at |T | = 10 shows the best performance among all the models. The weighting models show the small topic number at |T | = 10 to be better with both w j and w s obtained respectively from the co-citations homophily networks and cosine similarity based on the Node2vec embedding of the homophily networks. We did not add the results for the fuzzy aggregation for different topic number as the results are similar to the simple weighting schemes. As a conclusion, if available we recommend to use both prior cases and statute to generate topic models; otherwise, the prior cases if available and finally the statute. We recommend also to use small topic numbers e.g., 10.

To understand the uncertainty of topic models on our data, we calculated the standard deviation. [9] Table 6 shows standard deviation results for the baseline models and our best model, with respect to the coherence evaluation metric with |T | = 10. Our standard deviation results show a consistent small score, meaning that the results are stable. [8] As the three t-norm weighting methods show similar performance compared to the four simple weighting methods as shown in Table 3, we did not compare them here. [9] We did not conduct bootstrapping, because in our context, bootstrapping may not be straightforwardly applicable. We would need a very specific sampling method that would preserve the citation graph structure and its modularity for homophily to be relevant. To understand the performance of the topic models qualitatively, we also analyzed the output topic words. Table 7 shows topic word examples from the best performing model (i.e., the weighting model with w = w P j + w S j and |T | = 10) along with their topic IDs. We can see that these topic words are very informative and related to the areas of law in Canada. [10] Given these topic words, we can easily predict the topics of patent, immigration and refugee, contract, property, human right, labour and employment, and procedural. Through these predicted topics, we can easily understand the topics of important legal cases covered in the COLIEE dataset.

We further investigated the difference between topic similarity (ψ) and homophily (w x ) as outputs of our models. The difference for the resulting weights is shown in Figure 6. We can see that we have similar shapes of the topic in the perspective of their best weight used w n . However, if we investigate the most similar cases, different results may be obtained. With the cases-based topic similarity, with ψ = 0.65, In both examples, the applicants asked a judicial review for a humanitarian and compassionate relief. One application was accepted (#3451) and the other one rejected (#1276). [10] https://en.wikipedia.org/wiki/Law\_of\_Canada\#Procedural\_law

Conclusion

We presented a novel analysis method to the COLIEE corpus of the law article.

Thanks to homophily, we improve topic modeling using the citation structure even without having access to the cited content. We built networks composed by thousands of cases and references. The references belonged to two types of citations, i.e., prior cases, and statutes laws. We explored these two types of citations to investigate citation homophily among cases. We further proposed a cutting model and a weighting model for using these references to improve the RTM. Experiments indicated that the weighting model outperforms the cutting model, which significantly improves topic modeling performance. In addition, the predicted topics are very informative for legal case analysis. We publish both our data and codes online [11] for further research.

For future, first, we intent to use a multilayer network model with Detangler [START_REF] Renoust | Detangler: Visual analytics for multiplex networks[END_REF] to visualize the overlapping of topics based on topics content and similarity in order to evaluate the capacity of our topic models to relate similar documents. Second, we plan to combine all the different matter contained in the data, such as counselors. At last, we plan to further investigate the extraction of new links in the dataset with our topic modeling, allowing to explore the homophily between the cited cases and laws.

We have so far applied this method in the context of legal documents for a large purpose dedicated to computational law only. However, this work invites further investigation of a more general method of homophily-based topic modeling that would fit a larger set of application contexts, including citation network, but could also extend other relations implying homophily.

Declarations

Figure 2

 2 Figure 2 Community structure through citations (left: G, middle: G P , right: G S ).

Figure 3

 3 Figure 3 Network projection and community (left: G P , middle: G S , right: G ).

Figure 4

 4 Figure 4 Node2vec embedding of the networks using UMAP (left: G P , middle: G S , right: G ).

Figure 5

 5 Figure 5 Statistical model of LDA (left) and RTM (right).

  lists the coherence scores for the baseline LDA, RTM (w → ∞,) and the cutting model using different number of topics |T | of 200, 100, 50, and 10 with the best weight setting for G P , G S and G , respectively. It performs the best at |T | = 50 when using G P , and |T | = 10 when using G S and G . Though depending

  w n ≥ 1, and w j ≥ 0.05 for G P the closest cases are Bargig v. Can. (M.C.I.) (2015) case #4127 and Barrak v. Can. (M.C.I.) (2008) #4984. These cases are about immigration. To be more specific, both cases are investigating exception requests under the Immigration and Refugee Protection Act, and both were rejected under insufficient humanitarian and compassionate grounds. With the cases-based topic similarity, with ψ = 0.48, w n ≥ 1, and w j ≥ 0.5 for G S the closest cases are Can-Am Realty Ltd. v. MNR (1993) case #4580 and Deconinck v. MNR (1988) case #475. These cases are about tax. In both cases, a taxpayer is contesting a tax assessment, but one case accepted the plaintiff's appeal (#475) while rejected the other (#4580). With the statutes and case-based topic similarity, with ψ = 0.019 for G the closest cases are Diabate v. Can. (M.C.I.) (2013) case #3451 and De Araujo v. Can. (M.C.I.) (2007) case #1276. These cases are also about immigration.

Figure 6

 6 Figure 6 Comparison of the best wn and w j against the topic similarity ψ.

Table 1

 1 An example of reference identification within legal cases.

	Prior case	New Brunswick (Board of Management) v. Dunsmuir, [2008] 1 S.C.R. 190;
		372 N.R. 1; 329 N.B.R.(2d) 1; 844 A.P.R. 1, refd to. [para. 11]
	-→	New Brunswick (Board of Management) v. Dunsmuir, (2008)
	Statute	Immigration and Refugee Protection Act, S.C. 2001, c. 27, sect. 113(b) [para.
		14].]

-→

Immigration and Refugee Protection Act, S.C.

(2001)

Table 2

 2 Coherence scores of topics from G P and G S against wn, w j , cutting and weighting models.

		G P , wn		G P , w j		G S , wn		G S , w j
	LDA		0.131		0.131		0.131		0.131
	RTM	wn → ∞	0.159	w j → ∞	0.159	wn → ∞	0.159	w j → ∞	0.159
	Cutting	wn ≥ 100	0.166	w j ≥ 0.75	0.160	wn ≥ 10	0.167 w j ≥ 0.75	0.164
		wn ≥ 50	0.165	w j ≥ 0.50	0.166	wn ≥ 5	0.159	w j ≥ 0.50	0.164
		wn ≥ 5	0.167 w j ≥ 0.25	0.161	wn ≥ 0	0.164	w j ≥ 0.25	0.165
		wn ≥ 0	0.166	w j ≥ 0	0.166			w j ≥ 0	0.164
	Weighting			w P j	0.180			w S j	0.176
				w P s	0.170			w S s	0.171

Table 3

 3 Coherence, Cv, UMass, and C U CI scores of topics on G against cutting and weighting models.

	Model	G , wn, w j	Coherence	Cv	UMass	C U CI
	LDA		0.131	0.509	-0.457	-4.476
	RTM	w → ∞	0.159	0.611	-0.412	-1.078
	Cutting	w P n ≥ 0, w S n ≥ 0 w P n ≥ 5, w S n ≥ 10 w P j ≥ 0.50, w S j ≥ 0.25	0.163 0.163 0.167	0.616 0.619 0.613	-0.394 -0.395 -0.402	-1.061 -1.029 -1.057
	Weighting	w P j + w S j	0.165	0.620	-0.387	-1.031
		w P j × w S j	0.174	0.610	-0.419	-1.052
		T H,0 (w P j , w S j )	0.173	0.607	-0.416	-1.083
		T Y,1 (w P j , w S j )	0.174	0.622	-0.371	-1.030
		T E (w P j , w S j )	0.171	0.614	-0.395	-1.030
		w P s + w S s w P s × w S s T H,0 (w P s , w S s ) T Y,1 (w P s , w S s ) T E (w P s , w S s )	0.172 0.170 0.171 0.171 0.171	0.619 0.614 0.613 0.617 0.620	-0.387 -0.392 -0.402 -0.399 -0.387	-1.020 -1.050 -1.082 -1.055 -1.012

Table 4

 4 Coherence scores over different number of topics for G P , G S and G in the cutting model.

		G P , wn ≥ 5	G S , wn ≥ 10	G , (w P j ≥ 0.50, w S j ≥ 0.25)
	LDA |T | = 200	0.102	0.102	0.102
	LDA |T | = 100	0.113	0.113	0.113
	LDA |T | = 50	0.125	0.125	0.125
	LDA |T | = 10	0.151	0.151	0.151
	RTM(w → ∞) |T | = 200	0.159	0.159	0.159
	RTM(w → ∞) |T | = 100	0.165	0.165	0.165
	RTM(w → ∞) |T | = 50	0.169	0.169	0.169
	RTM(w → ∞) |T | = 10	0.174	0.174	0.174
	RTM |T | = 200	0.167	0.167	0.167
	RTM |T | = 100	0.166	0.174	0.173
	RTM |T | = 50	0.171	0.179	0.167
	RTM |T | = 10	0.159	0.182	0.180

Table 5

 5 Coherence scores of different weighting methods for the weighting model.

		w P j	w S j	w P j + w S j	w P j × w S j	w P s	w S s	w S s + w P s	w S s × w P s
	|T | = 200	0.180	0.176	0.165	0.174	0.170	0.171	0.172	0.170
	|T | = 100	0.166	0.175	0.176	0.169	0.170	0.169	0.171	0.168
	|T | = 50	0.172	0.171	0.164	0.167	0.175	0.170	0.170	0.174
	|T | = 10	0.190 0.181	0.192	0.187	0.177 0.180	0.175	0.179

Table 6

 6 Standard deviation for the baseline models and our best model, with respect to the coherence evaluation metric with |T | = 10.

		LDA	RT M (w = ∞) RT M (w = w P j + w S j )
	Coherence	0.151	0.174	0.192
	Std Dev.	0.0140	0.0143	0.0146

Table 7

 7 Examples of topic words and their predicated topic from the best performing model (i.e., the weighting model with w = w P j + w S j and |T | = 10).

	Topic ID Topic words	Predicated topic
	0	trademark, patent, claim, statement	Patent
	1	citizenship, resident, immigration, refuge	Immigration and refugee
	2	document, sale, product, person, subject, ship Contract
	4	property, land, right, aboriginal	Property
	5	act, human, right, protect, nation	Human right
	6	commission, employ, applicant	Labour and employment
	10	procedure, rule, action, report, file	Procedural

Acknowledgements

We thank the reviewers of Complex Networks 2019 for their constructive comments.

Funding

This work was supported by an interdisciplinary research funding from the Institute for Datability Science, Osaka University.

Availability of Data and Materials

The datasets generated during and/or analysed during the current study are available at https://figshare.com/articles/Improving_Topic_Modeling_through_ Homophily_for_Legal_Documents/12408104.

Competing Interests

The authors declare that they have no competing interests. [11] https://figshare.com/articles/Improving_Topic_Modeling_through_Homophily_for_Legal_Documents/ Authors' Contributions KA conducted most of the topic modeling experiments. CBEV performed the node2vec and fuzzy logic aggregation experiments. CC conceptualized this work and was a major contributor in writing the manuscript. BR mainly contributed to the legal network part and writing the manuscript. NO, NT, YN and HN frequently discussed this work and provided valuable feedback. All authors read and approved the final manuscript.

Abbreviations RTM: relational topic model; DAG: directed acyclic network; LDA: Latent Dirichlet allocation; COLIEE: Competition on Legal Information Extraction/Entailment; UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction; NPMI: normalized pointwise mutual information; PMI: pointwise mutual information;

Author details 1 Graduate of Information Science and Technology, Osaka University, Osaka, Japan. 2 Inria, IRISA, Rennes, France, cheikh-brahim.el-vaigh@inria.fr. 3 Institute for Datability Science, Osaka University, Osaka, Japana, {chu, renoust, takemura, n-yuta, nagahara}@ids.osaka-u.ac.jp. 4 Graduate School of Law and Politics, Osaka University, Osaka, Japan, greenaccess@law.osaka-u.ac.jp.


Ucamm ) C Engrand
email: cecile.engrand@u-psud.fr

E Charon

H Le- Roux

M Marinova

J Duprat

E Dartois

B Guérin

J Rojas

L Delauche

M Godard

D Troadec

EVIDENCE OF IRRADIATION IN INTERPLANETARY SPACE IN MINERALS FROM AN ULTRACARBONACEOUS ANTARCTIC MICROMETEORITE (

. We focus here on a mineral assemblage embedded in one UCAMM that exhibits evidence of ion irradiation, to get insight on the formation and evolution of this cometary particle.

.

. The assemblage consists of µm-to subµm sized Mg-rich pyroxenes, a large "triskell"-shaped Fe-sulfide, a few Mg-rich olivines, and minor Si-Al-Carich amorphous pockets and Fe-Ni metal. About 28 crystalline pyroxenes are identified whereas only 5 olivine crystals are observed in the section. GEMS are present embedded in the organic matter, close to the crystalline assemblage.

). Irradiation track lengths

range from 10 nm to ~ 1 µm (average 166 ± 113 nm, 1σ), for the ~ 100 nm thick FIB section. There is no preferred orientation of the tracks. The pyroxene grains located at the upper edge of the assemblage (labeled in light blue in Figure 1) contain both rims and tracks. Some tracks are observed across two adjacent pyroxene crystals. formation of the mineral aggregate. An igneous origin for this aggregate is suggested by the presence of small interstitial glassy pockets rich in Si-Al-Ca that are reminiscent of a mesostasis. The current track production rate at 1 AU determined by Berger & Keller [13] is (4.1 ± 1.2) x 10 4 tracks.cm - 2 .yr -1 (2π exposure). In these conditions, the average track concentration in DC06-43 would correspond to an exposure duration of (3.2 ± 0.9) x 10 5 years at 1 AU in current Sun conditions, or for up to ~ 3 Myrs at 3 AU, assuming that the solar energetic particle flux decays as a function of r -2 (r being the distance from the Sun) [START_REF] Hamilton | International Cosmic Ray Conference[END_REF]. In the case of pre-accretionary exposure, the required irradiation time may be shorter than previously calculated if the irradiation happened close to the more active young Sun. In that case, the transport mechanism of these minerals to the UCAMM forming region (expected to be beyond the nitrogen snow line [2]) should be at low temperature (< 500ºC) in order to preserve the irradiation tracks.

Irradiation tracks are also produced during the inter-planetary journey of the dust particles. For a 30 µm particle like DC06-43 travelling from a distance of 5 UA (from a JFC for instance), the calculated track production assuming a pure Poynting-Robertson (PR) drag would yield a concentration of ~ 2 x 10 9 tracks.cm -2 . An origin at 50 AU only increases the concentration to ~ 3 x 10 9 tracks.cm -2 , still a factor of 4 to 5 lower than the average value measured in DC06-43. This is comparable to the situation observed for 10 µm IDPs, for which an additional source of irradiation tracks is required [START_REF] Keller | [END_REF]. We have shown that the precursor of the organic matter of UCAMMs could be produced by irradiation of Nand CH4-rich ices by high energy galactic cosmic rays (GCR) at the surface of small bodies in the outer regions of the protoplanetary disk [2,[START_REF] Augé | C-coating[END_REF]. If the minerals were embedded under a few meters of these ices, GCR may also have contributed to the track production. Another explanation could also invoke a more complicated orbital evolution than simply governed by pure PR-drag. For instance, mean motion resonances with giant planets during the journey of UCAMMs toward the Earth could possibly trap the particles and increase the exposure duration of UCAMM to the interplanetary radiative environment.

Figure 1 :

 1 Figure 1: TEM micrograph (in bright field mode) of the FIB section of DC06-43. The mineralogy and nature of the components are annotated on the image (OM -organic matter, Pxpyroxene, Ol -olivine, FeS -iron sulfide, FeNi -iron-nickel metal, GEMS -Glass Embedded with Metals and Sulfides, Pt -platinum used for the sample preparation). Discussion: Calculations have been carried out using the Stopping and Range of Ions in Matter (SRIM) software [10] to estimate the range of damage caused by ion irradiation as a function of energy. The range of rim thicknesses from 20 to 100 nm is compatible with irradiation with energies ranging from 1 to 5 keV/nucleon by H to Fe ions. The track lengths (up to ~ 1 µm) require irradiation at much higher energy (>10 keV/nucleon). The rims are likely produced by low energy Solar Wind (SW) whereas the tracks result from irradiation by more energetic Solar energetic particles (SEP). As observed in lunar samples and IDPs [e.g. 11, 12], the irradiated rims are strongly depleted in Mg, likely due to a selective sputtering of Mg relative to Si. The presence of a continuous rim around the pyroxene in Figure 2 suggests a 4π irradiation of this mineral by SW before incorporation in the UCAMM.The occurrence of tracks across pyroxene grain boundaries shows that the irradiation happened after the

Figure 2 :

 2 Figure 2: (top) Example of pyroxene exhibiting a continuous irradiated rim (white marks). (bottom) Histogram of irradiated rim thicknesses measured on five pyroxenes.

Figure 3 :

 3 Figure 3: EDX profile through the mineral shown in Figure 2 top. (Left) EDX map showing the repartition of Si (blue) and Mg (green). (Right) intensity of Mg and Si across the mineral. The Mg depletion in the rim is very clear on both sides of the profile.

Figure 4 :

 4 Figure 4: Selected TEM micrographs of pyroxenes showing irradiation tracks, with their respective track concentrations given in insets.

Acknowledgments: Funding from CNRS, PNP, CNES, LabEx P2IO, IN2P3 DIM-ACAV and ANR (COMETOR), and logistic support from IPEV-PNRA in Antarctica are acknowledged.


Yangwei Deng

Hui Chen

Xinfeng Tao

Sylvain Trépout

Jun Ling
email: lingjun@zju.edu.cn

Min-Hui Li
email: min-hui.li@chimieparistech.psl.eu

Synthesis and self-assembly of poly(ethylene glycol)-block-poly(N-3-(methylthio)propyl glycine) and their oxidation-sensitive polymersomes

Keywords:

Amphiphilic block copolymers PEG-b-PMeSPG were synthesized, and their self-assemblies studied. PEG-b-PMeSPG polymersomes loaded with glucose oxidase showed oxidation-sensitive disruption upon H2O2 production in the presence of Dglucose.

have been frequently reported as smart drug carriers [24][25][26]. In a previous work, we synthesized a thioether-bearing polypeptoid, poly(N-3-(methylthio)propyl glycine) (PMeSPG), and prepared polymersomes from PMeSPG-b-PSar block copolymers [27]. As the hydrophobic thioethers can be oxidized into hydrophilic sulfoxides, the obtained polymersomes showed oxidation-responsive bursting in the presence of hydrogen peroxide (H2O2). However, the H2O2 used as external stimulus in the study had much higher concentration than the micromolar H2O2 level in vivo, which limited its realistic use. The production of H2O2 in-situ in the target site by oxidantgenerating enzymes and corresponding substrates is considered to be a more practical way [28]. In this work, we report on polymersomes made of poly(ethylene glycol)-block-poly(N-3-(methylthio)propyl glycine) (PEG-b-PMeSPG), which can encapsulate glucose oxidase (GOx), an enzyme which can oxidize glucose and generate H2O2. We demonstrate the disruption of polymersomes after the incubation with D-glucose. PEG-b-PMeSPG block copolymers with different PMeSPG chain lengths were synthesized by ring-opening polymerization (ROP) of N-3-(methylthio)propyl glycine N-thiocarboxyanhydride (MeSPG-NTA) initiated by aminoterminated PEG [29,30]. Their self-assembly was performed by nanoprecipitation and double emulsion method. The influence of the organic cosolvent in nanoprecipitation on the nanostructures was discussed in detail. PEG-b-PMeSPG polymersomes loaded with GOx were prepared by double emulsion method followed by extrusion treatment. Finally, the oxidation-responsive disruption of polymersomes was achieved after the incubation with D-glucose with the in-situ production of H2O2.

The monomer MeSPG-NTA was prepared as described in our previous work [27]. The synthesis of PEG-b-PMeSPG was performed through the ROP of N-3-(methylthio)propyl glycine N-thiocarboxyanhydride (MeSPG-NTA) using amino-terminated PEG with molecular weight (MW) of 2000 Da (PEG45-NH2) as a macromolecular initiator (Fig. 1). With various feeding ratios of monomers to initiator, three amphiphilic block copolymers PEG-b-PMeSPG, EM1 (PEG45-b-PMeSPG17), EM2 (PEG45-b-PMeSPG40) and EM3 (PEG45-b-PMeSPG71), were obtained with different MWs and hydrophilic/hydrophobic ratios (Table 1). The MW distributions (Ɖ) of the copolymers were controlled within 1.2. The compositions of blocks were determined by the characteristic signals in 1 H NMR spectrum (Fig. S1A for spectrum of EM3 and calculation in Supporting information). The SEC traces (Fig. S1B in Supporting information) exhibited monomodal MW distributions for all three copolymers, which shifted from right to left as the degree of polymerization (DP) of PMeSPG block increased. These results suggested good controllability of the polymerization. As the most common approach for self-assembly of block copolymers, nanoprecipitation was first employed to prepare PEG-b-PMeSPG nanoparticles dispersed in water. N,N-dimethylformamide (DMF) and tetrahydrofuran (THF) were chosen as the organic cosolvents, respectively, to dissolve the copolymers at a concentration of 5 g/L. Then water (2 mL) was added to the solution (1 mL) at the rate of 0.2 mL/h, driving the hydrophobic blocks to assemble together to avoid their contact with water and form finally nanostructures. The organic solvent was removed at the end by dialysis against water, leading to aqueous dispersions of polymer nanoparticles. The morphologies were analyzed by cryogenic electronic microscopy (Cryo-EM).

We discuss first the polymer nanoparticles formed using DMF as the nonselective cosolvent. As revealed by Cryo-EM images (Fig. 2A and Figs. S2A-C in Supporting information), nanospheres with diameters around 20-90 nm were obtained from EM1 with the hydrophilic ratio of 45%. Along with the spheres, a few unilamellar vesicles were also observed (Figs. S2B andC) with diameters around 50-130 nm, and membrane thickness around 9 nm. EM2 with hydrophilic ratio of 26% formed spherical and worm-like micelles, whose diameters were around 18-21 nm (Fig. 2C, Figs. S2D-F in Supporting information). EM3 with hydrophilic ratio of 16% and the largest MW (12300 Da) in these three copolymers self-assembled into polymersomes with broad size distribution (diameters = 50-550 nm) and membrane thickness around 12-14 nm (Fig. 2E and Figs. S2G-I in Supporting information). Simultaneously, lamellae were observed together with vesicles, and many of them were in disk-like shapes (Figs. S2H and I in Supporting information). Lamellae also had a diversity of size in the range of 250-2000 nm. Their thickness could be measured in the folded region of lamellae, which was found to be around 12 nm, similar to that of the vesicular membrane. As a widely accepted mechanism, vesicles were formed by the closure of open lamellae, when the edge energy of the lamellae meets with the bending energy for closed vesicles [31]. Intermediate structures are often observed in the self-assembly of polymers as "frozen" metastable states, because of the relatively high molecular weights of polymers. Therefore, the coexistence of folded lamellae, incomplete vesicles and closed vesicles in Fig. S2I could be considered as an evidence for the above mechanism of vesicle formation.

When THF was used as the organic cosolvent instead of DMF in nanoprecipitation, the morphologies of the nanostructures selfassembled from the three copolymers were totally different. EM1 formed many long fibers and ribbons (Fig. 2B and Figs. S3A-C in Supporting information). Fibers had the width around 5-6 nm, ribbons around 21-23 nm. EM2 formed lamellae with irregular and fragmented shapes (Fig. 2D and Figs. S3D-F in Supporting information), of which the membrane thickness was measured to be around 10-11 nm. For EM3 (Fig. 2F and Figs. S3G-I in Supporting information), lamellae with large area (over 2 μm) were observed predominantly, with the membrane thickness around 14-16 nm. The self-assembled nanostructures and their sizes discussed above were summarized in Table S1 (Supporting information).

The overall morphology of nanoparticles self-assembled by amphiphiles in water depends firstly on their packing parameter p [11,32]. p is a geometric parameter, p = V/a0l, where V is the volume of the hydrophobic part, l its length, and a0 the equilibrium area at the core/corona interface. As p increases from below 1/3 to equaling to 1, the preferred morphology changes from sphere, to cylinder, vesicle, and finally flat lamella. In the case of amphiphilic block copolymers, p is strongly related to the ratio of hydrophilic/hydrophobic blocks. For the copolymers PEG45-b-PMeSPGn, p increases when DP of the hydrophobic block PMeSPG increases. Indeed, the copolymers EM1, EM2 and EM3 showed a typical morphological change of "spheres-cylinders-vesicles" with DMF as the organic cosolvent. From the same copolymer, different morphologies of self-assemblies were obtained by choosing different organic solvents. It indicated the effect of solvent on copolymer self-assembly. This observation can be explained by the different polymer-solvent interactions. The solubility parameter δ is an important parameter to characterize the polymer-solvent interactions. δ is contributed by three components, as presented by Eq. ( 1), where δd represents the dispersion force, δp the polar interaction, and δh the hydrogen bonding. The miscibility between polymer and solvent can be predicted by the solubility parameter difference P-S [START_REF] Hansen | Hansen Solubility Parameters: A User's Handbook[END_REF][START_REF] Van Krevelen | Properties of Polymers: Their Correlation with Chemical Structure; their Numerical Estimation and Prediction from Additive Group Contributions[END_REF], as calculated by Eq. ( 2).

𝛿 2 = 𝛿 𝑑 2 + 𝛿 𝑝 2 + 𝛿 ℎ 2 (1)
𝛿 𝑃-𝑆 = ((𝛿 𝑑,𝑃 -𝛿 𝑑,𝑆 ) 2 + (𝛿 𝑝,𝑃 -𝛿 𝑝,𝑆 ) 2 + (𝛿 ℎ,𝑃 -𝛿 ℎ,𝑆 ) 2 ) 1/2

(2)

The solubility parameters and their components of the DMF, THF, PEG and PMeSPG were listed in Table S2 (Supporting information). Among them the values of DMF and THF were reported by Hansen et al. [START_REF] Hansen | Hansen Solubility Parameters: A User's Handbook[END_REF], while the values of PEG and PMeSPG were calculated by a group contribution method (Hoy's method) [START_REF] Van Krevelen | Properties of Polymers: Their Correlation with Chemical Structure; their Numerical Estimation and Prediction from Additive Group Contributions[END_REF][START_REF] Özdemir | [END_REF][START_REF] Hoy | Tables of Solubility Parameters[END_REF]. By putting these values into the Eq. ( 2), we were able to obtain the differences between the solubility parameters of polymer and solvent P-S. Generally, a smaller P-S indicates a better miscibility between polymer and solvent [START_REF] Van Krevelen | Properties of Polymers: Their Correlation with Chemical Structure; their Numerical Estimation and Prediction from Additive Group Contributions[END_REF]. For PEG, PEG-DMF = 4.46 MPa 1/2 and PEG-THF = 4.54 MPa 1/2 , while for PMeSPG, PMePG-DMF = 6.89 MPa 1/2 and PMeSPG-THF = 7.43 MPa 1/2 . The relatively low values of PEG-DMF and PEG-THF suggest DMF and THF are both good solvents for PEG with nearly equivalent miscibility. By comparing PMePG-DMF and PMeSPG-THF, it can be concluded that the miscibility between PMeSPG and DMF is better than that between PMeSPG and THF. During the nanoprecipitation procedure, when water (a poor solvent for PMeSPG) was added into the system, the hydrophobic blocks self-assembled together to avoid their exposure to water. In the hydrophobic regions, organic solvents still coexisted during the water addition, and more solvent molecules remained in the copolymer aggregates in the case of DMF/water system than in the case of THF/water system. This difference in solvent content may further result in different self-assembled morphologies. Take EM3 for example, when water was added into the EM3/DMF solution, some DMF molecules stayed in the bilayer membranes formed by copolymers, and played the role of plasticizers. Consequently, the membrane was flexible and had low bending modulus, which could close like a pulled bag to form vesicles. However, when the nanoprecipitation of EM3 was proceeded in EM3/THF solution, the self-assembled bilayer membranes involved fewer solvent molecules than in EM3/DMF, and the relatively low flexibility of the membrane did not allow the formation of closed vesicles. (Figs. 2E andF)

The solubility behavior of copolymers in self-assembly experiments was in accordance with the calculated results. When the PEGb-PMeSPG copolymers were dissolved in organic cosolvents for nanoprecipitation, we did find that all the copolymers dissolved easily in DMF, while much longer time was taken for the copolymers to be dissolved in THF, especially for EM3, heating was necessary for dissolution. Similar cases were reported on the solubility of other polypeptoids. PSar, also called as poly(N-methyl glycine), was soluble in DMF, but insoluble in THF [2]. Schlaad et al. also found that polypeptoids including poly(N-allyl glycine) and poly(N-3-(glycerolthio)propyl glycine) had obviously poorer solubility in THF than in DMF [START_REF] Robinson | [END_REF]. Especially poly(N-3-(glycerolthio)propyl glycine), which was synthesized from thiol-ene coupling between poly(N-allyl glycine) and 1-thiolglycerol, had thioether-containing sidechains like PMeSPG, and was hardly dissolved in THF but well dissolved in DMF. In general, there was no polypeptoid reported, which exhibited better solubility in THF than in DMF. Besides nanoprecipitation, double emulsion method is another interesting approach to prepare polymersomes with the advantage of high yield, where the bilayer vesicles were formed from water-in-oil-in-water (W/O/W) double emulsions [38,39]. In the previous work, we used this method successfully to prepare polypeptoid vesicles [27]. Here we prepared EM2 polymersomes by double emulsion method (experimental details in Supporting information). Giant vesicles were first obtained with the diameter of 2-10 m, as observed by confocal laser scanning microscopy (CLSM), where the vesicles were visualized by Nile Red loaded in the membrane (Fig. 3A). Extrusion treatment was employed on the giant vesicle dispersion [START_REF] Zhang | Proc. Natl. Acad. Sci[END_REF], using a mini lipid extruder (Avanti® Mini-Extruder, Avanti Polar Lipids, USA) with 200 nm polycarbonate membranes, which yielded nano-polymersomes (Fig. 3C). The change in sizes and distributions were analyzed by dynamic light scattering (DLS), which revealed a multi-peak distribution curve (Fig. 3B) transforming into a nearly monomodal curve, with a main peak around 160 nm (Fig. 3D).

Since the pendant thioether groups on hydrophobic blocks of PEG-b-PMeSPG can be oxidized into hydrophilic sulfoxide groups, the PEG-b-PMeSPG polymersomes have the potential to be disrupted under the action of oxidation stimuli. The GOx-encapsulated polymersomes were then prepared by adding GOx/PBS to EM2/chloroform solution when performing the double emulsion (See Supporting information for detailed procedure). After extrusion treatment, nano-polymersomes were obtained with the concentration of 0.5 g/L for both polymer and GOx. D-Glucose (25 mmol/L) was then introduced to the dispersion of GOx-loaded EM2 nanopolymersomes. DLS was performed to monitor the change in the polymersomes in the following 240 min at 37 C. The loaded glycoprotein enzyme GOx is a dimer consisting of 2 equal subunits with a molecular mass of 80 kDa each. Each subunit contains one flavin adenine dinucleotide (FAD, a cofactor) moiety and one iron. FAD works as the initial electron acceptor in redox reaction and can be reduced into FADH2. During the oxidation of D-glucose into gluconolactone by the catalysis of GOx, H2O2 is generated in presence of dissolved oxygen. It is H2O2 that transfers the hydrophobic thioether groups of PMeSPG into hydrophilic sulfoxides [27], eventually inducing the destabilization of polymersomes (Fig. 4A). In DLS, the count rates represent the scattering intensity and indicate the size and/or concentration of particles. As revealed in Fig. 4B, the count rates decreased mildly in the first 60 min, afterwards from 60 min to 140 min, the count rates dropped abruptly from around 900 kcps to 200 kcps. The count rates decreased slightly after 140 min, and fell to 110 kcps at 240 min. The size and size distribution were also assessed during the incubation with D-glucose. As shown in Fig. 4C, the hydrodynamic diameter of the polymersomes kept stable at around 145 nm in the first 60 min, and then decreased significantly from 145 nm to around 70 nm during 80 to 160 min. The diameter kept around 70 nm steadily until 240 min. Meanwhile, the polydispersity index (PDI, Fig. 4D) kept stable at around 0.2 in the first 60 min, and then started to increase, from 0.2 to around 0.5 from 80 min to 160 min. The PDI reached to 0.58 at 240 min. Note that from t = the time 180 min, the PDI value fluctuated intensely. As visualized by Cryo-EM (Fig. S4 in Supporting information), micelles and aggregates with various shapes and sizes below 100 nm took the majority in the sample after the incubation.

The evolution of GOx-loaded polymersomes observed in DLS as a function of the incubation time with D-glucose observed could be explained as follows. The glass transition temperature (Tg) of PMeSPG was dertermined by differential scanning calorimetry (DSC) analyses (PMeSPG45 and PMeSPG85, Fig. S5 in Supporting information). A rather low Tg around 20-30 C was measured, which was below both the body temperature and the temperature of incubation here. Therefore, hydrophobic blocks PMeSPG in the vesicle membrane had good mobility at 37 °C, which might favor the diffusion of glucose across the membrane and got involved in the oxidation process to eventually induce polymersome disruption [START_REF] Napoli | [END_REF]. When the vesicles were dissembled and GOx was released, the free enzymes had lower concentration than that of the encapsulated enzymes [START_REF] Napoli | [END_REF], therefore the redox reaction slowed down. Finally, the system contained smaller but more stable micelles composed of incompletely oxidized copolymers. In conclusion, we present here the synthesis and self-assembly of the amphiphilic block copolymers PEG-b-PMeSPG, with three different chain lengths of the hydrophobic block PMeSPG. Nanoprecipitation with DMF and THF as the organic cosolvents was used first to prepare the polypeptoid nanoparticles. Various morphologies such as spheres, fibers, lamellae and vesicles were observed for different copolymers depending on the hydrophilic/hydrophobic ratio and the cosolvent. Double emulsion method followed by extrusion treatment was employed then to prepare EM2 polymersomes and GOx-loaded EM2 polymersomes. GOx-loaded polymersomes could be disrupted under the action of oxidants generated in-situ in the presence of D-glucose. These polymersomes may find applications in drug delivery and can also be used for the design of micro/nano-reactors, biosensors and biodetectors.

Fig. 1 .

 1 Fig. 1. Synthesis of PEG-b-PMeSPG by ROP of MeSPG-NTA.

Fig. 2 .

 2 Fig. 2. Cryo-EM images of self-assemblies obtained by nanoprecipitation from EM1 (A, B), EM2 (C, D) and EM3 (E, F). DMF (A, C and E) and THF (B, D and F) were chosen as the organic cosolvents in nanoprecipitation experiments, respectively.

Fig. 3 .

 3 Fig. 3. (A) CLSM image and (B) DLS distribution curve of giant polymersomes EM2 from double emulsion; (C) Cryo-EM image and (D) DLS distribution curve of the EM2 self-assemblies obtained using double emulsion method followed by extrusion treatment. In CLSM image, Nile Red was loaded in the membrane for visualization.

Fig. 4 .

 4 Fig. 4. (A) Schematic illustration of the GOx-catalyzed conversion of D-glucose into gluconolactone with production of H2O2, transferring the pendant hydrophobic thioether groups of EM2 into hydrophilic sulfoxides; the evolution of (B) count rates, (C) hydrodynamic diameter (Z-average diameter) and (D) size distribution (PDI) of GOx-loaded EM2 polymersomes (0.5 mg copolymer, 0.5 mg GOx in 1 mL PBS) as a function of time of incubation with 25 mM glucose at 37 °C.

Table 1

 1 Syntheses of PEG-b-PMeSPG copolymers. DP, MW and PEG weight ratio (fPEG,wt) were calculated from the integrals of characteristic signals in1 H NMR spectra (Supporting information); b Determined by SEC calibrated with PEG standards (eluent: 0.01 mol/L LiBr/DMF at the rate of 1 mL/min).

	Copolymers	[M]/[I]	Yield	DPMeSPG/DPEG a	fPEG,wt a	Mn, NMR a	Mn, SEC b	Ɖ b
	EM1	14	91.1%	17/45	45%	4500	3600	1.14
	EM2	39	83.2%	40/45	26%	7800	4900	1.19
	EM3	82	82.5%	71/45	16%	12300	7700	1.15
	a

Acknowledgments We acknowledge the French National Research Agency (No. ANR-16-CE29-0028), the National Natural Science Foundation of China (No. 21674091) and the Joint Foundation of Shaanxi Province Natural Science Basic Research Program and Shaanxi Coal Chemical Group Co., Ltd. (No. 2019JLM-46) for financial support. Y. Deng thanks the China Scholarship Council for funding his Ph.D. scholarship in France.