Cécile Bothorel

Laurent Brisson

Inna Lyubareva

How to Choose Community Detection Methods in Complex Networks The Case Study of Ulule Crowdfunding Platform

Keywords: Social Networks Analysis, Community Detection, Choice of method, Complex Networks, Online cooperation, Crowdfunding Shared project count (mean)

Discovering community structure in complex networks is a mature field since a tremendous number of community detection methods have been introduced in the literature. Nevertheless, it is still very challenging for practitioners to choose in each particular case the most suitable algorithm which would provide the richest insights into the structure of the social network they study. Through a case study of the French crowdfunding platform, Ulule, this paper demonstrates an original methodology for the selection of a relevant algorithm. For this purpose we, firstly, compare the partitions of 11 well-known algorithms. Then, bivariate map based on hub dominance and transitivity is used to identify the partitions which unveil communities with the most interesting size and internal topologies. These steps result in three community detection methods relevant for our data. Finally, we add the socioeconomic indicators, meaningful in the framework of the crowdfunding platform, in order to select the most significant algorithm of community detection, and to analyze the cooperation patterns among the platform's users and their impact on success of fundraising campaigns. In line with previous socio-economic studies, we demonstrate that the social concept of homophily in online groups really matters. In addition, our approach puts in light that crowdfunding groups may benefit from diversity.

Introduction

In recent years socio-economic research on online groups and communities often proposes to extend the traditional approach and to encompass social network analysis modelling relationships by edges in the graph. The joint analysis of the two types of data-socio-economic and network structure-makes possible to provide important insights on the group functioning and to reveal properties of a social network that are not immediately obvious, e.g. the existence of sub-networks or communities operating within the global network.

For instance, in the context of crowdfunding platforms, many studies focus on the directly observable interactions among the participants of individual projects and show their role for the success of fundraising campaigns [START_REF] Kuppuswamy | Crowdfunding creative ideas: The dynamics of project backers[END_REF][START_REF] Agrawal | Entrepreneurial finance and the flatworld hypothesis: Evidence from crowdfunding entrepreneurs in the arts[END_REF][START_REF] Zheng | The role of multidimensional social capital in crowdfunding: A comparative study in china and us[END_REF]. However, few empirical studies focus on relational structures which are not explicitly stated in the context of crowdfunding platform [START_REF] Inbar | Community impact on crowdfunding performance[END_REF] and question whether relational circles can go beyond individual projects, by broadening the initial social capital of projects' leaders, and form a platform-level cross-project social network. Does the participation of the members of this cross-project network guarantee higher success rates of crowdfunding campaigns? These questions have strong managerial and economic implications for the platforms: Could this networking lead to the formation of a core group with an intense participation? Should the platforms further develop the mechanisms for cross-project networking? How should project leaders surround themselves to give themselves every chance of success? The study of these questions necessitates an appropriate methodology based on the detection of implicit communities. Therefore, the question arises of how to choose an appropriate community detection algorithm, relevant to the particular context.

As a matter of fact, many community detection methods exist. They have different ways to divide a network into multiple subsets of densely connected nodes, and hence result in different community structures. They may rely on different notions of community, and even with the same notion (eg. Newman defines a community as a "group of vertices with a higher-than-average density of edges connecting them" [START_REF] Newman | Modularity and community structure in networks[END_REF]), algorithms may optimize different objective functions or use different heuristic to get efficient implementations. The authors of the methods compare their efficiency in terms of computational properties, such as complexity, as well as in terms of validation metrics: modularity, rand index, conductance, etc [START_REF] Fortunato | Community detection in networks: A user guide[END_REF], describing resulting partitions quality. Although, as demonstrated in some comparative surveys [START_REF] Dao | Community structure: A comparative evaluation of community detection methods[END_REF][START_REF] Fortunato | Community detection in networks: A user guide[END_REF][START_REF] Jebabli | Community detection algorithm evaluation with ground-truth data[END_REF], choosing a community detection method is still a problematic task, especially if we are not aware of the underlying mechanisms.

Some recent papers have attempted to provide guidance on algorithm selection using criteria such as the mixing parameter of a network, computation time, or overlap with a simulated community structure [START_REF] Dao | Community structure: A comparative evaluation of community detection methods[END_REF][START_REF] Zheng | The role of multidimensional social capital in crowdfunding: A comparative study in china and us[END_REF]. Others provide more intuitive criteria such as the size of the communities [START_REF] Dao | Estimating the similarity of community detection methods based on cluster size distribution[END_REF] or descriptive network-oriented metrics on structural patterns within communities [START_REF] Dao | Community structures evaluation in complex networks: A descriptive approach[END_REF][START_REF] Dao | An empirical characterization of community structures in complex networks using a bivariate map of quality metrics[END_REF]. N. Smith et al. argue that such criteria should not be applied alone, but in conjunction with business-oriented objectives: the "best" method depends on the context, on the research question, i.e. on how the communities will be used [START_REF] Smith | A Guide for Choosing Community Detection Algorithms in Social Network Studies: The Question Alignment Approach[END_REF].

The current paper addresses this problem. It proposes an original methodology to guide practitioners in their choice of methods in connection to a specific research question. We first pre-select some candidate methods, and qualify their results, i.e. the resulting partitions, through qualitative characteristics. Then, final users may compare the few pre-selected partitions with comprehensive measures and finally select the most relevant one regarding the research question.

The case of the Ulule crowdfunding platform is used as example. In this case study, we suggest that some non-directly observable communities may exist within the cross-project social network. However, as in many exploratory studies, we have no preliminary information about their number or structures. Our goal is to discover these communities, to describe their internal organizations and to explore whether there is a relationship between particular communities' organization and the success of fundraising campaigns. This last element is the central research question of the case study.

This paper is organized as follows: In Section 2, we introduce the Ulule platform and its cross-project social network. In Section 3, we present a methodology which permit us to select, among 11 methods, three most relevant and convergent algorithms to discover the communities. In Section 4, by introducing additional socio-economic attributes of communities, we choose the most suitable method and discuss the relative success of different communities for fundraising campaigns. Last Section concludes.

Ulule crowdfunding platform

Ulule platform and its network

Crowdfunding represents a model of participatory financing, which has been used by an increasing number of companies, associations and individuals since the early 2010s. Its principle consists for a project "pitched" by its creator, to collect money from a large public. Thanks to numerous interaction mechanisms between participants (comments, news, promotion systems, etc.), these platforms play the role of facilitators of social capital.

To better understand the networking role of crowdfunding platforms, we use empirical data for the period 2010-2016 from the Ulule platform-one of the main crowdfunding platforms in France and Europe, which pays particular attention to the strengthening of its proper platform's community1 . After data cleaning, we keep 19,544 projects, of which 11,900 were successfully funded, and 7,644 failed. They globally attracted 876,758 contributors, who contributed a total of 47.75 million euros.

Graph of co-contributions

While 99.7% of Ulule contributors are one-time funders, 0.3% of them are not only very active by contributing to more than three different projects, but they regularly "meet", in each fundraising campaign, the same participants also contributing to these projects. We suggest that these active contributors, involved in a least 3 projects with at least 1 other contributor, are candidates to form the cross-project platform's network. On average, they contribute to 14 different Ulule projects, for an average total amount of over 600 euros per contributor. With these active Ululers, we define a non-oriented graph of co-contributions, in which each edge (u,v) means that the users u and v have contributed to 3 or more projects together. There are 469 connected components. All of them contain less than 10 nodes, except the largest one. With its 2,081 nodes and 4,749 edges, this giant component proves that the social network, transverse to individual projects, really exists at the Ulule platform.

The members of the cross-project network share projects with 4.56 other members (average degree). We find a power law distribution of degrees (the estimated exponential coefficient is 2.27) which is a common property in online social networks. 25% of the members have a degree greater than 4, the maximum degree being 199. There are therefore contributors who co-finance projects with really important number of different contributors: 24 of them have more than 50 neighbors in the graph.

We notice that the distribution of the Ulule's thematic categories according to whether or not projects are financed by the members of the graph reveals interesting information. The improvement in the success rate of campaigns of projects belonging to the graph is observed for all categories (28.6% overall improvement), especially for Games, Comics, Technology and Publishing. Indeed, these domains have a strong social component in the production and consumption of goods. These findings confirm our interest in studying communities of the active Ululers.

We find out that members of the obtained social graph vary a lot in terms of their social activity (node's degree, clustering coefficient and centralities) and contribution behavior (number of projects funded, average amount of contributions, specialization rate which quantify the variety of thematic categories addressed (projects' category) and their similarities with the neighboring nodes).

Multiple correspondence analysis on these attributes, followed by a hierarchical bottom-up classification, leads to 5 clusters of Ulule's active contributors. A detailed version is available [START_REF] Lyubareva | Une plateforme de crowdfunding et son réseau social : l'exemple ulule[END_REF], but for space reasons, we sum up the 5 profiles as follows:

• The Sponsors [START_REF] Newman | Finding and evaluating community structure in networks[END_REF] are the Ululers with a high degree and betweenness centrality. They may be considered as facilitors. Involved in 140 projects (in average), they are very active on various topics (not specialized).

• The Followers (653) are the users who arrive late, during the second half of the campaigns. They prefer very big projects (which average objective is more than 17k euros). Exhibiting a high closeness centrality, they frequently co-contribute with other contributors, in particular the Sponsors.

• The Precursors (538) are characterized by an early arrival in projects and especially before all their neighbors.

• The Collaborative Specialists (368), very highly thematically specialized, have also a very high clustering coefficient indicating a strong cohesion of links between neighbors. This may show a strong solidarity between some Ululers whose financing decisions are often collective, even if their fields of specialization are not necessarily identical. They don't contribute to a very large number of projects and are not attracted by the size of the projects.

• The Specialists (504) are not high contributors as well in terms of number of projects, nor in volume of contributions. They are not particularly highly connected as well, but they are however passionate about very specific themes, the same themes than their neighbors, demonstrating therefore social homophily. These findings will be mobilized in the Section 4 to refine the final choice of the community detection method in connection with the case study research question.

A methodology to choose community detection methods

There are many approaches to perform community detection based on different paradigms, including cut, internal density clustering, stochastic equivalence, flow models, etc [START_REF] Fortunato | Community detection in networks: A user guide[END_REF]. The purpose is not to provide an exhaustive overview here. We refer the reader to surveys like [START_REF] Fortunato | Community detection in networks: A user guide[END_REF][START_REF] Schaub | The many facets of community detection in complex networks[END_REF] to get details about these different approaches. In this work we focus on well-known methods, apply them to our casespecific graph and show how different the partitions produced can be, making the choice of a method non obvious.

We carefully kept a large variety of approaches as summarized in Table 1. While Edge betweenness is based on edge centrality detection in order to split networks into several communities, Louvain and Fast greedy optimize modularity by iteratively folding nodes into meta-nodes. Spectral method is also based on modularity, but identifies the community structure by finding leading eigenvectors corresponding to largest eigenvalues of a modularity matrix. Some approaches are based on a dynamic distance: for Walktrap, if two nodes are in the same community, the probability that a random walker will move from one to another in only a few movements is very high (notion of trap) and consequently the distance is low. Conclude combines a similar random walk-based distance to agglomerate nodes, and the local optimization of modularity inspired by the Louvain method selects iteratively the best agglomeration. Infomap relies on finding a configuration that maximizes the compression of random walks represented by an encoded binary sequence. Inspired by epidemic spreading mechanisms, a more basic and direct distance is used by the Label propagation and its variant SLPA, where a node should belong to the same community as most of its immediate neighbors. In the same vein, concepts have also been borrowed from theoretical physics with the Spin glass model which may be seen as an alternative to modularity maximization; the idea is to consider nodes as spin states and to minimize the energy of the configuration of spins to reach a stable state. Finally we can cite a statistical inference approach, DCSBM which uses stochastic blockmodels to infer the likeliness that a given observed network (and its latent block structure) is generated from a compatible model, and then suggest the most likely set of model parameters.

Methods are often compared with classic quantitative measures from information theory domain, such as NMI, the Normalized Mutual Information [START_REF] Chakraborty | Metrics for community analysis: A survey[END_REF], which evaluates their agreement to arrange nodes into similar clusters. Recent studies introduced more intuitive, and very simple quantitative metrics such as the size of the communities [START_REF] Dao | Estimating the similarity of community detection methods based on cluster size distribution[END_REF] where methods are similar if they produce similar community size distribution. In [START_REF] Dao | Community structures evaluation in complex networks: A descriptive approach[END_REF], the authors propose metrics dedicated to graph analysts. They describe the communities with structural measures, in order to quality the communities from a topological point of view. Each community is qualified with well-known organizational patterns, such as star-based structures, cliques (this will be detailed later, in Figure 3).

In this paper, we propose to practitioners a methodology based on both these quantitative and qualitative metrics. We show how to use them in order to compare the methods; through the case study from Ulule, and its specific business questionwhich communities lead to successful fundraising-, we show how to select the most relevant method:

 The initial step is to run community detection algorithms. We used 11 methods described in Table 1 and 11 partitions (where each node belongs to only one community) were obtained.  Step 1 aims at the choice of a subset of methods. We compare the partitions with validation metrics. Since we have no preliminary knowledge on the platform's communities, we are looking for the methods making consensus findings in order to build a robust foundation for subsequent steps.  As proposed in the previous literature, Step 2 brings an additional intuitive criterion, the size of communities, to refine the subset of consensual methods.

 Step 3 characterizes the consensual partitions with qualitative measures that are relevant for the current business-oriented problem. As our case study focuses on organizational patterns involving nodes, we will use bivariate maps based on graph structural indicators, such as the hub dominance and the clustering coefficient.  The last step introduces specific "business" indicators related to the current problem to finally make the choice decision (section 4).

Initiate the selection with consensual partitions

After the initial step, that is run the 11 methods with their default parameters), we obtain 11 partitions. In the first selection step, we compare how nodes are arranged into clusters: we compute the Normalized Mutual Information (NMI) often used in community detection because it allows the comparison of partitions even where nodes are assigned to a different number of clusters. We apply its normalized variant with values in [-1, 1] which is popular in the field of community analysis [START_REF] Chakraborty | Metrics for community analysis: A survey[END_REF]. The Figure 1b shows how our 11 partitions are astonishingly similar from a NMI point of view where all scores are positive, ranging from 0,26 to 0,76, with a large majority above 0,5. Eight among them demonstrate scores greater than 0.6 when compared to each other, with a very consensual group consisting of four methods: Edge Betweenness, SLPA, Fast greedy and Walktrap with scores higher than 0.7. There are three slightly different methods: Spectral, Label Propagation and Spin Glass that produce more specific partitions, which are all different from each other. While this difference may be easily explained by the fact that Spectral and Spin Glass implement inherently distinctive mechanisms, the results of Label Propagation, being a variant of SLPA, are quite surprising. As an intermediate result, on the basis of the NMI scores, Edge Betweenness, SLPA, Fast greedy, and Walktrap converge on their clustering task. To put it differently, knowing a random node's affiliation in Edge Betweenness partition, in our example, gives a high probability to successfully deduce its membership in the 3 other partitions. Louvain is also quite close to this group and as it is frequently used, we don't want to discard it right now.

These 5 methods reach a quite good consensus. We argue that in exploratory studies, as the current one, where practitioner has no a priori knowledge about communities that she wants to analyze, it is important to identify such robust clustering, demonstrating an agreement between different methods. This step is determinant for the further exploration of the research question.

The size of communities as qualitative choice

We propose then to complete the NMI analysis by adding information about the size of the discovered communities (Figure 1a). One can notice two types of partitions. Some communities are large with tens or sometimes thousands of members. Other partitions, on the contrary, exhibit small (2 or 3 members) to medium-sized communities (around 10-20 nodes). These two classes of methods are depicted in Figure 1c, where the similarity score compares the distribution of sizes of communities (score introduced in [START_REF] Dao | An empirical characterization of community structures in complex networks using a bivariate map of quality metrics[END_REF]). Regarding large communities, Louvain, Spin glass and Spectral produce an interesting variety of large to very large communities, whereas DCSBM only produces huge ones. We observe the same disparity when we focus on partitions with smaller groups. SLPA and Walktrap produce a lot of communities with approximately 10 nodes, without notable variety of sizes. On the contrary, Fast greedy's and Edge Betweenness partitions have a more flat profile in Figure 1a: most of their communities are small to medium, but some of them have also more than 100 nodes.

Therefore the consensual methods-Louvain, Edge Betweenness, SLPA, Fast greedy, and Walktrap-produce different distributions of sizes of communities, we have to choose whether we give priority to small-medium communities or to large ones. This example demonstrates that whereas in some contexts, the size criteria may be enough to make a choice, such information is not always sufficient. Taking into account that our research question focuses on the communities' forms and their efficiency for the fundraising campaigns, we add topological indicators in order to differentiate the partitions from the organizational perspective.

Structural classification of consensual partitions

In order to characterize organizational patterns within communities, which is our business-oriented objective here in the case study, we propose the use of structural measures applied to communities such as internal link density, average centrality of nodes, average degree. Such indicators are interesting to be combined in bivariate map [START_REF] Dao | An empirical characterization of community structures in complex networks using a bivariate map of quality metrics[END_REF] to describe structural patterns. For example, ploting a bivariate map with the mean out degree fraction (meanODF) paired with its standard deviation (stdODF) allows to explore different situations regarding the openness of communities and the cooperation between groups of Ululers [START_REF] Dao | Community structures evaluation in complex networks: A descriptive approach[END_REF]. However, Hub dominance and Transitivity are particularly relevant when one considers internal patterns of organization like cooperation, because their combination leads to wellknown patterns depicted on Figure 3:  Hub dominance: Internal edges of a community can be distributed in various ways around its nodes, either concentrating around a few highly centralized nodes, or uniformly distributed over the nodes. The Hub dominance metric identifies the level of centralized organization around well-connected nodes. The higher this metric of a community, the more likely it has a hub-like structure. Hub dominance can be considered as a normalized version of degree centrality. High Hub dominance leads to the well-known star-based patterns as depicted on Figure 3.  Transitivity: Very similar to the clustering coefficient [START_REF] Dao | An empirical characterization of community structures in complex networks using a bivariate map of quality metrics[END_REF], Transitivity reflects the probability that adjacent vertices of a vertex are connected. This metric is usually employed to evaluate modular structures (grids) or clique dominance in networks (Figure 3). For example, high Transitivity coupled with similar spheres of interests (or other attributes) among individuals often indicates the existence of social homophily, especially in online groups, also known as the proverb "birds of a feather flock together". Figure 2 plots the communities in the Hub dominance vs Transitivity space. One can see that large communities are concentrated in the same area. The methods responsible of those large communities (e.g. Louvain) indeed produce a priori very few different structural patterns. With low Hub dominance and low Transitivity, most communities could be considered as "string-based" structures (Figure 3a). In order to detect whether or not cooperation exists within large communities, we should have to zoom-in to extract dense sub-zones, i.e. apply again a detection of community to each community, and then project the new smaller communities in our bivariate map.

Conversely, Edge Betweenness, Walktrap and Fastgreedy which produce small and medium-sized communities seem to generate various directly observable types of organizations. Indeed the points are distributed in 3 of the 4 areas of our bivariate map in Figure 2. We especially find a lot of groups in the upper part of the map. This means that their members are organized around hubs, but in 2 different manners. When the Transitity is high, we find some clique-based organizations (Figure 3d), where Ululers are (nearly) all contributing with each other to common projects. When the Transitivity is low, the organizations are mimicking star-based structures, with a very high centralization (Figure 3c). Ululers in these groups are less involved in horizontal cooperation, but seem to follow influencers (Ululers with large degrees) who concentrate common projects with a lot of poorly connected contributors.

With these new insights, Edge Betweenness and Walktrap seem to be very good candidates: (i) they belong to the consensual group of methods previously shown, producing a robust partition; and (ii) they offer various internal organizational forms. Louvain, without offering diversified topological structures, demonstrates however interesting properties of high Hub dominance in its large communities. For this reason, we propose to keep Louvain to further explore the typology of its communities.

To summarize, Table 3 describes the obtained communities using several conventional network metrics. First, the three partitions do not appear very distinguishable with regards to these traditional indicators. Interestingly, it seems that the size of the groups does not really influence their average degree. Moreover, the average closeness centrality and the average density also remain stable regardless of the algorithm used. Second, this confirms the need for more qualitative measures, such as the bivariate map that we proposed here above, to cover the specificities of different community detection methods and to be able to choose among them.

Method

Introduction of case-specific data and typology of communities

At this step, in accordance with the objective alignment approach [START_REF] Smith | A Guide for Choosing Community Detection Algorithms in Social Network Studies: The Question Alignment Approach[END_REF], we introduce additional information on the contributors' profiles, specific to the current study, which was presented in the Section 2. This helps us to compare classifications of communities, generated by three partitions, and to choose one relevant community detection method. Depending on particular context and available data, other socio-economic indicators may be mobilized for the communities' classification.

Final selection of the method

To classify the communities of Louvain, Edge Betweenness and Walktrap methods, regarding the distribution of contributors' profiles, the following clustering methods were used: (i) a principal component analysis (2 dimensions) followed by an agglomerated hierarchical clustering (Euclidean distance, Ward method of variance minimization), (ii) the K-means method and (iii) a decision tree. The decision tree produces clusters of communities (Families in Figure 4) almost identical to the K-means method (completeness = 0.964; Adjusted Rand Index : ARI = 0.854). Proximity to clusters produced by the component analysis is not obvious when considering these measures (completeness = 0.513; ARI = 0.398), but nevertheless, the distribution of the types of contributors remains relatively close.

We observe similar distributions of contributors' profiles across the different community families when we compare the 3 community detection methods and their clusters of communities. The same variety of community forms can be seen:

 Family 1 (with the Sponsors): very large balanced communities composed of all the profiles. They have the particularity of having attracted the Sponsors and a lot of Followers.  Family 2 (Specialists): communities very clearly dominated by Specialists, whether collaborative or not.  Family 3 (Followers, Precursors): small and even micro communities rather dominated by Precursors and to a lesser extent Followers. Therefore, regardless the clustering method and the community detection method, the three afore-mentioned families of communities are clearly detected. The only distinguishing feature is the representation of each community family depending on the community detection method. Edge Betweenness proposes the most balanced clusters (Table 4), while Louvain produces mainly Family 1 and 2 items, and not surprisingly, most of the 167 (small) communities produced by Walktrap are from Family 3.

On this basis we choose the Edge Betweenness partition, composed of 72 communities of various sizes, exhibiting substantial examples of each family (see the exact distribution of its Ululers' profiles in Figure 4).

Community structures and projects success

Once we know the community detection method, it is possible to describe the communities and their success rates in fundraising campaigns. Table 5 uses relational (clustering coefficient, the average of the betweenness and closeness centralities of the members, mean degree) and socio-economic attributes (volume of funding, number of interactions via comments, thematic specialization and project success rate) in order to characterize collaboration rates and organization of communities.. The three families of communities differ crucially in terms of collaboration. On crowdfunding platforms, it can take different forms: sharing projects within a community (weight of links in the Ulule social network), cohesion of members of a community around the same projects (clustering coefficient), or communication via a feedback system (comments). As presented in the Table 5, each family of communities in the case of the Ulule platform favors one of these forms.

Family

The combination of two collaborative aspects-the weight of links in the graph (average number of shared projects) and the clustering coefficient-can inform us about community organization. For example, in Family 1, the fact that the average number of shared projects is very high but the members are poorly connected to each other (low clustering coefficient) highlights a highly centralized organization around a few central actors. We probably find here the communities with low Transitivity and high Hub dominance (probably star-based communities as seen previously). The Sponsors may play the structuring roles of these communities.

The communities in Family 2 are made up of members who share the same interests. These thematic groups, which are strongly connected and supportive, collectively take their decisions on project funding and contribute on average more than other communities.

Finally, in Family 3, communities are also structured around certain themes but without clear specialization. Members of these communities communicate a lot through the feedback system (comments).

Comparative analysis of success rates of families of communities

Since the thematic categories are not represented in a balanced way in the three Families of Communities, we cannot directly compare the respective effectiveness of community forms for the different themes. Nevertheless, if we look in more detail at the communities that have a 100 percent success rate for all our indicators (Table 6), there are 13 communities belonging to Families 2 and 3 (respectively 9 and 4 communities). These two Families have very close thematic choices.

These best communities are small in size, quite specialized and finance projects on average on 4 different themes. The projects are very different in terms of the objectives to be achieved (all types of amounts are represented), but less than 25 % of the projects have a larger than average scope, so we have rather modest projects.

Surprisingly, the 9 least successful communities in the graph also belong to Families 2 and 3. What can be noticed in the Table 7 is that the most and the least successful communities share quite similar characteristics (size, thematic specialization). What clearly differentiates the best performing communities is the clustering coefficient and the number of comments, which are significantly higher. In other words, the thematic specialization that characterizes Family 2 and partially Family 3, does not in itself guarantee success of crowdfunding projects. To reach significant economic performance of fundraising campaigns, they must be coupled with strong social involvement and cohesion of its members.

With regard to Family 1, characterized by a high centralization and a high thematic openness, we notice a relatively high success rate. Thematic diversity also attracts participants in communities.

Conclusion

Community detection makes it possible to identify very diverse groups in a social network. This paper demonstrates a methodology to choose one relevant community detection algorithm, among 11 well-known ones, providing fruitful insights into the cooperation forms not directly observable on a crowdfunding platform. The choice of a particular community detection method is not an easy or a neutral choice. As demonstrated in the paper, depending of partition methods, practitioners obtain a range of various community types, which will drastically change the final results of their analysis. This paper substantiates that an accurate way to choose one suitable method is a complex task. Especially in the context of exploratory studies it necessitates the combination of a range of techniques, e.g. in our case, partitions' similarities, qualitative criteria and structural indicators (stringbased, star-based or clique-based organizations of communities).

In line with N. Smith et al. [START_REF] Smith | A Guide for Choosing Community Detection Algorithms in Social Network Studies: The Question Alignment Approach[END_REF] this study substantiates that the choice of a method is determined by the research context and problematics. Additional techniques, specific data and indicators allow to narrow down the scope of available options in the methods choice. Their alignment with the practitioners' research question plays a crucial role for the final choice of a particular method. In the framework of the case study presented in this paper, the choice of the Edge Betweenness method results from the analysis of socio-economic characteristics and the exploration of the distribution of Ululers' profiles. This way, we have identified 3 robust families of platform's communities and their distinctive features, i.e. organization, number of participants, collaboration intensity, thematic specialization, and performance in the fundraising campaigns. Depending on the context and available data, different socio-economic indicators may be mobilized to obtain the communities' classification and a range of further business-oriented questions may be addressed: e.g. precise distribution of string-based, star-based or clique-based forms in communities' families, life circle and evolution dynamics of the communities and many others.

Figure 1 :

 1 Figure 1: The similarity between community detection methods in terms of (a) Community size distributions, (b) NMI, (c) size fitting quality,

Figure 2 :

 2 Figure 2: Structural description of communities in terms of Hub dominance and Transitivity (consensual methods)

Figure 3 :

 3 Figure 3: A categorization of internal community structure according to two topological property

Figure 4 :

 4 Figure 4: Typology of communities (Edge Betweenness, Decision Tree).

Table 1 :

 1 A summary of community detection methods used to study community structure.

	Method	Approach	Reference
	Louvain	Multilevel modularity	Blondel et al. (2008) [2]
	Fast greedy	Modularity optimization Clauset et al. (2004) [4]
	Spectral	Vector partitioning	Newman and Girvan (2004) [18]
	Spin glass	Energy model	Reichardt and Bornholdt (2006) [21]
	DCSBM	Stochastic blockmodels	Karrer and Newman (2011) [13]
	Walktrap	Dynamic distance	Pons and Latapy (2005) [19]
	Conclude	Dynamic distance	Meo et al. (2014) [16]
	Edge betweenness Edge centrality detection Girvan and Newman (2002) [10]
	Infomap	Information compression Rosvall and Bergstrom (2008) [22]
	Label propagation Topological closeness	Raghavan et al. (2007) [20]
	SLPA	Topological closeness	Xie and Szymanski (2011) [25]

Table 3 :

 3 Conventional topological metrics to describe communities: number of communities generated (in brackets) and average statistics calculated on the whole set of generated communities.

	(Communities count)	Member count (mean)	Degree (mean)	Clustering coefficient (mean)	Betweenness (mean)	Closeness (mean)
	Louvain (22)	90.48	3.63	0.26	2,871.45	0.24
	Edge Bet. (72)	28.90	2.96	0.20	2,234.84	0.23
	Walktrap (167)	12.39	2.29	0.15	1,695.03	0.22

Table 4 :

 4 Sizes of communities and their contributions by Family (Edge Betweenness, Decision Tree).

	Family	Commun ity (count)	Member count per community (mean)	Member contributio n (mean in €)	Project goal (mean in €)	Project count per member (mean)	Project count per communit y (mean)
	1	8	161.5	41.4	8,050	14.4	469.5
	2	28	19.1	47.7	8,501	13.3	124.8
	3	36	7.0	41.3	9,589	13.7	42.5

Table 5 :

 5 Communities characteristics

		Member count per community (mean)	Project count	Shared project count (mean)	Theme count per community (mean)	Comment count per project (mean)	Community Clustering Coefficient (mean)
	1	161.5	1,642	4.3	13.4	73.2	0.25
	2	19.1	781	3.7	4.1	137.3	0.32
	3	7.0	547	3.4	5.8	176.0	0.15

Table 6

 6 Communities with high success rates vs. communities with lower success rates. The indicators are averaged by community. Sub-table"All" shows the indicators for all 72 communities.

https://www.ulule.com/about/ulule/

Acknowledgements

The authors of this study would like to thank Arnaud Burgot, the CEO of Ulule, very warmly for providing us with the data from the platform and for giving us his confidence in carrying out this work.


Carla Atallah

Hélène Greige-Gerges

Catherine Charcosset
email: catherine.charcosset@univ-lyon1.fr

Claude Bernard Lyon

Development of cysteamine

Keywords: Cysteamine, Encapsulation, Freeze-drying, Liposomes, Stability

published or not. The documents may come

Introduction

Cysteamine (Cyst) is an aminothiol compound synthesized by human body cells and derives from coenzyme A degradation [START_REF] Besouw | Cysteamine: an old drug with new potential[END_REF][START_REF] Gallego-Villar | Cysteamine revisited: repair of arginine to cysteine mutations[END_REF] (Fig. 1). Cyst is described to have a radioprotective [START_REF] Bacq | Protection against x-rays and therapy of radiation sickness with betamercaptoethylamine[END_REF], an anti-cancer [START_REF] Apffel | Tumor Rejection in Experimental Animals Treated with Radioprotective Thiols[END_REF], and anti-malarial [START_REF] Fortin | Cysteamine, the Molecule Used To Treat Cystinosis, Potentiates the Antimalarial Efficacy of Artemisinin[END_REF] effects. It can be used for hyperprolactinemia treatment by reducing the size of pituitary glands and depleting plasma prolactin [START_REF] Sagar | The mechanism of action of cysteamine in depleting prolactin immunoreactivity[END_REF]. This molecule represents the only treatment of cystinosis disease, which is characterized by cystine accumulation in many tissues [START_REF] Elmonem | Cystinosis: a review[END_REF]. In lysosomes, cysteamine reacts with cystine to form a mixed disulfide of half-cystine and Cyst, which then leaves the cells via a lysine transporter [START_REF] Bozdağ | Formulation and in vitro evaluation of cysteamine hydrochloride viscous solutions for the treatment of corneal cystinosis[END_REF]. It can also be used as a depigmenting agent [START_REF] Chavin | Some potent melanin depigmentary agents in the black goldfish[END_REF][START_REF] Farshi | Efficacy of cysteamine cream in the treatment of epidermal melasma, evaluating by Dermacatch as a new measurement method: a randomized double blind placebo controlled study[END_REF][START_REF] Frenk | Selective action of mercaptoethylamines on melanocytes in mammalian skin: experimental depigmentation[END_REF][START_REF] Mansouri | Evaluation of the efficacy of cysteamine 5% cream in the treatment of epidermal melasma: a randomized doubleblind placebo-controlled trial[END_REF]. Cyst is freely soluble in water (23.5 mg/mL), has a foul odor and a bitter taste, and presents a weak pharmacokinetic profile (Lahiani-Skiba et al., 2007). Besides, Cyst is unstable in aqueous solutions due to the rapid oxidation of the sulfhydryl group, leading to the formation of cystamine. This reaction is catalyzed by metal ions such as Cu 2+ , Fe 3+ , and Zn 2+ and is stimulated in alkaline pH [START_REF] Atallah | Challenges for cysteamine stabilization, quantification, and biological effects improvement[END_REF]. To increase its stability and improve its biological effects, Cyst has been encapsulated in liposomes [START_REF] Butler | Depletion of Cystine in Cystinotic Fibroblasts by Drugs Enclosed in Liposomes[END_REF][START_REF] Jaskierowicz | Oral Administration of Liposome-entrapped Cysteamine and the Distribution Pattern in Blood, Liver and Spleen[END_REF][START_REF] Jeitner | Possible oncostatic action of cysteamine on the pituitary glands of oestrogen-primed hyperprolactinaemic rats[END_REF], cyclodextrins (Lahiani-Skiba et al., 2007), and emulsions [START_REF] Gresham | Use of a sustained-release multiple emulsion to extend the period of radio protection conferred by cysteamine[END_REF].

Liposomes are spherical vesicles that comprise one or more lipid bilayer structures enclosing an aqueous core. They are used to encapsulate active compounds for various applications [START_REF] Laouini | Preparation, Characterization and Applications of Liposomes: State of the Art[END_REF]. Their biocompatibility, biodegradability, and low toxicity make them very appropriate carriers for many drugs [START_REF] Zylberberg | Pharmaceutical liposomal drug delivery: a review of new delivery systems and a look at the regulatory landscape[END_REF]. Cyst has been encapsulated in liposomes to enhance its absorption through the intestinal wall [START_REF] Butler | Depletion of Cystine in Cystinotic Fibroblasts by Drugs Enclosed in Liposomes[END_REF], its effectiveness in cystinosis [START_REF] Jaskierowicz | Oral Administration of Liposome-entrapped Cysteamine and the Distribution Pattern in Blood, Liver and Spleen[END_REF], and the period of prolactin depletion action [START_REF] Jeitner | Possible oncostatic action of cysteamine on the pituitary glands of oestrogen-primed hyperprolactinaemic rats[END_REF]. To the best of our knowledge, the liposomes encapsulating Cyst were not characterized in previous studies, and the effect of encapsulation on Cyst stability was not evaluated.

The freeze-drying is a promising approach to increase the shelf life of liposomes incorporating active agents. Among several cryoprotectants (sucrose, trehalose, glucidex 6D and 19D), Sebaaly et al. demonstrated that hydroxypropyl beta-cyclodextrin (HP-β-CD) was the most effective cryoprotectant of liposomes during the freeze-drying of eugenol-loaded Phospholipon 90H (PH 90H) liposomes [START_REF] Sebaaly | Effect of composition, hydrogenation of phospholipids and lyophilization on the characteristics of eugenol-loaded liposomes prepared by ethanol injection method[END_REF]. Besides, Gharib et al. proved that HP-β-CD protects hydrogenated but not unsaturated liposomes during freeze-drying [START_REF] Gharib | Hydroxypropyl-ßcyclodextrin as a membrane protectant during freeze-drying of hydrogenated and nonhydrogenated liposomes and molecule-in-cyclodextrin-in-liposomes: Application to trans-anethole[END_REF].

In this study, blank and Cyst-loaded liposomes composed of PH 90H and Lipoid S100 were prepared by the ethanol injection method. The liposomal suspensions were characterized for their size, polydispersity index (PdI), and zeta potential by diffusion light scattering (DLS) as well as for their morphology using transmission electron microscopy (TEM). The concentrations of Cyst, phospholipid, and cholesterol (Chol) in liposomes were determined spectrophotometrically by the Ellman method with some modifications, Bartlett method and Chol quantitation kit, respectively. The encapsulation efficiency (EE%) and the loading ratio (LR %) of Cyst in liposomes were also determined. Freeze-drying of PH 90H liposomes was realized according to the optimized protocol previously described [START_REF] Gharib | Hydroxypropyl-ßcyclodextrin as a membrane protectant during freeze-drying of hydrogenated and nonhydrogenated liposomes and molecule-in-cyclodextrin-in-liposomes: Application to trans-anethole[END_REF], and the characteristics of the reconstituted vesicles were determined. Moreover, the stability of free and encapsulated Cyst was assessed at different temperatures (4, 25, and 37°C) and in the presence and absence of room light. Finally, the storage stability of the freeze-dried form of encapsulated Cyst was evaluated after four months of storage at 4°C.

Materials and methods

Materials

Cyst, monobasic phosphate, cystamine dihydrochloride, DL-dithiothreitol, and metformin hydrochloride were purchased from Sigma-Aldrich (Buchs, Switzerland). 5,5′-Dithiobis(2nitrobenzoic acid) (DTNB), ammonium molybdate were purchased from Sigma-Aldrich (St Louis, Missouri, USA). Ethylenediaminetetraacetic acid (EDTA), dibasic phosphate, sulfuric acid, and hydrogen peroxide were purchased from Sigma-Aldrich (Steinheim, Germany). PH 90 H (90% soybean PC, 4% lysoPC, 2% triglycerides, 2% water, and 0.5% ethanol) and Lipoid S100 (94% soybean PC, 3% lysoPC, 0.5% N-acyl-PE, 0.1% PE, 0.1% phosphatidylinositol, 2% water, 0.2% ethanol) were obtained from Lipoid GmbH (Ludwigshafen, Germany). Chol was obtained from the Fisher chemical (Loughborough, UK), and sodium bisulfite was purchased from Combi Blocks (San Diego, California, USA). 4-Amino-3-hydroxy-1-naphthalene sulfonic acid was purchased from Sigma-Aldrich (Bangalore, India). Chol quantitation kit was purchased from SPINREACT (Barcelona, Spain). HP-β-CD-oral grade (MS=0.85) was purchased from Roquette (Lestrem, France).

Sodium dodecyl phosphate and phosphoric acid were obtained from Sigma-Aldrich (China).

All other chemicals were of analytical grade.

Cyst quantification

The quantification of Cyst was done using two methods. For the quantification of Cyst in liposomal suspension, a modified Ellman method was used. First, a stock solution of Cyst was prepared in water (1 mg/mL) and then diluted to obtain final concentrations of Cyst ranging between 2.5 and 100 µg/mL. Ellman's reagent was prepared by dissolving 4 mg DTNB in 1 mL of reaction buffer (pH 8.0). The latter is composed of 0.1 M sodium phosphate and 1 mM EDTA. 250 µL of water (blank) or Cyst solution was added to 1.25 mL reaction buffer, 1.25 mL of methanol, and 50 µL Ellman's reagent solution. This reaction mixture was then vortexed and incubated at room temperature for 15 min. The absorbance was measured at 412 nm using the Spectrophotometer UV5 Mettler Toledo (Columbus Ohio, USA). The calibration curve was constructed by plotting the absorbance against the concentration of Cyst ranging from 2.5 to 100 µg/mL.

For the stability studies, the simultaneous quantification of Cyst and its degradation product cystamine was needed. An ion-pair chromatography with an isocratic mode consisting of acetonitrile: water containing 0.1% phosphoric acid and SDS (4 mM) (v/v, 45:55) was used. 20 µL of the samples were injected into a reverse-phase Kinetex C18 column (4.6 × 100 mm, 2.6 μm). The flow rate was set at 1.0 mL/min, the column temperature was maintained at 25°C, and the detection wavelength was 215 nm. Metformin was used as an internal standard with a concentration of 1 µg/mL. Linearity was between 2.5 and 100 µg/mL for Cyst and between 5 and 100 µg/mL for cystamine.

Preparation of liposomes

Liposomes were prepared by the ethanol injection method, as described by [START_REF] Sebaaly | Effect of composition, hydrogenation of phospholipids and lyophilization on the characteristics of eugenol-loaded liposomes prepared by ethanol injection method[END_REF]. Saturated (PH 90H) or unsaturated lipids (Lipoid S100) (10 mg/mL) and Chol (5 mg/mL) were dissolved in absolute ethanol to obtain the organic phase (10 mL). Then, the organic phase was injected into the aqueous phase (20 mL) containing different concentrations of Cyst (0; 0.5; 0.75; 1; 2.5; 5 mg/mL), using a syringe pump (Fortuna optima, GmbH-Germany), at a temperature above the transition temperature of the phospholipid (55°C for PH 90H and -20°C for Lipoid S100) and under magnetic stirring at 400 rpm. The phospholipid:Chol:Cyst molar ratios ranged between 1:0.98:1 and 1:0.98:10; for PH 90H and between 1:1.17:1.17 and 1:1.17:11.7 for Lipoid S100. The contact between the organic solution and the aqueous phase leads to a spontaneous liposome formation. The liposomal suspensions were then left for 15 min at 25 °C under stirring (400 rpm). The ethanol was removed by rotary evaporation (Heidolph GmbH, Germany) under reduced pressure at 40 °C and the obtained liposomal solutions were stored at 4 °C for further characterization. Each preparation was performed in triplicate.

Characterization of the liposomal suspensions

Determination of size, pdI and zeta potential by dynamic light scattering

Malvern Zetasizer Nanoseries (Zetasizer Nano ZS; Malvern Instruments Ltd, France) was used to determine the mean size of the vesicles. All batches were diluted with ultrapure water.

The particle-size distribution data were collected using the DTS (nano) software (version 5.10) provided with the instrument. The polydispersity index, which indicates the particle size distribution, ranges from 0 (monodispersed) to 1 (very broad distribution). Zeta potential was calculated using Smoluchowski's equation from the electrophoretic mobility of liposomes.

All measurements were carried out at 25°C after 2 min of equilibration and were performed in triplicate. Data were expressed as the mean values ± SD.

Morphological characterization by transmission electron microscopy

The morphology of the liposomal suspensions was imaged by transmission electron microscope (TEM) (CM 120; Philips, Eindhoven, Netherlands) operating at an accelerating voltage of 120 kV. A drop of a solution of the liposomal suspension with or without dilution was placed onto a carbon-coated copper grid for 2 min, and the remaining liquid was removed using a filter paper. The grid was placed on negative staining using a 1% sodium silicotungstate solution during 30 s. The excess amount of sodium silicotungstate solution was removed using a filter paper, and dried samples were observed.

Determination of phospholipid:Chol:Cyst molar ratios in the final liposome structures

The concentrations of incorporated compounds (phospholipids, Chol, and Cyst) in lipid vesicles were obtained by subtracting the concentrations of free compounds from their total concentrations determined in the suspensions. The suspensions underwent centrifugation at 21382xg during 1 h at 4°C using Vivaspin 500 centrifugal concentrator (Sartorius Stedim Biotech, Germany, MWCO = 10 kDa). The filtrates contained the free compounds.

Phospholipid quantification assay

Bartlett's method was used to determine free and total concentrations of phospholipids in the various liposomal suspensions. This method is based on several steps; first, the digestion of the organic phosphate (0.5 mL from total liposomal suspensions, filtrates, and standard solutions of phosphorus) using diluted sulfuric acid (5 mM) at 200°C for 1h. Then, the oxidation to the inorganic phosphates was done in the presence of 10% hydrogen peroxide (0.1 mL) for 30 min at 200°C. A phosphomolybdic complex was formed after the addition of ammonium molybdate (4.6 mL) and the reduction of this complex by the 4-amino-3hydroxyl-1-naphthalene sulfonic acid (0.2 mL) at 100°C for 15 min. A blue solution appeared and was measurable at 815 nm spectrophotometrically.

The incorporation ratio (IR) of phospholipids was calculated as follows: ncorporated mass of phospholipids nitial added mass of phospholipids 100 Eq. ( 1)

where the incorporated mass of phospholipids stands for the phospholipid mass determined in the liposome structure; the initial mass of phospholipids represents the mass initially used to prepare liposomes.

Chol quantification assay

The quantification of Chol (free form or total amount present in the suspension) was performed after its enzymatic hydrolysis and oxidation using the Chol CHOD-POD kit (SPINREACT, Spain). Chol in the samples was oxidized by Chol oxidase into 4-cholestenona and hydrogen peroxide. The latter reacts with 4-aminophenazone in the presence of peroxidase to form the colorimetric indicator quinonimine, having an absorbance proportional to Chol concentration in the sample. Standard solutions (0 to 2 mg/mL) were prepared from Chol stock solution (2 mg/mL); the latter was prepared in 10% Triton X-100. Samples from the liposome suspensions were diluted in 10% Triton X-100 and sonicated for 20 min at room temperature to ensure the dissolution of vesicles and the release of incorporated Chol. 1 mL of the working reagent containing buffer and enzyme was then added to 10 μL of each sample (liposome suspension, filtrate, and Chol standard solution) and incubated for 10 min at room temperature. The absorbance was monitored at a wavelength of 505 nm.

The incorporation ratio (IR) of Chol was calculated as follows:

ncorporated mass of hol nitial added mass of hol 100 Eq. ( 2)

Where the incorporated mass of Chol stands for the Chol mass determined in the liposome structure, the initial mass of Chol represents the mass initially used to prepare liposomes.

Encapsulation efficiency and loading ratio of Cyst

The EE and LR% of Cyst in liposomes were determined using the optimized Ellman method.

250 µL of the Cyst standard solution, the filtrate containing free Cyst, or the diluted solution of the liposomal suspension were added to 1.25 mL phosphate buffer, 1.25 mL of methanol, and 50 µL of Ellman's reagent. This method allows the determination of the free ([ yst] free )

and the total ([Cyst] tot ) Cyst concentrations. EE% was calculated as the following equation:

EE cyst [ yst] tot [ yst] free [ yst] tot 100
where [Cyst] tot and [Cyst] free correspond to the concentration of total and free Cyst in the liposomal suspension, respectively.

LR% of Cyst was calculated as follows:

L m total m free m initial 100 Eq. ( )

where m total and m free are the mass of total and free Cyst in the liposomal suspension and m initial is the mass of Cyst initially added to the aqueous phase during the preparation of the liposomes.

Stability study

The formulation PH 90H: Chol: Cyst (1:0.98:10) was selected to compare the stability of Cyst loaded in liposome to that of free Cyst. The liposomal suspension (5 mL) was centrifuged, the supernatant was eliminated, and the pellet was re-suspended in water. The concentration of the encapsulated Cyst was determined after the destruction of liposomes by sonication; Cyst solutions with a similar concentration were prepared in water and placed at 4, 25, and 37°C in the dark. Also, the same concentration of free Cyst was placed in light at 25°C. At different times, aliquots were removed, and the concentration of Cyst remaining in solution and that of formed cystamine were determined by ion-pair chromatography.

The percentage of remaining Cyst was calculated as follows:

emaining yst ( )

[ yst] t

[ yst] t0 100 Eq. ( 5)

where is the concentration of remaining Cyst determined at time t and is the initial Cyst concentration.

The percentage of formed cystamine was calculated as follows:

ormed cystamine ( )

[ ystamine] t

[ ystamine] f 100 Eq. ( 6)

where is the concentration of formed cystamine determined at time t and is the final cystamine concentration obtained after the total conversion of cysteamine to cystamine.

Freeze-drying of liposomes

The freeze-drying of blank and Cyst loaded liposomes composed of PH 90H and Chol was performed according to Gharib et al. [START_REF] Gharib | Hydroxypropyl-ßcyclodextrin as a membrane protectant during freeze-drying of hydrogenated and nonhydrogenated liposomes and molecule-in-cyclodextrin-in-liposomes: Application to trans-anethole[END_REF]. Blank liposomes and Cyst loaded liposomes (PH 90H: Chol: Cyst molar ratios of 1:0.98:1 and 1:0.98:10) were freshly prepared; 5 mL were then ultracentrifuged at 170000 g for 1 h at 4°C. The supernatant was removed, and the pellet was reconstituted in an aqueous solution of 50 mM HP-β-CD (2 mL). The samples were kept at -20°C overnight then placed into the drying chamber of Cryonext 23020 freeze-dryer (Trappes, France), pre-cooled to -20°C, then lowered to -40°C with a slow cooling profile of 0.5° /min. The product was stabilized for 30 min at -38°C before the vacuum was applied. Primary drying was executed at a pressure of 150 μbar for 3 h at -10°C, then the temperature of the drying chamber was progressively increased to 5°C for 6 h at 250 μbar, to reach finally 10° at 350 μbar for 9 h. The temperature was adjusted at a product temperature higher than that of the sublimation temperature of the water. During primary drying, the ice crystals are sublimated, resulting in the porous cake of the freeze-concentrated matrix. A secondary drying step for 10 h at 20° and 100 μbar pressure was applied. This step is crucial to decrease the residual water content of the amorphous matrix. Finally, the vials were removed from the freeze-dryer, closed with rubber caps, and stored at 4°C. The lyophilized liposomes were then reconstructed with ultra-pure water to its original volume (5 mL) before characterization and further analysis.

Storage stability of freeze-dried forms

After four months of storage at 4°C, freeze-dried liposomes were resuspended in 5 mL water;

the particle size, PdI, and zeta potential values of the obtained suspensions were determined.

The remaining Cyst concentrations were measured using ion-pair chromatography. The

Results and Discussion

Liposomes characterization

Size, polydispersity index, and zeta potential

The characterization of liposomes composed of saturated (PH 90H) and unsaturated (Lipoid S100) phospholipids and Chol was realized by DLS. The mean particle size, polydispersity index (PdI), and zeta potential values were determined (Table 1).

Two populations were observed in blank and Cyst-loaded liposomes. The first population was nanometric and represented more than 94% of the suspensions. The second population was of a micrometric size and represented a negligible percentage varying between 1 and 6%.

The size of blank PH 90H liposomes was 211 ± 8 nm, while that of blank Lipoid S100 liposomes was lower (151 ± 11 nm). Similar results were obtained by Gharib et al. [START_REF] Gharib | Drug-in-cyclodextrin-inliposomes as a carrier system for volatile essential oil components: Application to anethole[END_REF].

The PdI values were close to 0.2 for all the formulations, signifying that the liposomes had homogeneous size. The zeta potential indicates the vesicle's surface charge; this parameter reflected the liposome stability since it shows repulsive forces between particles [START_REF] Domingues | What can light scattering spectroscopy do for membrane-active peptide studies?[END_REF]. All liposomal formulations had a negative charge due to the presence of the phosphate group at the surface of the vesicles [START_REF] Ascenso | Novel tretinoin formulations: a drug-in-cyclodextrin-in-liposome approach[END_REF]. Liposomes incorporating Cyst presented zeta potential values lower than blank ones. A variation of the pH between the supernatants of blank liposomal formulation (pH 7) and Cyst loaded liposomes (pH 9.5) was noticed. Also, the positively charged amino group of Cyst can interact with the negative charges of phosphate groups of lipid membranes. The decrease of zeta potential values could be due to the negative sulfur group of cysteamine attached to the surface. Indeed, after the incubation of Cyst with the blank liposomes, the centrifugation of the mixture, and the reconstitution of the pellet with water, the vesicles presented lower zeta potential values than blank liposomes indicating that electrostatic interaction between Cyst and liposomes surface may occur.

Berleur et al. studied the interaction of Cyst with dipalmitoylphosphatidylcholine (DPPC)

using differential scanning calorimetry, electron spin resonance, and turbidimetry. They demonstrated that Cyst interacts with the DPPC membranes in their polar head region. At a low concentration (10 -4 M), Cyst behaved like a divalent cation (amine and SH groups), creating electrostatic bridges with the negatively charged phosphate groups of the polar heads of lipids leading to an increase in membrane stability. These bridges are disrupted when Cyst is present at high concentration (10 -2 and 10 -1 M) inducing a decrease in the lipid membrane rigidity; Cyst acted like a monovalent cation and displacement of the slightly charged SH extremity by the amine of Cyst is occurred [START_REF] Berleur | The binding of the radioprotective agent cysteamine with the phospholipidic membrane headgroup-interface region[END_REF].

Incorporation ratios of phospholipid and Chol, encapsulation efficiency and loading ratio of Cyst

The quantification of phospholipids showed a high incorporation ratio of phospholipids in all the formulations since the IR values were above 94 ± 1.41% (Table 2). Our results are in accordance with those of Sebaaly et al. for blank PH 90 liposomes [START_REF] Sebaaly | Effect of composition, hydrogenation of phospholipids and lyophilization on the characteristics of eugenol-loaded liposomes prepared by ethanol injection method[END_REF]. Cyst did not affect the incorporation ratio of phospholipids since it is a hydrophilic molecule, and therefore, it is expected to be encapsulated in the aqueous core of the liposomes. Concerning

Chol incorporation, the IR of Chol was higher for Lipoid S100 blank liposome (95 ± 2.0%) than for PH 90H blank liposome (80 ± 0.0%). Similar results were reported by Azzi et al., where the values of hol were higher for Lipoid S100 (85.2 ± 2. ) when compared to PH 90H liposomes (8 .2 ± 3.8 ) [START_REF] Azzi | First evaluation of drug-in-cyclodextrin-in-liposomes as an encapsulating system for nerolidol[END_REF]. Regarding Chol incorporation into Cystloaded liposomes, the IR% values were slightly decreased for Cyst-loaded PH 90H liposomes compared to blank liposomes. While for Lipoid S100-liposomes, a noticeable decrease of the IR of Chol was observed at high Cyst to lipid molar ratios (phospholipid:Chol:Cyst of 1:1.17:5.89 and 1:1.17:11.78). The mechanism underlying this decrease is unclear.

On the other hand, EE and LR% of Cyst decreased while increasing the concentration of Cyst in both types of liposomes. The higher EE and LR% values were obtained for the liposomal suspensions containing the lowest concentration of Cyst (Table 2). At this same Cyst concentration, the EE and the LR% of Cyst were higher for unsaturated liposomes when compared to saturated ones. The preparation of PH 90H liposomes required heating above the main transition temperature of the phospholipid (55°C), which could accelerate the oxidation of the product. The EE% values of Cyst were generally lower (less than 40%) than those determined for hydrophobic bioactive molecules [START_REF] Azzi | First evaluation of drug-in-cyclodextrin-in-liposomes as an encapsulating system for nerolidol[END_REF][START_REF] Sebaaly | Preparation and characterization of clove essential oil-loaded liposomes[END_REF] using the same liposome formulations. A low encapsulation efficiency usually results when encapsulating a hydrophilic drug [START_REF] Eloy | Liposomes as carriers of hydrophilic small molecule drugs: Strategies to enhance encapsulation and delivery[END_REF] because of its diffusion in and out of the lipid membrane. Thus, the drug is difficultly retained inside the liposomes (Çağdaş et al., 2014). The EE% values of caffeine were 7, 18, and 30% in Lipoid S100, Phospholipon 90G (phosphatidylcholine 95%) liposomes (Tuncay Tanriverdi, 2018), and egg or soy lecithin [START_REF] Budai | Liposomes for Topical Use: A Physico-Chemical Comparison of Vesicles Prepared from Egg or Soy Lecithin[END_REF]. Also, the EE% value of ciprofloxacin was 9% in liposomes composed of 1,2distearoyl-sn-glycero-3-phosphoglycerol-distearoylphosphatidylcholine-Chol (5:5:5 molar ratio) [START_REF] Oh | Formulation and efficacy of liposomeencapsulated antibiotics for therapy of intracellular Mycobacterium avium infection[END_REF]; that of mitomycin C was 31% in Phospholipon 80:Chol (9:6 molar ratio) [START_REF] Chetoni | Liposomeencapsulated mitomycin C for the reduction of corneal healing rate and ocular toxicity[END_REF] and the EE% value of ascorbic acid was 40% in phosphatidylcholine liposomes [START_REF] Serrano | Phosphatidylcholine liposomes as carriers to improve topical ascorbic acid treatment of skin disorders[END_REF].

Liposomes morphology

TEM images of blank liposomes and Cyst-loaded liposomes are shown in Fig. 2. These images correlated with the DLS results and showed the formation of nanometer-sized vesicles. Oligolamellar spherical-shaped vesicles were obtained for blank and Cyst-loaded liposomes.

Stability study

The stability of free and encapsulated Cyst was assessed at different temperatures (4, 25, and 37°C) and in dark and light. Aliquots were withdrawn at different time intervals, and the concentration of Cyst was determined by ion-pair chromatography. The results were expressed as the percentage of Cyst remaining in the solution as a function of time. The plots showed a straight line indicating that the oxidation of Cyst followed a zero-order reaction, and the slope corresponds to -k.

The stability of Cyst decreased when the temperature increased (Fig. 3). Cyst was oxidized after 15 h at 4°C in the dark with a degradation rate of 5.25 ± 0.09 µg.mL -1 .h -1 (Table 3).

Whereas at 37°C in the dark, the total oxidation of Cyst was achieved after 8h of incubation with a degradation rate of 11.52 ± 0.16 µg.mL -1 .h -1 . Pescina et al. confirmed the temperature dependence of Cyst oxidation after measuring the stability of the molecule at -20, 4, and 25°C [START_REF] Pescina | Effect of pH and penetration enhancers on cysteamine stability and trans-corneal transport[END_REF].

The effect of light on Cyst stability was also evaluated, and the results showed that there wasn't any significant difference between the stability of incubated yst in light or dark (Fig. 4). The degradation rate of Cyst was slightly higher at 25°C in light (10.68 ± 0.57 µg.mL -1 .h - 1 ) than in dark (10.06 ± 0.07 µg.mL -1 .h -1 ).

The encapsulation of Cyst slowed down the oxidation of Cyst in all the studied conditions.

For example, at 4°C in the dark, the total oxidation was delayed for 43 h for the encapsulated Cyst compared to the free Cyst. The protection factor, defined as the degradation rate of the free Cyst divided by the degradation rate of the encapsulated Cyst, was also calculated.

Following the encapsulation of Cyst in liposomes, protection factors from its oxidation were of 2.96, 4.7, 5, and 4.14 after storage at 4°C, 25°C at dark, 25°C in light, and 37°C, respectively (Table 3). Whereas, the use of cyclodextrin didn't affect the stability of Cyst [START_REF] Pescina | Effect of pH and penetration enhancers on cysteamine stability and trans-corneal transport[END_REF]. Besides, Dixon et al. found that the addition of antioxidants except the enzyme catalase had an insignificant effect on the prevention of Cyst oxidation [START_REF] Dixon | Novel approaches for improving stability of cysteamine formulations[END_REF].

To our knowledge, the quantification of Cyst and its degradation product cystamine is done for the first time in liposomal suspension. Cyst was totally converted to cystamine. The latter was the only product present in the liposomal suspension after the total oxidation of Cyst. The encapsulation delayed this conversion in all the studied conditions (Fig. 5).

Freeze-drying

Gharib et al. succeeded to freeze-dry PH 90H liposomes formulations using HP-β-CD (50 mM) as a cryoprotectant [START_REF] Gharib | Hydroxypropyl-ßcyclodextrin as a membrane protectant during freeze-drying of hydrogenated and nonhydrogenated liposomes and molecule-in-cyclodextrin-in-liposomes: Application to trans-anethole[END_REF]. Two formulations of Cyst loaded liposomes (PH 90H:Chol:Cyst 1:0.98:1 and 1:0.98:10 molar ratios) were selected for freeze-drying. The liposomes were then reconstituted in water and characterized for their morphology, homogeneity, size, surface charge, and LR% (Table 4).

The morphology of the liposomes was preserved after lyophilization (Fig. 6); the vesicles appeared to be oligolamellar and spherical. The presence of HP-β-CD did not affect liposome lamellarity or morphology.

The size of blank and Cyst PH 90H loaded liposomes was maintained after lyophilization regardless of the concentration of Cyst. A slight increase of the pdI values was observed after lyophilization and reconstitution for blank and Cyst-loaded liposomes. The zeta potential value was conserved for the blank liposomes; nevertheless, it significantly decreased for Cystloaded liposomes from -42 to -19 mV for the formulation PH 90H:Chol:Cyst molar ratio of 1:0.98:1 and from -48 to -33 mV for the formulation PH 90H:Chol:Cyst molar ratio of 1:0.98:10. The variation of the zeta potential was due to the loss of Cyst bound to the outer side of the membranes of the vesicles during lyophilization. Furthermore, the loading ratio of Cyst after the freeze-drying decreased from 24 to 8% in PH 90H:Chol:Cyst (1:0.98:1) and from 16 to 5% in PH 90H:Chol:Cyst (1:0.98:10). After lyophilization, liposomal suspension retained 33% of Cyst present in the PH 90H liposomal suspension. A significant decrease in the loading rate of anethole after freeze-drying in anethole loaded liposomes was also observed previously [START_REF] Gharib | Hydroxypropyl-ßcyclodextrin as a membrane protectant during freeze-drying of hydrogenated and nonhydrogenated liposomes and molecule-in-cyclodextrin-in-liposomes: Application to trans-anethole[END_REF].

Storage stability of freeze-dried forms

The mean particle size, PdI, and zeta potential values of the reconstituted liposomes after freeze-drying and after four months of storage at 4°C were evaluated (Table 4). The two formulations used in this study were stable with adequate size, PdI, and zeta potential values.

Besides, the total concentrations of Cyst in the reconstituted liposomes after freeze-drying were determined after four months and compared to those obtained at t 0 . The percentage of remaining Cyst was 16% for the formulation PH 90H:Chol:Cyst (1:0.98:10). This result was satisfactory, especially when considering the high instability of Cyst; Cyst was oxidized entirely to Cystamine after 15 h in aqueous solution and 60 h in liposomal suspensions after storage at 4°C (Fig. 3A). The storage conditions are critical in the case of Cyst. Taking into account its instability, the eye-drop Cystaran TM (Cyst ophthalmic solution 0.44%) has to be stored at -20°C, and administered hourly during the daytime. After its opening, it should be used within one week [START_REF] Huynh | Cysteamine ophthalmic solution 0.44% for the treatment of corneal cystine crystals in cystinosis[END_REF]. Cyst hydrochloride (0.5%) formulated ophthalmic preparation is easily oxidized within the first week after storage at +4°C, rendering the preparation less effective [START_REF] Reda | Effect of Storage Conditions on Stability of Ophthalmological Compounded Cysteamine Eye Drops[END_REF]. In comparison with the aqueous liposomal suspension, the freeze-dried form conserved Cyst for a more extended period. The elimination of the water from the liposomal suspension is the factor that induced this improvement of Cyst stability. The presence of water will lead to the degradation of Cyst to cystamine [START_REF] Gana | An Integrated View of the Influence of Temperature, Pressure, and Humidity on the Stability of Trimorphic Cysteamine Hydrochloride[END_REF].

Conclusion

In this study, we proved that the encapsulation of Cyst in conventional liposomes improved its stability. Cyst-loaded liposomes were nanometric and presented a negative surface charge.

In addition to its incorporation in vesicles, Cyst binds to the surface of the vesicle. The loading ratio of Cyst in liposomes varied between 12 and 24% depending on Cyst to lipid molar ratio. Besides, in this work, we succeeded in preparing freeze-dried liposomes that retain a noticeable amount of Cyst after four months of storage at 4°C. These results could help to develop various pharmaceutical forms of Cyst and enlarge its applications.

Figures Legends

  concentrations of Cyst in the liposomal suspension before and after storage of dried liposomes at 4°C for 4 months. The experiment was carried out in triplicate.

Figure 1 :

 1 Figure 1: Structure of Cyst.

Figure 2 :

 2 Figure 2: TEM images for blank PH 90H liposomes (A), Cyst-loaded PH 90H liposomes (PH

Figure 3 :

 3 Figure 3: Stability study of free and encapsulated Cyst in the dark at 4°C (A), 25°C (B), and

Figure 4 :

 4 Figure 4: Stability study of free and encapsulated Cyst at 25°C in the dark (A) and in light

Figure 5 :

 5 Figure 5: Percentage of formed cystamine in free and Cyst-loaded liposomes in the dark at

Figure 6 :

 6 Figure 6: TEM images for blank (A) and Cyst-loaded-PH 90H liposomes (B) after freeze-

Figure

  Figure(s)

Figure 6

 6 Figure 6

Table 1 :

 1 Size, pdI, and zeta potential of blank and Cyst loaded PH 90H and Lipoid S100 liposomes.

Table 3 :

 3 Degradation rate constants of free and encapsulated Cyst at 4°C (dark), 25°C (dark and light), and 37°C (dark).

	Formulations	Temperature (°C)	Degradation rate constant k (µg. mL -1 .h -1 )	Protection factor
	Free Cyst Cyst loaded liposome	4 (dark)	5.25 ± 0.09 1.77 ± 0.18	-2.96
	Free Cyst Cyst loaded liposome	25 (dark)	10.06 ± 0.07 2.14 ± 0.12	-4.7
	Free Cyst Cyst loaded liposome	25 (light)	10.68 ± 0.57 2.12 ± 0.07	-5
	Free Cyst	37 (dark)	11.52 ± 0.16	-
	Cyst loaded liposome		2.78 ± 0.05	4.14

Table 4 :

 4 Size, polydispersity index, zeta potential values for blank and Cyst-loaded liposomes and loading rate of Cyst in liposomes made from PH 90H before and after freeze-drying at t 0 and after four months of storage in powder form (bold). The values obtained after lyophilization were compared to those before lyophilization.

			Before freeze-drying			After freeze-drying
		Size (nm)	pdI	Zeta potential (mV)	LR%	Size (nm)	pdI	Zeta (mV) potential	LR%
	Blank PH 90H liposome	187 ± 42	0.15 ± 0.08 -17 ± 2.8		172 ± 14 0.23 ± 0.03 -18 ± 2.5
	PH 90H:Chol:Cyst 1:0.98:1	181 ± 19	0.12 ± 0.01 -37 ± 0.2	24 ± 2	166 ± 2 231 ± 36	0.15 ± 0.05 0.29 ± 0.05	-19 ± 0.8 -18 ± 3.5	8 ± 1.2 0 ± 0
	PH 90H:Chol:Cyst 1:0.98:10	187 ± 18	0.18 ± 0.06 -48 ± 4	16 ± 4	193 ± 27 226 ± 15	0.32 ± 0.02 0.23 ± 0.02	-33 ± 1.4 -34 ± 0.5	5 ± 0.5 0.8 ±0.11

Acknowledgements

Authors thank the esearch unding Program at the Lebanese University and the "Agence Universitaire de la rancophonie, projet P S " for supporting the project (2018-2020).

Authors also thank Géraldine Agusti for the TEM measurements. Electron microscopy studies have been done at the " entre Technologique des Microstructures" -Claude Bernard University of Lyon.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 1:1.17:1.17 94 ± 2.0 156 ± 13 6 ± 2.0 5 ± 0.2 -32.0 ± 0.41 0.3 ± 0.0 1:1.17:1.77 97 ± 1.0 150 ± 3 3 ± 1.0 5 ± 0.3 -34.0 ± 3.70 0.3 ± 0.0 1:1.17:2.36 97 ± 1.0 149 ± 1 3 ± 1.0 5 ± 1.0 -34.3 ± 1.45 0.3 ± 0.0 1:1.17:5.89 97 ± 1.0 152 ± 11 3 ± 1.0 5 ± 0.2 -42.5 ± 1.17 0.3 ± 0.1 1:1.17:11.78 97 ± 1.6 160 ± 8 3 ± 1.8 4 ± 0.2 -39.2 ± 1.30 0.2 ± 0.0


W Liu

M Chiao

M R Collier

T Cravens

M Galeazzi

D Koutroumpa

K D Kuntz

R Lallement

S T Lepri

D Mccammon

K Morgan

F S Porter

S L Snowden

N E Thomas

Y Uprety

E Ursino

B M Walsh

THE STRUCTURE OF THE LOCAL HOT BUBBLE

Keywords: ISM, bubbles -ISM, structure -X-rays, diffuse background -X-rays, ISM

DXL (Diffuse X-rays from the Local Galaxy) is a sounding rocket mission designed to quantify and characterize the contribution of Solar Wind Charge eXchange (SWCX) to the Diffuse X-ray Background and study the properties of the Local Hot Bubble (LHB). Based on the results from the DXL mission, we quantified and removed the contribution of SWCX to the diffuse X-ray background measured by the ROSAT All Sky Survey (RASS). The "cleaned" maps were used to investigate the physical properties of the LHB. Assuming thermal ionization equilibrium, we measured a highly uniform temperature distributed around kT =0.097 keV±0.013 keV (FWHM)±0.006 keV (systematic). We also generated a thermal emission measure map and used it to characterize the three-dimensional (3D) structure of the LHB which we found to be in good agreement with the structure of the local cavity measured from dust and gas.

INTRODUCTION

The diffuse soft X-ray background observed at 1/4 keV in the ROSAT R12 band (Snowden et al. 1997) is dominated by a local source that shows no sign of absorption by cool interstellar gas (Juda et al. 1991). One optical depth at 1/4 keV is roughly 1×10 20 HI cm -2 , a quantity reached within 50 pc at average interstellar densities. An irregular "local cavity" extending about 100 pc from the Sun was shown by the Copernicus satellite to be almost entirely devoid of cool gas (Savage & Jenkins 1972;Jenkins & Meloy 1974;Knapp 1975). If filled with 10 6 K gas at a reasonable pressure, the cavity could produce observed "local" X-rays (Sanders et al. 1977). The portion of the local cavity filled with this hot gas was dubbed the Local Hot Bubble (LHB) (Sanders galeazzi@physics.miami.edu et al. 1977;Tanaka & Bleeker 1977;Cox & Snowden 1986), and the enhanced X-ray emitting areas at intermediate latitudes were found to correlate well with minima in the measured neutral gas column (Snowden et al. 1990), as if the cool gas had been displaced by the hot gas. ROSAT demonstrated that a smaller portion of the soft X-ray background is due to the Galactic halo (Burrows & Mendenhall 1991;Snowden et al. 1991). Emission from the hot Galactic halo contributes significantly only in areas of low absorption at intermediate and high Galactic latitudes.

This simple picture was upset by the discovery of diffuse X-ray emission from within the solar system due to Solar Wind Charge eXchange (SWCX), which could provide some or all the soft diffuse X-ray emission at 1/4 keV (Cravens 2000;Cravens et al. 2001;Robertson & Cravens 2003;Lallement 2004;Koutroumpa et al. 2009). SWCX emission is generated when the highly charged solar wind ions interact with the neutral ma-Liu, et al.

terials within the solar system, gaining an electron in a highly excited state which then decays emitting an X-ray photon with the characteristic energy of the ion. In order to improve our understanding of the local diffuse X-ray emission and the structure of the LHB, it is essential to remove the contamination of the SWCX. However, despite many efforts, an accurate estimation of the SWCX is quite difficult, especially in the 1/4 keV band, due to the poorly known cross sections for producing the many X-ray lines from SWCX, limited data on heavy ion fluxes in the Solar Wind, and the general spectral similarity of SWCX and thermal emission (Cravens 2000;Lallement 2004;Koutroumpa et al. 2006;Snowden et al. 2009;Henley & Shelton 2008;Crowder et al. 2012;Yoshino et al. 2009). Efforts to estimate the SWCX contribution to historical measurements for the diffuse X-ray background, such as in the ROSAT All-Sky Survey (RASS) are even more problematic due to the limited solar wind data (see Kuntz et al. 2015, for further discussion).

DXL (Galeazzi et al. 2011(Galeazzi et al. , 2012;;[START_REF] Thomas | Gamma-Ray Space Instrumentation for Astronomy XVIII[END_REF]) is a sounding rocket mission designed to quantify and characterize the contribution of SWCX to the diffuse Xray emission. To separate the SWCX contribution, DXL uses the spatial signature of SWCX emission due to the "helium focusing cone", a higher neutral He density region downwind of the Sun (Michels et al. 2002;Snowden et al. 2009;[START_REF] Thomas | Gamma-Ray Space Instrumentation for Astronomy XVIII[END_REF]. By comparing the DXL data and the RASS data along the DXL scan path, our team measured the broad band averaged cross sections and provided a significantly more accurate empirical estimate of the SWCX emission. DXL estimated the total SWCX contribution to be ∼ 40% of the X-ray flux at 1/4 keV in the Galactic plane (Galeazzi et al. 2014), supporting the previous picture of a hot bubble filling the local interstellar medium in all directions and accounting for the remaining ∼60% in the plane. Based on the results from Galeazzi et al. (2014), Snowden et al. (2014) showed that the gas pressure from the remaining local emission is in pressure equilibrium with the local interstellar clouds, eliminating the long standing pressure problem of the LHB (Jenkins 2009).

In this paper we reevaluate the properties of the LHB based on the RASS data (Snowden et al. 1997) combined with the estimate of the SWCX contribution from DXL. We focused on the R1 (∼0.11-0.284 keV) and R2 (∼0.14-0.284 keV) data, as the LHB contribution to the R4 (∼0.44-1.01 keV) and R5 (∼0.56-1.21 keV) bands is negligible. In § 2 we describe how to remove the SWCX emission from the RASS data, and to estimate the LHB temperature and emission measure, § 3 contains the results, and conclusions are in § 4.

2. DATA ANALYSIS Snowden et al. (1998Snowden et al. ( , 2000) ) used the shadows cast by nearby (100-200 pc) clouds to estimate and remove the contribution from background emission (Galactic halo and extragalactic components) to the RASS R1 and R2 maps, producing clean "local" maps (<100-200 pc). With the advance in X-ray telescopes, the shadow technique is now feasible for individual pointing for spectroscopy study to disentangle the foreground and background (Galeazzi et al. 2007;Smith et al. 2007;Gupta et al. 2009;Henley et al. 2015;Liu et al. 2016;Ursino et al. 2016). These "local" maps should contain only the contribution from SWCX, both heliospheric (S( , b, t)) and geocoronal (G), and the LHB (L( , b)). For each RASS band we can therefore write the total flux, F ( , b), as:

F ( , b, t) = S( , b, t) + L( , b) + G (1)
Following the procedure from [START_REF] Uprety | [END_REF], based on the models of Koutroumpa et al. (2006), the heliospheric component can be written as

S( , b, t) = β(t) × N ( , b) (2)
where

β(t) = n p (R 0 , t)v rel (t)α He (3)
depends on the solar wind properties and the cross section with neutrals (n p (R 0 , t) is the proton density at R 0 = 1AU , v rel is the relative speed between solar wind and neutral flow, and α He is the compound cross-section for Helium), and

N ( , b) = n He R 2 ds + α H α He n H R 2 ds (4)
where n R 2 ds is the integrated neutral column density along the line of sight, weighted by one over the distance from the Sun squared, and α H α He is the ratio between cross sections with H and He. [START_REF] Uprety | [END_REF] combined data from the same part of the sky from DXL and RASS and found the best fit parameter for β(t) for each RASS band for given values of G and α H /α He . Therefore, the heliospheric SWCX contribution to any RASS band for any direction can be directly estimated given the neutral distribution. Figure 1 shows the Aitoff-Hammer projection of the H and He neutral integral during the RASS campaign, calculated based on a well determined model for the interstellar neutral distributions within the solar system (Lallement et al. 1985a,b;Lallement 2004;Koutroumpa et al. 2006). The sharp edges visible in this maps are due to abrupt shifts in vantage point around the Earth's orbit during the ROSAT survey, since the survey comes back to its starting point after six months and there were missed sections that were backfilled at later times. [START_REF] Uprety | [END_REF] extensively discuss the use of different combinations of G and α H /α He and their effect of the systematic error of their results. For this investigation, we use their best fit parameters for β(t) shown in their Table 2, combined with the neutral distribution shown in Figure 1 to generate maps of SWCX contribution to both R1 and R2. The model SWCX maps were then subtracted from the local maps of Snowden et al. (1998Snowden et al. ( , 2000) ) to produce the "clean" LHB maps in the R1 and R2 band which contain only the LHB emission. We note that these maps are different from those shown in [START_REF] Uprety | [END_REF]. [START_REF] Uprety | [END_REF] subtracted the SWCX contribution for the total R1 and R2 bands, producing "clean" astrophysical maps, containing both local and non-local components. Figure 2 shows the Aitoff-Hammer projection of the total LHB emission in R1+R2 band, and Figure 3 shows the projection of the R2/R1 ratio of the LHB (notice that all the maps are smoothed since the LHB emission is assumed to vary smoothly over large angular scales and our analysis is insensitive to any variation on finer scales).

Assuming that the LHB is in collisional equilibrium and can be well represented by a single, unabsorbed thermal component, it is possible to estimate the temperature of the LHB in any given direction based on the ratio of the R2/R1 bands. Unfortunately, none of the current thermal models available in XSPEC 1 are particularly accurate in the R1 and R2 bands. The Raymond-Smith model (Raymond & Smith 1977) estimates the emission of a large number of weak lines that are known to be present, but it lacks accurate excitation rates and wavelengths. The Mekal model (Mewe et al. 1985;Kaastra & Mewe 1993) is identical in treatment of ionization balance with Raymond-Smith model, but has many more lines and updated Fe L calculations. The APEC model (Smith et al. 2001) includes only transitions for which accurate atomic rates are available and lacks many lines at low energy. In Figure 4 we plot the R2/R1 ratio as a function of temperature using the Raymond-Smith, Mekal, and APEC models with Anders & Grevesse (1989) abundance table. It is immediately evident that the curve for APEC model is quite different from the other two and therefore any conclusion will depend on the model used. However, as it turns out, the systematic effect introduced by the choice of model is not large as our LHB data, as we will show in the next section, are clustered in the region where the curves nearly overlap.

We also point out that in the RASS maps there are brighter regions associated with additional X-ray emission from extended sources, e.g., the Cygnus Loop, Vela SNRs, the Galactic halo beyond the Draco Clouds,

w i t h S W C X w i t h o u t S W C X C o u n t R 2 / R 1 Figure 5.
The distribution of R2/R1 ratio before subtracting the SWCX (in black) and after subtracting the SWCX (in red).

and the Monogem Ring. We have excluded them in our study by both setting an upper limit in the RASS R4+R5 value and manually removing regions associated with known structures unrelated to the LHB.

3. RESULTS

The LHB Temperature

In Figure 5 we show the distribution of the R2/R1 ratio over the whole sky before (in black) and after (in red) removing the SWCX contribution from the RASS data. The peak of R2/R1 is shifted from ∼ 1.09 to 0.86, with a FWHM of 0.16. This corresponds to a temperature of kT =0.099 keV using the APEC model, 0.103 keV for Raymond-Smith model and 0.091 keV for Mekal model. The difference between APEC and Raymond-Smith models is very small since the R2/R1 ratio of 0.86 is in a region where the three lines in Figure 4 are very close to each other. Combining the three results, we estimate the peak temperature as kT =0.097 keV, with a systematic error of 0.006 keV. This systematic error also includes the systematic uncertainties in the SWCX parameters discussed in [START_REF] Uprety | [END_REF]. Based on the the relation between the R2/R1, the temperature of the LHB is therefore cooler than previous estimates based on the maps without SWCX subtraction (Snowden et al. 1998;Kuntz & Snowden 2000). In the left of Figure 6 we also show the Aitoff-Hammer projection of the temperature of the LHB. The distribution of the LHB temperature is quite uniform with a FWHM of 0.013 keV. In the figure, the areas affected by non-LHB extended sources have been interpolated and circled in black.

The LHB Emission Measure and Size

We also used the new R1 and R2 maps to extract the Emission Measure (EM) of the emitting plasma for each direction in the sky based on the APEC model. The Aitoff-Hammer projection of the LHB EM is shown on the right of Figure 6. The emission measure is generally larger towards high latitude while smaller at low latitude in the northern hemisphere. In the southern hemisphere the emission measure is small from 0 • < l < 180 • . It is small at low latitude and large at high latitude from 180 • < l < 360 • . Over the whole sky, the distribution ranges from ∼0.8×10 -3 cm -6 pc to ∼6.5×10 -3 cm -6 pc.

Assuming that the electron density in the LHB is constant, we can use the EM to estimate the size of the LHB. For constant electron density, the emission measure is expressed as EM = n e n p L, where n e and n p are the electron and proton densities, L is the path length through the LHB emitting plasma. Adopting the electron value of n e = 4.68 × 10 -3 cm -3 (Snowden et al. 2014), we estimated the extension of the LHB in all directions and we built its three-dimensional structure.

Figures 7 and8 show the extension of the LHB along great-circle cuts through the Galactic pole and Galactic plane. The dash lines correspond to regions contaminated by distant bright sources (Snowden et al. 1998).

We also compared our results with measurements at other wavelenghts. Figure 9 shows the distribution of the local Inter-Stellar Medium (ISM) in the Galactic plane from reddening data (Lallement et al. 2014). The superimposed black line represents the contour of the LHB from our measurement which is the same as in Figure 8. The shape of the LHB matches the boundary of the local cavity very well after removing the contribution of SWCX showing that the LHB and the local cavity are closely correlated. Figure 10 shows the same data as Figure 9 but in the vertical plane and on a smaller scale. Although there is no clear boundary information of the local cavity toward high latitude, our contour matches well the local cavity at low latitude.

Based on our reconstruction of the LHB, we also calculated the total energy currently enclosed in the LHB as 3.38×10 50 ergs which is about 15.6 times smaller than estimated without removing the SWCX contribution to the RASS maps. We note that, while this is consistent with the energy released in a single supernova explosion, the LHB has been cooling away for millions of years, and its size and longevity remain inconsistent with a single SN explosion (Cox & Anderson 1982;Cox & Snowden 1986;Smith & Cox 2001).

There are a few systematics which affects our results. The first is the choice of model to convert the R1 to R2 ratio to temperature, which has already been discussed, and contributes a systematic error of kT =0.006 keV to the estimate of the LHB temperature. We also investigated the effect of using different abundance tables. For example, we used the abundance table by Wilms et al. (2000) and foud it has a smaller effect than the choice of model, contributing a systematic error of kT =0.003 keV. The second major source of systematic uncertainty is the assumption made to derive the value of β(t) by fitting the DXL data, namely the choice of α H /αHe and G. As Liu, et al. discussed in [START_REF] Uprety | [END_REF], however, although the value of β(t) varies significantly for different combinations of α H /αHe and G, the total SWCX contributions are very similar. We tested different combinations and, as expected, we found that the distribution of R2/R1 ratio after subtracting the SWCX is generally similar, and the systematic uncertainty on the peak of the distribution is ∆ R2 R1 = 0.040, corresponding a temperature difference of kT =0.003 keV. Another possible source of systematics is the fact that the RASS R1 rate is systematically lower than other 1/4 keV all-sky surveys, e.g., the University of Wisconsin sky survey (Dan McCammon, private communication, see also in McCammon & Sanders (1990)). Considering an 18% correction on the R1 band, to match the RASS results with previous surveys, the peak of the R2/R1 distribution would then be shifted to 0.73, corresponding to a lower temperature of 0.088 keV.

SUMMARY

Based on the data from the DXL sounding rocket mission, we quantified and removed the SWCX contribution to the foreground diffuse X-ray emission, and obtained a "cleaned" map of the LHB emission from the RASS data. Assuming that the LHB is in thermal ionization equilibrium, we measured the temperature of the LHB from the R2/R1 ratio, and estimated its emission measure over the whole sky. We found that the estimated temperature of the LHB is cooler after the contamination of the SWCX is removed. Assuming the LHB has a constant electron density, we also estimated the size of the LHB in each direction and built a three-dimensional model of the LHB, which matches quite well with maps of the local cavity from reddening data. This work was supported by NASA award numbers NNX11AF04G and NNX09AF09G.

Figure 1 .

 1 Figure 1. The Aitoff-Hammer projection of the neutral integral distribution for H (left) and He (right) in units of cm -3 AU -1 .

Figure 2 .

 2 Figure 2. The Aitoff-Hammer projection of the total "cleaned" LHB emission in the RASS R1+R2 band in RU after removing both the non-local components and SWCX contribution.

Figure 3 .

 3 Figure 3. The Aitoff-Hammer projection of the R2/R1 band ratio of the "cleaned" LHB.

1Figure 4 .

 4 Figure 4. R2/R1 ratio as a function of temperature from Raymond-Smith model (black), Mekal model (blue), and APEC model (red) with Anders & Grevesse (1989) abundance table.

Figure 6 .

 6 Figure 6. The Aitoff-Hammer projection of the LHB temperature (left) in keV and the LHB emission measure in cm -6 pc. Regions affected by non-LHB extended sources have been interpolated and circled in black.

Figure 7 .

 7 Figure 7. The radius of the LHB in great-circle cuts through the Galactic poles along the labeled longitude. The red dashed line corresponds to directions of non-LHB bright extended sources.

805 Figure 8 .Figure 9 .Figure 10 .

 8058910 Figure 8. The radius of the LHB in great-circle cuts through the Galactic plane. The red dashed line corresponds to directions of non-LHB bright extended sources.