APS/123-QED Collaborative tagging as a tripartite network R. Lambiotte* and M. ausloost SUPRATECS, Universite de liege B5 Sart-Tilman, B-4000 Liege, Belgium ( Dated:09/07/2005) abstract We describe online collaborative communities by tripartite networks, the nodes being persons items and tags. We introduce projection methods in order to uncover the structures of the networks e communities of users, genre families. To do so, we focus on the correlations between the nodes depending on their profiles, and use percolation techniques that consist in removing less correlated links and observing the shaping of disconnected islands. The structuring of the network is visualised by using a tree representation. The notion of diversity in the system is also discussed PACS numbers: 89.75 Fb. 89.65.Ef. 64.60.Ak Electronic address: Renaud. Lambiotte @ulg. ac be t Electronic address: Marcel. Ausloos Qulg ac be
arXiv:cs/0512090v2 [cs.DS] 29 Dec 2005 APS/123-QED Collaborative tagging as a tripartite network R. Lambiotte ∗ and M. Ausloos † SUPRATECS, Universit´e de Li`ege, B5 Sart-Tilman, B-4000 Li`ege, Belgium (Dated: 09/07/2005) Abstract We describe online collaborative communities by tripartite networks, the nodes being persons, items and tags. We introduce projection methods in order to uncover the structures of the networks, i.e. communities of users, genre families... To do so, we focus on the correlations between the nodes, depending on their profiles, and use percolation techniques that consist in removing less correlated links and observing the shaping of disconnected islands. The structuring of the network is visualised by using a tree representation. The notion of diversity in the system is also discussed. PACS numbers: 89.75.Fb, 89.65.Ef, 64.60.Ak ∗Electronic address: Renaud.Lambiotte@ulg.ac.be †Electronic address: Marcel.Ausloos@ulg.ac.be 1
I. INTRODUCTION Recently, new kinds of websites have been dedicated to the sharing of people's habits and tastes, examples including their preferences in music, scientific articles, movies, websites These sites allow members to upload from their own computer a library that characterises their habits in the corresponding topic (an iTunes music library for instance), and next to create a web page containing this list of items. Additionally, the website proposes the users to discover new content by comparing their taste with that of other users, thereby helping them discover new musics/books/websites.. that should(statistically) fit their profile This method rests on a feedback between the users and a central server, and is usually called collaborative filtering. The emergence of these collaborative websites answers the needs of Internet users to retrieve useful and coherent informations from the millions of pages and data that form the Web. Let us stress that the use of statistical methods in order to make coherent suggestions from a user profile is common in commercial websites, i.e Amazon. The main particularities of colaborative systems are:(i) their non-commercial purpose, even though the frontier with commercial companies is more and more vague(see for instance the acquisition of del icio us by Yahoo in November 2005 ); (ii)their transparency, namely these sites are relatively open and do not hide the profiles of each user, contrary to Amazon for instance. From a scientific point of view, this transparency opens perspectives in order to perform large scale experiences(including thousands of people) on taste formation quantitative sociology, musicology. The available data also suggest alternative methods in order to perform large scale classifications of music/science/internet. Those sub-divisions should be based on the intrinsic structure of the audience of the items In parallel with this sharing and statistical comparing of content, collaborative websites usually propose tagging possibilities. This process, called"folksonomy"(short for "folk taxonomy) means that the websites allow users to publicly tag their shared content, the key point being that their tag is not only accessible to themselves, but also to the whole ensemble of users. For instance, in the case of music sharing habits, a group like The Beatles is described in different ways, i.e. pop, 60s, britpop., that depend on the different backgrounds, tastes, music knowledge or network of acquaintances. of the users Both methods, i.e. collaborative filtering(CF) and collaborative tagging(CT) lead to complex networks from which structures have to be extracted in order to deliver useful infor
I. INTRODUCTION Recently, new kinds of websites have been dedicated to the sharing of people’s habits and tastes, examples including their preferences in music, scientific articles, movies, websites... These sites allow members to upload from their own computer a library that characterises their habits in the corresponding topic (an iTunes music library for instance), and next to create a web page containing this list of items. Additionally, the website proposes the users to discover new content by comparing their taste with that of other users, thereby helping them discover new musics/books/websites... that should (statistically) fit their profile. This method rests on a feedback between the users and a central server, and is usually called collaborative filtering. The emergence of these collaborative websites answers the needs of Internet users to retrieve useful and coherent informations from the millions of pages and data that form the Web. Let us stress that the use of statistical methods in order to make coherent suggestions from a user profile is common in commercial websites, i.e. Amazon. The main particularities of collaborative systems are: (i) their non-commercial purpose, even though the frontier with commercial companies is more and more vague (see for instance the acquisition of del.icio.us by Yahoo in November 2005); (ii) their transparency, namely these sites are relatively open and do not hide the profiles of each user, contrary to Amazon for instance. From a scientific point of view, this transparency opens perspectives in order to perform large scale experiences (including thousands of people) on taste formation, quantitative sociology, musicology... The available data also suggest alternative methods in order to perform large scale classifications of music/science/internet. Those sub-divisions should be based on the intrinsic structure of the audience of the items. In parallel with this sharing and statistical comparing of content, collaborative websites usually propose tagging possibilities. This process, called ”folksonomy” (short for ”folk taxonomy”) means that the websites allow users to publicly tag their shared content, the key point being that their tag is not only accessible to themselves, but also to the whole ensemble of users. For instance, in the case of music sharing habits, a group like The Beatles is described in different ways, i.e. pop, 60s, britpop..., that depend on the different backgrounds, tastes, music knowledge or network of acquaintances... of the users. Both methods, i.e. collaborative filtering (CF) and collaborative tagging (CT) lead to complex networks from which structures have to be extracted in order to deliver useful infor- 2
mations to users. In this work, we discuss methods that lead to the identification of a priori unknown collective behaviours, and to a hierarchical representation of the network struc- turing. To do so, we focus on empirical data extracted from websites specialised in musi e. g. audioscrobbler. com and musicmobs. com, and in scientific articles, i.e. citeulike. com. We how that the tagging collaborative process leads to a tripartite network, i.e. a network with three different kinds of nodes(the users, the items and the tags) and where the links relate three nodes of different kinds. The next step of the analysis consists in projecting the tri- partite network on lower order networks. To do so, we evaluate the correlations between the items/tags, depending on their use. Filtering methods 1, i.e. percolation idea-based(PIB methods, allow to uncover the collective behaviours. The resulting hierarchical structure of the network leads to a statistical definition of the notion of genre, and draws a direct link between collaborative filtering and taxonomy. Finally, we discuss methods for measuring the diversity of people 2 II. CLASSIFICATION METHODS In this section, we give a short review of the usual strategies that can be used to classify and organise content 3, as well as their main differences 1. Taxonomies Taxonomies include the Dewey Decimal classification for libraries, computer directory systems, the Linnean system of classifying living things. By construction, a taxonomy is hierarchical and exclusive. In these systems, each item is associated to one category which belongs to a more general category, each category belonging to a more general one until the root of the tree is attained. For instance. the music artists Charlie parker and Charles Mingus can reasonably be classified in the categories Bebop and Free Jazz, both of them belonging to the category Jazz. By construction, taxonomies lead to an automatic tructuring of content into hierarchical structures, that allow users to search with different levels of specificity
mations to users. In this work, we discuss methods that lead to the identification of a priori unknown collective behaviours, and to a hierarchical representation of the network structuring. To do so, we focus on empirical data extracted from websites specialised in music, e.g. audioscrobbler.com and musicmobs.com, and in scientific articles, i.e. citeulike.com. We show that the tagging collaborative process leads to a tripartite network, i.e. a network with three different kinds of nodes (the users, the items and the tags) and where the links relate three nodes of different kinds. The next step of the analysis consists in projecting the tripartite network on lower order networks. To do so, we evaluate the correlations between the items/tags, depending on their use. Filtering methods [1], i.e. percolation idea-based (PIB) methods, allow to uncover the collective behaviours. The resulting hierarchical structure of the network leads to a statistical definition of the notion of genre, and draws a direct link between collaborative filtering and taxonomy. Finally, we discuss methods for measuring the diversity of people [2]. II. CLASSIFICATION METHODS In this section, we give a short review of the usual strategies that can be used in order to classify and organise content [3], as well as their main differences. 1. Taxonomies Taxonomies include the Dewey Decimal classification for libraries, computer directory systems, the Linnean system of classifying living things... By construction, a taxonomy is hierarchical and exclusive. In these systems, each item is associated to one category, which belongs to a more general category, each category belonging to a more general one until the root of the tree is attained. For instance, the music artists Charlie Parker and Charles Mingus can reasonably be classified in the categories Bebop and Free Jazz, both of them belonging to the category Jazz. By construction, taxonomies lead to an automatic structuring of content into hierarchical structures, that allow users to search with different levels of specificity. 3
2. Tagging systems Tagging systems are non-hierarchical and non-exclusive. They consist in associating to each item a list of keywords, all the keywords being considered at the same level. Tagging systems are especially adapted for content that is not easily categorisable into exclusive categories, and for situations when no hierarchical difference exists between categories. Let us take the example of music. In addition to the usual genre classification of a music group a listener may consider additional terms describing its mood, i.e. Sad, Nervous, Happy a taxonomical description requires a hierarchical organisation, i.e. a music group is placed in a directory Jazz/Sad or in a directory Sad/Jazz. In a case when the importance of each characteristic is not clear, such a hierarchy is obviously not adequate, and may lead to problems in order to retrieve all relevant items. For instance a music group placed in Sad/Jazz is not found is the hierarchically higher category Jaz Collective description Usually, the choice of the set of tags available is done by an authority, such as a librarian or an editor, while the attribution of these tags is performed by the same authority or by the creators of the item, i.e. the authors of a scientific paper It is only recently that websites have led to the emergence of collaborative tagging, also called folksonomy. Contrary to the usual tagging classification systems, folksonomy is: (i) anarchic: the choice for the keywords is not restrained by any carcan(contrary to PACS classifications in physics literature for instance), but may include any word composed of letters (ii)democratic: the tagging is equivalently performed by a large ensemble of persons, and not by a central one In itself, folksonomy is especially suitable for systems where no authority is present in order to organise the classifications. That is one of the reasons why it is gaining popularity on the web. The democratic aspect of the method also leads to a very rich description for each item. Namely items that are tagged by many persons are usually characterised by a spectrum of tags, revealing the diverse levels of descriptions associated to them. Nonetheless the richness of the methods may also be a weakness in practice, in order to retrieve useful
2. Tagging systems Tagging systems are non-hierarchical and non-exclusive. They consist in associating to each item a list of keywords, all the keywords being considered at the same level. Tagging systems are especially adapted for content that is not easily categorisable into exclusive categories, and for situations when no hierarchical difference exists between categories. Let us take the example of music. In addition to the usual genre classification of a music group, a listener may consider additional terms describing its mood, i.e. Sad, Nervous, Happy... A taxonomical description requires a hierarchical organisation, i.e. a music group is placed in a directory Jazz/Sad or in a directory Sad/Jazz. In a case when the importance of each characteristic is not clear, such a hierarchy is obviously not adequate, and may lead to problems in order to retrieve all relevant items. For instance, a music group placed in Sad/Jazz is not found is the hieriarchically higher category Jazz. 3. Collective description Usually, the choice of the set of tags available is done by an authority, such as a librarian or an editor, while the attribution of these tags is performed by the same authority or by the creators of the item, i.e. the authors of a scientific paper. It is only recently that websites have led to the emergence of collaborative tagging, also called folksonomy. Contrary to the usual tagging classification systems, folksonomy is: (i) anarchic: the choice for the keywords is not restrained by any carcan (contrary to PACS classifications in physics literature for instance), but may include any word composed of letters. (ii) democratic: the tagging is equivalently performed by a large ensemble of persons, and not by a central one. In itself, folksonomy is especially suitable for systems where no authority is present in order to organise the classifications. That is one of the reasons why it is gaining popularity on the web. The democratic aspect of the method also leads to a very rich description for each item. Namely items that are tagged by many persons are usually characterised by a spectrum of tags, revealing the diverse levels of descriptions associated to them. Nonetheless, the richness of the methods may also be a weakness in practice, in order to retrieve useful 4
information from a database for instance. This is due to the very large number of tags associated to each item, as well as to the use of terms that have not been optimised by an authority, such as synonyms or words written with several orthograph 4. Beyong the words Finally, collaborative filtering( CF) is also a democratic method of classification, but contrary to the above classifying methods, it does not require the use of words in order to attribute a category to items. Usually, Cf uses statistical methods in order to link items depending on the people who use it. In the case of music, for instance, music groups are related if they have common audiences. Consequently, the observed categories rest on a more subtle description of items than the limited use of tagging words. This is especially true in music where more and more groups prefer to be associated to influential groups rather thantobeingcategorisedinusualsubdivisions4.Inawebsitelikewww.myspacecomfor instance, the attributes of an artist encompass such a list of influences III. METHODOLOGY A. Tripartite structure The structure of collaborative websites can be viewed as a tripartite network. Namely, it is a network composed of three kinds of nodes: i) the persons or users u; ii)the items i that can be music groups or scientific articles; iii) the tags I that are used by the person u to describe the item i. Depending on the systems under consideration, a person can use one or several tags on each item. The resulting network can be represented by a graph where edges run between the item i and the user u, passing through the tag 1. moreover, a weight is attributed to each link depending on the number of tags given by u to i. For instance, if u uses two tags for i, the weight of the links is 3 Let us note ny the number of users, nit the number of items, and nr the number of tags in the considered sample. Consequently, each listener u can be characterised by the nIt X nT
information from a database for instance. This is due to the very large number of tags associated to each item, as well as to the use of terms that have not been optimised by an authority, such as synonyms or words written with several orthographs. 4. Beyong the words Finally, collaborative filtering (CF) is also a democratic method of classification, but, contrary to the above classifying methods, it does not require the use of words in order to attribute a category to items. Usually, CF uses statistical methods in order to link items depending on the people who use it. In the case of music, for instance, music groups are related if they have common audiences. Consequently, the observed categories rest on a more subtle description of items than the limited use of tagging words. This is especially true in music where more and more groups prefer to be associated to influential groups rather than to being categorised in usual subdivisions [4]. In a website like www.myspace.com, for instance, the attributes of an artist encompass such a list of influences. III. METHODOLOGY A. Tripartite structure The structure of collaborative websites can be viewed as a tripartite network. Namely, it is a network composed of three kinds of nodes: i) the persons or users µ; ii) the items i that can be music groups or scientific articles; iii) the tags I that are used by the person µ to describe the item i. Depending on the systems under consideration, a person can use one or several tags on each item. The resulting network can be represented by a graph where edges run between the item i and the user µ, passing through the tag I. Moreover, a weight is attributed to each link depending on the number of tags given by µ to i. For instance, if µ uses two tags for i, the weight of the links is 1 2 . Let us note nU the number of users, nIt the number of items, and nT the number of tags in the considered sample. Consequently, each listener µ can be characterised by the nIt ×nT 5
众 △ FIG. 1: Tripartite structure of the tagging system. In this example, user u owns two items i1 and i2 that are respectively tagged by two keywords, (I1, I2), and three keywords(I2, I3, I4) matrix 0 1/2 1/2….0 /3….1/3 1/3 where oil denotes the weight of tag I in its description of i, so that > ai= l if u owns i and zero otherwise. Each item and each tag is also characterised by similar matrices that we note y and a respect B. Projecting method A common way to simplify the analysis of multi-partite networks consists in projecting them on lower order networks, i.e. unipartite or bipartite networks 1. Bipartite networks In the following, we only focus on the correlations between two kinds of nodes, for instance between the users and the items. To do so, we first reduce the tripartite network to a bipartite one by summing over all nodes of one kind, thereby neglecting possible correlations between the three kinds of nodes. Such neglected correlations may include the role of the
FIG. 1: Tripartite structure of the tagging system. In this example, user µ owns two items i1 and i2 that are respectively tagged by two keywords, (I1, I2), and three keywords (I2, I3, I4). matrix σ µ : σ µ = 0 ... 1/2 ... 1/2 ... 0 ... ... ... ... ... ... ... ... 1/3 ... 1/3 ... ... 1/3 ... ... ... ... ... ... ... (1) where σ µ iI denotes the weight of tag I in its description of i, so that P I σ µ iI = 1 if µ owns i and zero otherwise. Each item and each tag is also characterised by similar matrices that we note γ i and α I respectively. B. Projecting method A common way to simplify the analysis of multi-partite networks consists in projecting them on lower order networks, i.e. unipartite or bipartite networks. 1. Bipartite networks In the following, we only focus on the correlations between two kinds of nodes, for instance between the users and the items. To do so, we first reduce the tripartite network to a bipartite one by summing over all nodes of one kind, thereby neglecting possible correlations between the three kinds of nodes. Such neglected correlations may include the role of the 6
specialisation of a user on the way he tags items. For instance, a Jazz lover will have a tendency to use more specific tags for Jazz bands, because of i) his knowledge of more specific tags, and ii) his need of specialised tags in order to retrieve informations from the many Jazz songs composing his library By using the above reduction method, the bipartite network users-item is obtained by summing over all tags, so that each listener p is now described by thethe nir-vector ol, 可=( the index running over all items, and where a,=2ar. The items are characterised by the nr- vector元n=(…,1,…0,…,1,…, These vectors are signatures of the users/ items,that account for their interests/audience. In the case of music, we call these vectors the music signatures of people and groups. In the following, we also focus on the bipartite network em-tag, where the information about users has been eliminated. It is accordingly defined by summing over all users, and leads to the vectors i and a 2. Unipartite networks In order to project the bipartite network on a unipartite one, we look at the correlations between two nodes of the same kind, relatively to his behaviour with another kind. For instance,one may look how persons u and A are correlated by using common items. To do so, we introduce the symmetric correlation measure where air. oi denotes the scalar product between the two nrt-vector, and I its associated norm. This correlation measure, that corresponds to the cosine of the two vectors in the nI dimensional space, vanishes when the persons have no common item, and is equal to l when their item libraries are strictly identical. In Eq 3, we use the subscript CF for collaborative filtering, as this quantity is good candidate for measuring the similitude of users dependin on their profiles In the following, we also look at the correlations CCe, that measures the correlations of groups depending on their common audiences, and ccr (CT for collaborative tagging) that measures the correlations of the tags depending on the items to which they are attributed
specialisation of a user on the way he tags items. For instance, a Jazz lover will have a tendency to use more specific tags for Jazz bands, because of i) his knowledge of more specific tags, and ii) his need of specialised tags in order to retrieve informations from the many Jazz songs composing his library. By using the above reduction method, the bipartite network users-item is obtained by summing over all tags, so that each listener µ is now described by the the nIt-vector σ µ |I : σ µ |I = (..., 1, ..., 0, ..., 1, ...), (2) the index running over all items, and where σ µ |I = P I σ µ iI . The items are characterised by the nU -vector γ i |I = (..., 1, ..., 0, ..., 1, ...). These vectors are signatures of the users/items, that account for their interests/audience. In the case of music, we call these vectors the music signatures of people and groups. In the following, we also focus on the bipartite network item-tag, where the information about users has been eliminated. It is accordingly defined by summing over all users, and leads to the vectors γ i |µ and α I |µ . 2. Unipartite networks In order to project the bipartite network on a unipartite one, we look at the correlations between two nodes of the same kind, relatively to his behaviour with another kind. For instance, one may look how persons µ and λ are correlated by using common items. To do so, we introduce the symmetric correlation measure: C µλ CF = σ µ |I .σ λ |I |σ µ |I ||σ λ |I | ≡ cos θµλ (3) where σ µ |I .σ λ |I denotes the scalar product between the two nIt-vector, and || its associated norm. This correlation measure, that corresponds to the cosine of the two vectors in the nItdimensional space, vanishes when the persons have no common item, and is equal to 1 when their item libraries are strictly identical. In Eq.3, we use the subscript CF for collaborative filtering, as this quantity is good candidate for measuring the similitude of users depending on their profiles. In the following, we also look at the correlations C ij CF , that measures the correlations of groups depending on their common audiences, and C IJ CT (CT for collaborative tagging) that measures the correlations of the tags depending on the items to which they are attributed. 7
t=0 t=2 4 4 4 21 FIG. 2: Branching representation of a squared correlation matrix of 13 elements. At each increasing step(t=0, 1, 2) of the filter o, links are removed, so that the network decomposes into isolated islands. These islands are represented by squares, whose size depends on the number of nodes in the island. Starting from the largest island, branches indicate a parent relation between the islands.Moreover,we indicate one characteristic item for each node of the tree representation.To do so, we look at all the elements belonging to the corresponding island, and chose the item i that maximises 2, Cu, where the sum is performed over all items of the island. In this explicative case it is the item 4 at t=0, the items 4 and 1l at t=l 3. Structure analysis At this level, the search for structures requires the analysis of large correlation d the uncovering of connected blocks that could be identified as fami- lies/genres/communities. In order to extract families of alike elements from the correlation matrix C, we define the filter coefficient E [0, 1[ and filter the matrix elements so that
FIG. 2: Branching representation of a squared correlation matrix of 13 elements. At each increasing step (t=0,1,2) of the filter φ, links are removed, so that the network decomposes into isolated islands. These islands are represented by squares, whose size depends on the number of nodes in the island. Starting from the largest island, branches indicate a parent relation between the islands. Moreover, we indicate one characteristic item for each node of the tree representation. To do so, we look at all the elements belonging to the corresponding island, and chose the item i that maximises P j C ij , where the sum is performed over all items of the island. In this explicative case, it is the item 4 at t = 0, the items 4 and 11 at t = 1... 3. Structure analysis At this level, the search for structures requires the analysis of large correlation matrices, and the uncovering of connected blocks that could be identified as families/genres/communities. In order to extract families of alike elements from the correlation matrix C, we define the filter coefficient φ ∈ [0, 1[ and filter the matrix elements so that 8
FIG.3:Graphrepresentationofthetagscorrelationmatrixobtainedfromwwwciteulike.com Only the 120 most used tags have been considered in the dataset. The values of the filter parameter o are 0.1(before the percolation transition) and o =0.25(for which the system has decomposed into disconnected islands). The graphs were plotted thanks to the visone graphical tools 5 Cg=l if Cu>o and Cg=0 otherwise. Starting from =0.0, namely a fully connected network, increasing values of the filtering coefficient remove less correlated links and lead to the shaping of well-defined islands, completely disconnected from the main island a branching representation of the community structuring [1 is used to visualise the process(see Fig. 2 for the sketch of three first steps of an arbitrary example). To do so, we start the procedure with the lowest value of o =0.0, and we represent each isolated island by a square whose surface is proportional to its number of internal elements. Then, we increase slightly the value of e.g. by 0.05, and we repeat the procedure. From one step to the next step, we draw a bond between emerging sub-islands and their parent island. The filter is increased until all bonds between nodes are eroded(that is, there is only one node left in each island ). Let us note that islands composed of only one element are not depicted for the sake of clarity. Applied to the above correlation matrix Cu, the tree structure gives some insight into the specialisation by following branches from their source(top of the figure toward their extremity(bottom of the figure By construction, the above procedure unambiguously attributes to each element a hi- erarchical set of categories. Consequently, starting from collaborative filtering that is a non-exclusive and non-hierarchical process, we have arrived to an exclusive and hierarchical
FIG. 3: Graph representation of the tags correlation matrix obtained from www.citeulike.com. Only the 120 most used tags have been considered in the dataset. The values of the filter parameter φ are 0.1 (before the percolation transition) and φ = 0.25 (for which the system has decomposed into disconnected islands). The graphs were plotted thanks to the visone graphical tools [5]. C ij φ = 1 if C ij > φ and C ij φ = 0 otherwise. Starting from φ = 0.0, namely a fully connected network, increasing values of the filtering coefficient remove less correlated links and lead to the shaping of well-defined islands, completely disconnected from the main island. A branching representation of the community structuring [1] is used to visualise the process (see Fig.2 for the sketch of three first steps of an arbitrary example). To do so, we start the procedure with the lowest value of φ = 0.0, and we represent each isolated island by a square whose surface is proportional to its number of internal elements. Then, we increase slightly the value of φ, e.g. by 0.05, and we repeat the procedure. From one step to the next step, we draw a bond between emerging sub-islands and their parent island. The filter is increased until all bonds between nodes are eroded (that is, there is only one node left in each island). Let us note that islands composed of only one element are not depicted for the sake of clarity. Applied to the above correlation matrix C ij , the tree structure gives some insight into the specialisation by following branches from their source (top of the figure) toward their extremity (bottom of the figure). By construction, the above procedure unambiguously attributes to each element a hierarchical set of categories. Consequently, starting from collaborative filtering that is a non-exclusive and non-hierarchical process, we have arrived to an exclusive and hierarchical 9
FIG. 4: Branching representation of the correlation matrix represented in Fig 3. The filtering with parameter ranging from 0. 1 to 0. 4 induces a snake of squares at each filtering level. The shape of the snake as well as its direction are irrelevant structure that may be viewed as a taxonomy. This relation could have helpful applications in order to automatically structure content in systems without a central authority. C. Data Analysis This work is based on the analysis of data retrieved from collaborative filtering websites We detail in the following the data obtained from each site 1. wwno audioscrobbler. com a database has been downloaded from audioscrobbler. com in January 2005. It consists of a listing of users(each represented by a number), together with the list of music groups that the users own in their library. In the original data set, there are 617900 different music groups and 35916 users. On average, each user owns 140 music groups in his/her library, while each 10
FIG. 4: Branching representation of the correlation matrix represented in Fig.3. The filtering, with parameter ranging from 0.1 to 0.4 induces a snake of squares at each filtering level. The shape of the snake as well as its direction are irrelevant. structure that may be viewed as a taxonomy. This relation could have helpful applications in order to automatically structure content in systems without a central authority. C. Data Analysis This work is based on the analysis of data retrieved from collaborative filtering websites. We detail in the following the data obtained from each site. 1. www.audioscrobbler.com A database has been downloaded from audioscrobbler.com in January 2005. It consists of a listing of users (each represented by a number), together with the list of music groups that the users own in their library. In the original data set, there are 617900 different music groups and 35916 users. On average, each user owns 140 music groups in his/her library, while each 10
