Detecting the Community Structure and Activity Patterns of Temporal Networks: A Non-Negative Tensor Factorization Approach
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{"title"=>"Detecting the community structure and activity patterns of temporal networks: A non-negative tensor factorization approach", "type"=>"journal", "authors"=>[{"first_name"=>"Laetitia", "last_name"=>"Gauvin", "scopus_author_id"=>"26664461700"}, {"first_name"=>"André", "last_name"=>"Panisson", "scopus_author_id"=>"9133831400"}, {"first_name"=>"Ciro", "last_name"=>"Cattuto", "scopus_author_id"=>"6701361048"}], "year"=>2014, "source"=>"PLoS ONE", "identifiers"=>{"pui"=>"373059596", "sgr"=>"84900316533", "issn"=>"19326203", "arxiv"=>"arXiv:1308.0723v3", "pmid"=>"24497935", "scopus"=>"2-s2.0-84900316533", "doi"=>"10.1371/journal.pone.0086028"}, "id"=>"131ed4b2-8742-3a36-b9bc-cb053880e7da", "abstract"=>"The increasing availability of temporal network data is calling for more research on extracting and characterizing mesoscopic structures in temporal networks and on relating such structure to specific functions or properties of the system. An outstanding challenge is the extension of the results achieved for static networks to time-varying networks, where the topological structure of the system and the temporal activity patterns of its components are intertwined. Here we investigate the use of a latent factor decomposition technique, non-negative tensor factorization, to extract the community-activity structure of temporal networks. The method is intrinsically temporal and allows to simultaneously identify communities and to track their activity over time. We represent the time-varying adjacency matrix of a temporal network as a three-way tensor and approximate this tensor as a sum of terms that can be interpreted as communities of nodes with an associated activity time series. We summarize known computational techniques for tensor decomposition and discuss some quality metrics that can be used to tune the complexity of the factorized representation. We subsequently apply tensor factorization to a temporal network for which a ground truth is available for both the community structure and the temporal activity patterns. The data we use describe the social interactions of students in a school, the associations between students and school classes, and the spatio-temporal trajectories of students over time. We show that non-negative tensor factorization is capable of recovering the class structure with high accuracy. In particular, the extracted tensor components can be validated either as known school classes, or in terms of correlated activity patterns, i.e., of spatial and temporal coincidences that are determined by the known school activity schedule.", "link"=>"http://www.mendeley.com/research/detecting-community-structure-activity-patterns-temporal-networks-nonnegative-tensor-factorization-a-4", "reader_count"=>169, "reader_count_by_academic_status"=>{"Unspecified"=>3, "Professor > Associate Professor"=>8, "Researcher"=>27, "Student > Doctoral Student"=>15, "Student > Ph. D. Student"=>55, "Student > Postgraduate"=>5, "Student > Master"=>27, "Other"=>7, "Student > Bachelor"=>9, "Lecturer"=>2, "Lecturer > Senior Lecturer"=>3, "Professor"=>8}, "reader_count_by_user_role"=>{"Unspecified"=>3, "Professor > Associate Professor"=>8, "Researcher"=>27, "Student > Doctoral Student"=>15, "Student > Ph. D. Student"=>55, "Student > Postgraduate"=>5, "Student > Master"=>27, "Other"=>7, "Student > Bachelor"=>9, "Lecturer"=>2, "Lecturer > Senior Lecturer"=>3, "Professor"=>8}, "reader_count_by_subject_area"=>{"Unspecified"=>11, "Agricultural and Biological Sciences"=>9, "Business, Management and Accounting"=>1, "Computer Science"=>75, "Decision Sciences"=>2, "Engineering"=>20, "Environmental Science"=>2, "Biochemistry, Genetics and Molecular Biology"=>1, "Mathematics"=>18, "Medicine and Dentistry"=>2, "Neuroscience"=>5, "Physics and Astronomy"=>18, "Social Sciences"=>5}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>2}, "Social Sciences"=>{"Social Sciences"=>5}, "Decision Sciences"=>{"Decision Sciences"=>2}, "Physics and Astronomy"=>{"Physics and Astronomy"=>18}, "Mathematics"=>{"Mathematics"=>18}, "Unspecified"=>{"Unspecified"=>11}, "Environmental Science"=>{"Environmental Science"=>2}, "Engineering"=>{"Engineering"=>20}, "Neuroscience"=>{"Neuroscience"=>5}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>9}, "Computer Science"=>{"Computer Science"=>75}, "Business, Management and Accounting"=>{"Business, Management and Accounting"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>1}}, "reader_count_by_country"=>{"United States"=>5, "Sri Lanka"=>1, "Japan"=>3, "United Kingdom"=>2, "Malaysia"=>1, "Switzerland"=>2, "Spain"=>4, "India"=>1, "Cuba"=>1, "Luxembourg"=>1, "China"=>1, "Finland"=>1, "Italy"=>2, "France"=>1, "Chile"=>1, "Germany"=>4}, "group_count"=>16}

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Figshare

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  • {"files"=>["https://ndownloader.figshare.com/files/1372295", "https://ndownloader.figshare.com/files/1372296", "https://ndownloader.figshare.com/files/1372297", "https://ndownloader.figshare.com/files/1372298", "https://ndownloader.figshare.com/files/1372300", "https://ndownloader.figshare.com/files/1372301", "https://ndownloader.figshare.com/files/1372302", "https://ndownloader.figshare.com/files/1372303", "https://ndownloader.figshare.com/files/1372304", "https://ndownloader.figshare.com/files/1372305", "https://ndownloader.figshare.com/files/1372307", "https://ndownloader.figshare.com/files/1372308", "https://ndownloader.figshare.com/files/1372309"], "description"=>"<div><p>The increasing availability of temporal network data is calling for more research on extracting and characterizing mesoscopic structures in temporal networks and on relating such structure to specific functions or properties of the system. An outstanding challenge is the extension of the results achieved for static networks to time-varying networks, where the topological structure of the system and the temporal activity patterns of its components are intertwined. Here we investigate the use of a latent factor decomposition technique, non-negative tensor factorization, to extract the community-activity structure of temporal networks. The method is intrinsically temporal and allows to simultaneously identify communities and to track their activity over time. We represent the time-varying adjacency matrix of a temporal network as a three-way tensor and approximate this tensor as a sum of terms that can be interpreted as communities of nodes with an associated activity time series. We summarize known computational techniques for tensor decomposition and discuss some quality metrics that can be used to tune the complexity of the factorized representation. We subsequently apply tensor factorization to a temporal network for which a ground truth is available for both the community structure and the temporal activity patterns. The data we use describe the social interactions of students in a school, the associations between students and school classes, and the spatio-temporal trajectories of students over time. We show that non-negative tensor factorization is capable of recovering the class structure with high accuracy. In particular, the extracted tensor components can be validated either as known school classes, or in terms of correlated activity patterns, i.e., of spatial and temporal coincidences that are determined by the known school activity schedule.</p></div>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "patterns", "temporal", "non-negative", "tensor", "factorization"], "article_id"=>922903, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>["https://dx.doi.org/10.1371/journal.pone.0086028.s001", "https://dx.doi.org/10.1371/journal.pone.0086028.s002", "https://dx.doi.org/10.1371/journal.pone.0086028.s003", "https://dx.doi.org/10.1371/journal.pone.0086028.s004", "https://dx.doi.org/10.1371/journal.pone.0086028.s005", "https://dx.doi.org/10.1371/journal.pone.0086028.s006", "https://dx.doi.org/10.1371/journal.pone.0086028.s007", "https://dx.doi.org/10.1371/journal.pone.0086028.s008", "https://dx.doi.org/10.1371/journal.pone.0086028.s009", "https://dx.doi.org/10.1371/journal.pone.0086028.s010", "https://dx.doi.org/10.1371/journal.pone.0086028.s011", "https://dx.doi.org/10.1371/journal.pone.0086028.s012", "https://dx.doi.org/10.1371/journal.pone.0086028.s013"], "stats"=>{"downloads"=>9, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Detecting_the_Community_Structure_and_Activity_Patterns_of_Temporal_Networks_A_Non_Negative_Tensor_Factorization_Approach_/922903", "title"=>"Detecting the Community Structure and Activity Patterns of Temporal Networks: A Non-Negative Tensor Factorization Approach", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372292"], "description"=>"<p>Reference score matrix containing the correct number of students in each class.</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "matrix", "containing", "students"], "article_id"=>922900, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.t003", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Reference_score_matrix_containing_the_correct_number_of_students_in_each_class_/922900", "title"=>"Reference score matrix containing the correct number of students in each class.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372289"], "description"=>"<p>Matrix obtained through NTF containing the number of students in each community projected over the different classes.</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "ntf", "containing", "students", "projected"], "article_id"=>922897, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.t004", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Matrix_obtained_through_NTF_containing_the_number_of_students_in_each_community_projected_over_the_different_classes_/922897", "title"=>"Matrix obtained through NTF containing the number of students in each community projected over the different classes.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372287"], "description"=>"<p>Each panel corresponds to one component obtained by non-negative tensor factorization of the school temporal network, with , and provides the activity level of the component as a function of the time of the day. For clarity, the panels only show the activity patterns for the first day of data (see Fig. S for the second day). Components that can be matched to classes are marked as <i>class</i>. The other three components that correspond to mixed classes exhibit activity patterns that can be understood in terms of gatherings in the social spaces of the school.</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "patterns", "extracted"], "article_id"=>922895, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.g006", "stats"=>{"downloads"=>3, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Activity_patterns_of_the_extracted_components_/922895", "title"=>"Activity patterns of the extracted components.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372288"], "description"=>"<p>Each panel corresponds to one of the three components of Fig. 6 that cannot be matched to school classes. The activity pattern of each component is compared with the time series of the co-location vector () for two choices of that correspond, respectively, to the cafeteria and the playground, i.e., the social spaces of the school. The horizontal axis is the time of the day, and the vertical axis has been rescaled for each curve so that its maximum is .</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "patterns", "components", "co-location"], "article_id"=>922896, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.g007", "stats"=>{"downloads"=>2, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Activity_patterns_of_components_vs_co_location_in_social_spaces_/922896", "title"=>"Activity patterns of components vs co-location in social spaces.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372265"], "description"=>"<p>Sample histogram of the membership weights for one component of the decomposition (one column of factor for ).</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems"], "article_id"=>922873, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.g005", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Distribution_of_the_membership_weights_/922873", "title"=>"Distribution of the membership weights.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372263"], "description"=>"<p>Left panel: core consistency curve. For each value of the number of components used for factorization, the core consistency values for the 5 best decompositions are reported (crosses). The solid line is a guide for the eye. A crossover between two regimes is visible for . Right panel: component-node matrix for components. Rows correspond to network nodes and columns to components. The matrix is obtained from the factor by classifying each node as belonging (lighter rectangles) or not belonging (dark blue rectangles) to a given component. The order of the nodes has been rearranged to expose the block structure of the matrix. Colors identify components, and the community structures that can be matched to school classes are annotated with the corresponding class name.</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "decomposition", "empirical", "temporal"], "article_id"=>922871, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.g004", "stats"=>{"downloads"=>0, "page_views"=>10, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_NTF_decomposition_of_an_empirical_temporal_network_/922871", "title"=>"NTF decomposition of an empirical temporal network.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372262"], "description"=>"<p>The original temporal network is represented as a three-way tensor, which is then decomposed by using non-negative tensor factorization. The complexity of the model (number of components ) is tuned by using quality indicators that provide information on the stability, coverage or redundancy of the decomposition.</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "detection", "non-negative", "tensor"], "article_id"=>922870, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.g003", "stats"=>{"downloads"=>1, "page_views"=>19, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Community_activity_structure_detection_via_non_negative_tensor_factorization_/922870", "title"=>"Community-activity structure detection via non-negative tensor factorization.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372258"], "description"=>"<p>The cube on the left is the original 3-way tensor, which is represented as the sum of rank-1 tensors (on the right), each generated as the outer product of three 1-dimensional vectors (thin rectangles). Each of the rank-1 terms on the right corresponds to one component.</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "kruskal"], "article_id"=>922867, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.g001", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Pictorial_representation_of_the_Kruskal_decomposition_/922867", "title"=>"Pictorial representation of the Kruskal decomposition.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-01-31 04:16:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/1372290"], "description"=>"<p>Each row of the table corresponds to one tensor decomposition, with a number of components ranging from to . For each decomposition, we report the core consistency, the number of classes that can be matched to the extracted components, the total number of nodes (students) these classes are known to comprise, the number of nodes that belong to the component (i.e., the class nodes covered by the component), and the ratio between the latter two columns, that is, the fraction of the known class structure that the method has recalled.</p>", "links"=>[], "tags"=>["Computer modeling", "algebra", "Algebraic topology", "Linear algebra", "Applied mathematics", "geography", "Human geography", "Spatial analysis", "Sociology", "Computational sociology", "social networks", "Social systems", "recovered", "non-negative", "tensor", "factorization"], "article_id"=>922898, "categories"=>["Mathematics", "Sociology"], "users"=>["Laetitia Gauvin", "André Panisson", "Ciro Cattuto"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0086028.t001", "stats"=>{"downloads"=>8, "page_views"=>14, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Class_structure_recovered_by_non_negative_tensor_factorization_as_a_function_of_the_number_of_components_/922898", "title"=>"Class structure recovered by non-negative tensor factorization as a function of the number of components.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-01-31 04:16:06"}

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Relative Metric

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