Micro-Macro Analysis of Complex Networks
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{"title"=>"Micro-macro analysis of complex networks", "type"=>"journal", "authors"=>[{"first_name"=>"Massimo", "last_name"=>"Marchiori", "scopus_author_id"=>"7003307946"}, {"first_name"=>"Lino", "last_name"=>"Possamai", "scopus_author_id"=>"55265723400"}], "year"=>2015, "source"=>"PLoS ONE", "identifiers"=>{"isbn"=>"10.1371/journal.pone.0116670", "scopus"=>"2-s2.0-84951575235", "pui"=>"601957125", "doi"=>"10.1371/journal.pone.0116670", "sgr"=>"84951575235", "pmid"=>"25635812", "issn"=>"19326203"}, "id"=>"363735bf-0fae-3acf-aef5-b3b3ec8a92d0", "abstract"=>"Complex systems have attracted considerable interest because of their wide range of applications, and are often studied via a \"classic\" approach: study a specific system, find a complex network behind it, and analyze the corresponding properties. This simple methodology has produced a great deal of interesting results, but relies on an often implicit underlying assumption: the level of detail on which the system is observed. However, in many situations, physical or abstract, the level of detail can be one out of many, and might also depend on intrinsic limitations in viewing the data with a different level of abstraction or precision. So, a fundamental question arises: do properties of a network depend on its level of observability, or are they invariant? If there is a dependence, then an apparently correct network modeling could in fact just be a bad approximation of the true behavior of a complex system. In order to answer this question, we propose a novel micro-macro analysis of complex systems that quantitatively describes how the structure of complex networks varies as a function of the detail level. To this extent, we have developed a new telescopic algorithm that abstracts from the local properties of a system and reconstructs the original structure according to a fuzziness level. This way we can study what happens when passing from a fine level of detail (\"micro\") to a different scale level (\"macro\"), and analyze the corresponding behavior in this transition, obtaining a deeper spectrum analysis. The obtained results show that many important properties are not universally invariant with respect to the level of detail, but instead strongly depend on the specific level on which a network is observed. Therefore, caution should be taken in every situation where a complex network is considered, if its context allows for different levels of observability.", "link"=>"http://www.mendeley.com/research/micromacro-analysis-complex-networks-2", "reader_count"=>15, "reader_count_by_academic_status"=>{"Professor > Associate Professor"=>2, "Researcher"=>1, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>8, "Student > Master"=>1, "Student > Bachelor"=>1, "Professor"=>1}, "reader_count_by_user_role"=>{"Professor > Associate Professor"=>2, "Researcher"=>1, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>8, "Student > Master"=>1, "Student > Bachelor"=>1, "Professor"=>1}, "reader_count_by_subject_area"=>{"Engineering"=>2, "Unspecified"=>1, "Biochemistry, Genetics and Molecular Biology"=>1, "Mathematics"=>1, "Psychology"=>2, "Social Sciences"=>1, "Computer Science"=>7}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>2}, "Social Sciences"=>{"Social Sciences"=>1}, "Psychology"=>{"Psychology"=>2}, "Computer Science"=>{"Computer Science"=>7}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>1}, "Mathematics"=>{"Mathematics"=>1}, "Unspecified"=>{"Unspecified"=>1}}, "reader_count_by_country"=>{"United States"=>1, "Italy"=>2, "Slovenia"=>1, "Australia"=>1}, "group_count"=>3}

Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/1877006"], "description"=>"<p>The online social networks of the Netherlands created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298444, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g010", "stats"=>{"downloads"=>0, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_Netherlands_8217_s_city_based_online_social_network_/1298444", "title"=>"The Netherlands’s city-based online social network.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877019"], "description"=>"<p>Effect of the abstraction process on the degree distribution <i>P</i>(<i>k</i>) for increasing values of fuzziness <i>f</i> for the Netherlands city-based online social network. We detected that the behavior starts from a small world scale-free configuration and is ideally maintained for <i>f</i> < 0.11. When <i>f</i> increases, it changes to uniform and finally to random (when <i>f</i> is maximum).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298457, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g021", "stats"=>{"downloads"=>0, "page_views"=>18, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_the_abstraction_process_on_the_degree_distribution_P_k_/1298457", "title"=>"Effect of the abstraction process on the degree distribution <i>P</i>(<i>k</i>).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877012"], "description"=>"<p>Effect of the telescopic abstraction for the physical <i>D</i><sub><i>p</i></sub>, topological <i>D</i><sub><i>t</i></sub> and metrical <i>D</i><sub><i>m</i></sub> diameter as a function of fuzziness <i>f</i>. All the values were normalized by the baseline values at <i>f</i> = 0 (i.e., no abstraction is applied). The top panels contain results of subways, the bottom ones of city-based online social networks and the US airline network.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298450, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g014", "stats"=>{"downloads"=>1, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_the_telescopic_abstraction_on_the_diameter_as_a_function_of_f_/1298450", "title"=>"Effect of the telescopic abstraction on the diameter as a function of <i>f</i>.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1876992"], "description"=>"<p>Example of grids applied on top of networks as a function of fuzziness. Leftmost grid has low fuzziness <i>f</i> = 0.125 whereas the rightmost has <i>f</i> = 1. The granularity of the spectrum in this example is equal to 7. In this paper, we only consider linear increase of <i>f</i>.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298430, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g004", "stats"=>{"downloads"=>0, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Sequence_of_box_covering_iteration_for_telescopic_analysis_/1298430", "title"=>"Sequence of box covering iteration for telescopic analysis.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1876989"], "description"=>"<p>Example of micro-macro analysis obtained by increasing (abstraction process to the macro world) fuzziness <i>f</i>. When <i>f</i> = 0, no abstraction is applied whereas at increasing values of <i>f</i>, the network will be more obfuscated and the structure will be simpler. In the extreme situation when <i>f</i> is maximum, <i>f</i> = 1 (not displayed in the figure), the original network will be collapsed into a one node graph.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298427, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g001", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_micro_macro_analysis_/1298427", "title"=>"Example of micro-macro analysis.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877002"], "description"=>"<p>The online social networks of India created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298440, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g008", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_India_8217_s_city_based_online_social_network_/1298440", "title"=>"India’s city-based online social network.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877015"], "description"=>"<p>Effect of the telescopic process on subways (leftmost column), the US airline and city-based online social networks (rightmost column) as a function of <i>f</i>. The statistical properties considered in these panels are topological and metrical <i>E</i><sub><i>glob</i></sub>. The abstraction process does not preserves the topological <i>E</i><sub><i>glob</i></sub> (top panels) while varying <i>f</i>. In particular, regardless of the network considered, the networks viewed at macro level are simpler and more efficient compared to micro view. Conversely, the situation is slightly different for metrical <i>E</i><sub><i>glob</i></sub> (bottom panels). In this case, the connection pattern of the system considered alters significantly the outcome of the abstraction process. In fact, we detected that the structure of subway networks allow a good preservation of the metrical efficiency in the spectrum whereas in city-based online social networks this feature is absent.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298453, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g017", "stats"=>{"downloads"=>0, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_the_telescopic_process_on_E_glob_/1298453", "title"=>"Effect of the telescopic process on <i>E</i><sub><i>glob</i></sub>.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877008"], "description"=>"<p>The log-log plots of the cumulative degree distributions <i>P</i><sub><i>cum</i></sub>(<i>k</i>) of subways (Boston, Milan, New York, Paris, a to d), the US airline (e) and city-based online social networks (letter f to j) of Italy, Australia, The Netherlands, India and the United Kingdom. The distributions are characterized by exponents <i>γ</i> of <i>P</i>(<i>k</i>) ∼ <i>k</i><sup>−<i>γ</i></sup> that is one plus the slope of <i>P</i><sub><i>cum</i></sub>(<i>k</i>) (in a log-log plot), i.e. <i>γ</i> = 1 + <i>γ</i><sub><i>cum</i></sub>. The coefficient is <i>γ</i> = 3.5 for subways networks, 2.6 for the US airline, 1.85 for Indian city-based online social network, 1.68 for the United Kingdom, 2.61 for Italy, 1.94 for Australia and 1.61 for the Netherlands. The coefficients for subways might not be precise due to the small dimension of the networks.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298446, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g012", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_P_cum_distribution_of_subways_transportation_and_social_networks_/1298446", "title"=>"<i>P</i><sub><i>cum</i></sub> distribution of subways, transportation and social networks.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877018"], "description"=>"<p>Effect of the telescopic analysis on topological and metrical normalized cost over efficiency for subways (leftmost column), the US airline and city-based online social networks (rightmost column). By dividing the cost of the networks by the global efficiency (that ranges between 0 and 1), we verified that subway networks are cheaper as well as very efficient, more than city based online social networks. This is evidence that subway network have an economic inborn principle that is maintained during the telescopic abstraction process.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298456, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g020", "stats"=>{"downloads"=>0, "page_views"=>23, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_the_telescopic_analysis_on_C_E_glob_/1298456", "title"=>"Effect of the telescopic analysis on <i>C</i>/<i>E</i><sub><i>glob</i></sub>.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1876996"], "description"=>"<p>How Boston (leftmost panels) and New York (rightmost panels) subway networks vary in the micro-macro spectrum according to increasing values of fuzziness. From the panel, it is clear that the spatial structure of the systems remain relatively unchanged in the first steps of the abstraction process.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298433, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g006", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_box_covering_on_real_networks_/1298433", "title"=>"Example of box covering on real networks.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877001"], "description"=>"<p>The online social networks of Australia created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298439, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g007", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Australia_8217_s_city_based_online_social_network_/1298439", "title"=>"Australia’s city-based online social network.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877014"], "description"=>"<p>Impact of the telescopic analysis on degree correlations <i>ρ</i> as a function of <i>f</i> for subway networks (left panel), the US airline and city-based online social networks (right panel). It is worth noting that the telescopic process yields disassortative networks regardless of the network. This means that in subways, the topological structure will be drastically changed whereas in the other networks the degree correlation tends to remain stable (at least will have the same sign).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298452, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g016", "stats"=>{"downloads"=>0, "page_views"=>18, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Impact_of_the_telescopic_analysis_on_the_degree_correlations_961_/1298452", "title"=>"Impact of the telescopic analysis on the degree correlations <i>ρ</i>.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1876991"], "description"=>"<p>One-step application of the abstraction process to a small graph. (a) Original graph <i>G</i>. Red (dashed) circles identify the group of nodes that will be merged together. (b) Output graph <i>G</i><sub><i>i</i></sub> in which nodes <i>c</i>, <i>f</i>, <i>e</i>, <i>h</i>, <i>l</i> and <i>n</i>, <i>m</i> are collapsed into new nodes <i>e</i>, <i>c</i>, <i>h</i> ∈ <i>V</i><sub><i>i</i></sub> respectively. Coordinates are the barycenter of collapsed nodes. Three edges are then removed because they connect the collapsed nodes: (<i>n</i>, <i>m</i>), (<i>c</i>, <i>e</i>), and (<i>f</i>, <i>e</i>).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298429, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g003", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_One_step_abstraction_process_/1298429", "title"=>"One-step abstraction process.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877004"], "description"=>"<p>The online social networks of Italy created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298442, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g009", "stats"=>{"downloads"=>0, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Italy_8217_s_city_based_online_social_network_/1298442", "title"=>"Italy’s city-based online social network.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877017"], "description"=>"<p>Effect of the telescopic analysis on topological and metrical cost (<i>c</i><sub><i>t</i></sub> and <i>c</i><sub><i>m</i></sub>) as a function of <i>f</i> for subways (leftmost column), the US airline and city-based online social networks (rightmost column). We note that our coarse graining process produces networks more expensive than detailed ones. This effect might be caused by the creation of redundant structures in macro level systems so that the whole cost will be higher. Even though both curves are positively correlated to <i>f</i>, the slope in subways networks is smaller compared to city-based online social networks. To verify whether this effect is not trivially caused by a low efficiency value, we will consider <i>C</i>/<i>E</i><sub><i>glob</i></sub> index (see <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0116670#pone.0116670.g020\" target=\"_blank\">Fig. 20</a>).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298455, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g019", "stats"=>{"downloads"=>1, "page_views"=>16, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_the_telescopic_analysis_on_c_t_and_c_m_/1298455", "title"=>"Effect of the telescopic analysis on <i>c</i><sub><i>t</i></sub> and <i>c</i><sub><i>m</i></sub>.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877007"], "description"=>"<p>The online social networks of the United Kingdom created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298445, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g011", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_United_Kingdom_8217_s_city_based_online_social_network_/1298445", "title"=>"United Kingdom’s city-based online social network.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877020"], "description"=>"<p>Datasets statistics of subways, the US airline and city-based online social networks: number of nodes <i>n</i> and edges <i>m</i> of the graphs, maximum degree <i>k</i><sub><i>max</i></sub> and average node degree ⟨<i>k</i>⟩, standard deviation of the degree <i>σ</i><sub><i>k</i></sub>, assortativity mixing by degree <i>ρ</i>, physical, topological and metrical diameter <i>D</i>, global and local efficiency <i>E</i><sub><i>glob</i></sub>, <i>E</i><sub><i>loc</i></sub>, costs and <i>C</i>/<i>E</i> property (defined as the ratio between cost and global efficiency). Both <i>topological</i> and <i>metrical</i> versions are calculated of the latter three indicators.</p><p>Statistical features of transportation and city-based online social networks.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298458, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai", "Zachary Wyman"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.t001", "stats"=>{"downloads"=>7, "page_views"=>34, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Statistical_features_of_transportation_and_city_based_online_social_networks_/1298458", "title"=>"Statistical features of transportation and city-based online social networks.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877010"], "description"=>"<p>Number of collapsed nodes <i>n</i> and edges <i>m</i> as a function of <i>f</i> in log-log axes. The values are normalized by the baseline values <i>n</i>(0) and <i>m</i>(0) respectively, obtained at <i>f</i> = 0 (i.e., no abstraction applied). The leftmost panels refer to subway networks whereas the rightmost refer to city-based online social networks and the US airline network. The decrease of <i>n</i> and <i>m</i> is clearly exponential, even though the rate is influenced by many factors like network size and node positions.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298448, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g013", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Number_of_collapsed_nodes_and_edges_as_a_function_of_f_in_log_log_axes_/1298448", "title"=>"Number of collapsed nodes and edges as a function of <i>f</i> in log-log axes.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1876990"], "description"=>"<p>Examples of different graph types.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298428, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g002", "stats"=>{"downloads"=>1, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Graph_types_/1298428", "title"=>"Graph types.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877016"], "description"=>"<p>Effect of the telescopic analysis on topological and metrical <i>E</i><sub><i>loc</i></sub> as a function of <i>f</i> in subways (leftmost panels), the US airline and city-based online social networks (rightmost panels). The left most panels show that <i>E</i><sub><i>loc</i></sub> is almost stable in the spectrum meaning that the local properties of the subway networks are preserved by the analysis. However, in networks with heterogeneous topological structure, the telescopic process will further increase <i>E</i><sub><i>loc</i></sub> resulting in the creation of systems that are densely connected at local level.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298454, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g018", "stats"=>{"downloads"=>2, "page_views"=>28, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_the_telescopic_analysis_on_E_loc_/1298454", "title"=>"Effect of the telescopic analysis on <i>E</i><sub><i>loc</i></sub>.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1877013"], "description"=>"<p>Effect of the telescopic analysis on the degree: maximum degree <i>k</i><sub><i>max</i></sub> (leftmost column), mean degree ⟨<i>k</i>⟩ (center column) and standard deviation <i>σ</i><sub><i>k</i></sub> (rightmost column) for subways (top panels), the US airline and city-based online social networks (bottom panels). All values were normalized relatively to the baseline value at <i>f</i> = 0 (where no abstraction is applied). The explanation of the results obtained is not so trivial. In general, the degree properties of the networks will be drastically modified as fuzziness increases. The degree tend to decrease linearly in subways whereas in airline and social-based networks the telescopic effect results in an exponential decrease.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298451, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g015", "stats"=>{"downloads"=>2, "page_views"=>21, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_the_telescopic_analysis_on_the_degree_/1298451", "title"=>"Effect of the telescopic analysis on the degree.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/1876993"], "description"=>"<p>Grid displacement issue when the distance between two nodes is less than fuzziness value. Wrong (a) and correct (b) grid displacement.</p>", "links"=>[], "tags"=>["network modeling", "spectrum analysis", "results show", "fuzziness level", "detail level", "Complex Networks Complex systems"], "article_id"=>1298431, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Massimo Marchiori", "Lino Possamai"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0116670.g005", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Box_covering_issue_/1298431", "title"=>"Box covering issue.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-01-30 03:07:43"}

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{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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