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Network Analysis for Understanding Dynamics: Wikipedia, Music & Chaos

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My presentation slides for the Invited Talk at IEICE Technical Committee on Information Network Science (NetSci), May 18, 2012. (Takashi Iba, http://twitter.com/taka_iba )

My presentation slides for the Invited Talk at IEICE Technical Committee on Information Network Science (NetSci), May 18, 2012. (Takashi Iba, http://twitter.com/taka_iba )

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  • 1. Network Analysis for Understanding Dynamics - Wikipedia, Music & Chaos - 時間発展のネットワーク分析 TAKASHI IBA 井庭  崇 Ph.D. in Media and Governance Associate Professor at Faculty of Policy Management, Keio University, Japan twitter in Japanese: takashiiba twitter in English: taka_iba
  • 2. Takashi Iba 井庭 崇Studying ... 1997 - Neural Computing (Optimization & Learning) The Science of Complexity (Social and Economic Systems) Research Methodology (Multi-Agent Modeling & Simulation) Software Engineering (Model-Driven Development, UML) Introduction to Complex Systems 2004 - (1998) in Japanese Sociology (Autopoietic Systems Theory) Network Analysis Pattern Language (A method to describe tacit knowledge) CreativityTeaching at SFC, Keio University Social Systems Theory Pattern Language Simulation Design Social Systems Theory Science of Complex Systems [Reality+ Series] (2011) in Japanese
  • 3. Network Analysis for Understanding Dynamics - Wikipedia, Music & Chaos - 時間発展のネットワーク分析 TAKASHI IBA 井庭  崇 Ph.D. in Media and Governance Associate Professor at Faculty of Policy Management, Keio University, Japan twitter in Japanese: takashiiba twitter in English: taka_iba
  • 4. Network Analysis for Understanding Dynamicswant to capture the process / dynamics ofphenomena as a whole.introducing network analysis in a new way.to build a directed network by connectingbetween nodes, based on the sequential orderof occurrence.a new viewpoint “dynamics as network”, whichis a distinct from “dynamics of network” and“dynamics on network.”
  • 5. Dynamics
  • 6. Dynamics
  • 7. When Network Scientists talk about dynamics... Dynamics of Networks the evolution of network structure in time Dynamics on Networks the information spread or interaction on networks
  • 8. Dynamics of NetworksStructural Change of Alliance Networks of Nations 1850 1946 2000 the evolution of network structure in time T. Furukawazono, Y. Suzuki, and T. Iba, "Historical Changes of Alliance Networks among Nations", NetSci07, 2007 古川園智樹, 鈴木祐太, 井庭 崇, 「国家間同盟ネットワークの歴史的変化」, 情報処理学会 第 58回数理モデル化と問題解決研究会, Vol.2006, No.29, 2006年3月, pp.93-96
  • 9. Dynamics of NetworksAlliance Networks of Nations the evolution of network structure in time
  • 10. When Network Scientists talk about dynamics... Dynamics of Networks the evolution of network structure in time Dynamics on Networks the information spread or interaction on networks
  • 11. Dynamics on NetworksIterated Games on Alliance Network of Nations, as an example the information spread or interaction on networks T. Furukawazono, Y. Takada, and T. Iba, "Iterated Prisoners Dilemma on Alliance Networks", NetSci07, 2007 古川園 智樹, 高田 佑介, 井庭 崇, 「ネットワーク上におけるジレンマゲーム」, 情報処理学会 ネットワーク生態学研究会 第2回サマースクール, 2006
  • 12. When Network Scientists talk about dynamics... Dynamics of Networks the evolution of network structure in time Dynamics on Networks the information spread or interaction on networks
  • 13. Dynamics of NetworksDynamics on NetworksDynamics as Networks
  • 14. Dynamics as Networksthe dynamics of system/phenomena are represented as networks
  • 15. Dynamics as NetworksSequential Collaboration Network of Wikipedians sequential an article order Editor A Editor A Editor B Editor B Editor C Editor C the dynamics of system/phenomena are represented as networks
  • 16. Dynamics as NetworksSequential Collaboration Network of Wikipedians The Sequential Collaboration Network of an Article on Wikipedia (English) “Star Wars Episode IV: A New Hope” the dynamics of system/phenomena are represented as networks
  • 17. Dynamics as NetworksSequential Collaboration Network of WikipediansThe Sequential Collaboration Networkof an Article on Wikipedia (English)“Australia” the dynamics of system/phenomena are represented as networks
  • 18. Dynamics as NetworksSequential Collaboration Network of Wikipedians The Sequential Collaboration Network of an Article on Wikipedia (English) “Damien (South Park)” the dynamics of system/phenomena are represented as networks
  • 19. Dynamics as Networksthe dynamics of system/phenomena are represented as networks
  • 20. Dynamics as Networksthe dynamics of system/phenomena are represented as networks Chord Networks of Music Chord-Transition Networks of Music Collaboration Networks of Wikipedia Collaboration Networks of Linux Co-Purchase Networks of Books, CDs, DVDs State Networks of Chaotic Dynamical Systems
  • 21. Chord Networks of Music Dynamics as Networksthe dynamics of system/phenomena are represented as networks
  • 22. Sequential Chord Network of Music
  • 23. Sequential Chord Network of Music(Carpenters)I NEED TO BE IN LOVE WE’VE ONLY JUST BEGUN TOP OF THE WORLD RAINY DAYS AND MONDAY I WON’T LAST A DAY GOODBY TO LOVE SING ONLY YESTERDAY WITHOUT YOU
  • 24. JAMBALAYA SOLITAIRE PLEASE MR.POSTMAN HURTING EACH OTHER (ON THE BAYOU)THERE’S A KIND OF HUSH BLESS THE BEASTS (THEY LONG TO BE) FOR ALL WE KNOW (ALL OVER THE WORLD) AND CHILDREN CLOSE TO YOU MAKE BELIEVE IT’SYESTERDAY ONCE MORE THOSE GOOD OLD DREAMS YOUR FIRST TIME
  • 25. Sequential Chord Network of Music [Integrated]Carpentersmajor 19 songsI NEED TO BE IN LOVEWE’VE ONLY JUST BEGUNTOP OF THE WORLDRAINY DAYS AND MONDAYGOODBY TO LOVESINGONLY YESTERDAYI WON’T LAST A DAY WITHOUT YOUSOLITAIREPLEASE MR.POSTMANJAMBALAYA (ON THE BAYOU)HURTING EACH OTHERTHERE’S A KIND OF HUSH (ALL OVER THE WORLD)FOR ALL WE KNOWBLESS THE BEASTS AND CHILDREN(THEY LONG TO BE) CLOSE TO YOUYESTERDAY ONCE MORETHOSE GOOD OLD DREAMSMAKE BELIEVE IT’S YOUR FIRST TIME
  • 26. Sequential Chord Network in-degree distributionof Music [Integrated] 1Carpenters 0.1major 19 songs p(k) 0.01 0.001 1 10 100 k weight distribution out-degree distribution 1 1 0.1 0.1 P(w) P(k) 0.01 0.01 0.001 0.001 1 10 100 1 10 100 w k
  • 27. Chord-Transition Networks of Music Dynamics as Networks the dynamics of system/phenomena are represented as networks
  • 28. Sequential Chord-Transition Network of Music
  • 29. Sequential Chord-Transition Network of MusicI NEED TO BE IN LOVE WE’VE ONLY JUST BEGUN TOP OF THE WORLD RAINY DAYS AND MONDAY I WON’T LAST A DAY GOODBY TO LOVE SING ONLY YESTERDAY WITHOUT YOU
  • 30. JAMBALAYA SOLITAIRE PLEASE MR.POSTMAN HURTING EACH OTHER (ON THE BAYOU)THERE’S A KIND OF HUSH BLESS THE BEASTS (THEY LONG TO BE) FOR ALL WE KNOW (ALL OVER THE WORLD) AND CHILDREN CLOSE TO YOU MAKE BELIEVE IT’SYESTERDAY ONCE MORE THOSE GOOD OLD DREAMS YOUR FIRST TIME
  • 31. Sequential Chord-Transition Network of MusicCarpentersmajor 19 songsI NEED TO BE IN LOVEWE’VE ONLY JUST BEGUNTOP OF THE WORLDRAINY DAYS AND MONDAYGOODBY TO LOVESINGONLY YESTERDAYI WON’T LAST A DAY WITHOUT YOUSOLITAIREPLEASE MR.POSTMANJAMBALAYA (ON THE BAYOU)HURTING EACH OTHERTHERE’S A KIND OF HUSH (ALL OVER THE WORLD)FOR ALL WE KNOWBLESS THE BEASTS AND CHILDREN(THEY LONG TO BE) CLOSE TO YOUYESTERDAY ONCE MORETHOSE GOOD OLD DREAMSMAKE BELIEVE IT’S YOUR FIRST TIME
  • 32. Sequential Chord-TransitionNetwork of Music [Integrated] in-degree distribution 1Carpentersmajor 19 songs 0.1 p(k) 0.01 0.001 1 10 k weight distribution out-degree distribution 1 1 0.1 0.1 P(w) p(k) 0.01 0.01 0.001 0.001 1 10 100 1 10 w k
  • 33. Papers & Presentations 広瀬 隼也, 井庭 崇, 「コードを基にした楽曲ネットワークの可視化と分析」, 情報処理学会 第5回ネットワーク生態学シンポジウム, 沖縄, 2009年3月 T. Iba, "Network Analysis for Understanding Dynamics", InternationalSchool and Conference on Network Science 2010 (NetSci2010), 2010 Collaborators Junya Hirose Former Graduate Student
  • 34. Collaboration Networks of Wikipedia Dynamics as Networks the dynamics of system/phenomena are represented as networks
  • 35. Editorial Collaboration Networks ofWikipedia Articles in Various Languages •The characteristics of collaboration patterns of all articles in a certain language. •The commonality and differences of collaboration patterns among Wikipedias written in various languages.
  • 36. Editorial Collaboration Networks ofWikipedia Articles in Various Languages n Method: Sequential collaboration network n Analysis 1: Comparison of 12 different languages n Analysis 2: Distribution of account and IP users n Analysis 3: Distribution of Featured Articles
  • 37. Editorial Collaboration Networks ofWikipedia Articles in Various Languages n Method: Sequential collaboration network n Analysis 1: Comparison of 12 different languages n Analysis 2: Distribution of account and IP users n Analysis 3: Distribution of Featured Articles
  • 38. n Method: Sequential collaboration network Building a sequential collaboration network, connecting a relation from editor A to editor B, if editor B follows on work done by editor A. order an article 1 Editor A Editor A 2 Editor B 3 4 Editor A Editor B Editor C Editor C 5
  • 39. Sequential Collaboration Network of Article“Collaborative Innovation Networks” in English WikipediaThe number of Nodes = 51Average path length = 6.399
  • 40. Sequential Collaboration Network of Article “Basel”in English WikipediaThe number of Nodes = 594Average path length = 6.577
  • 41. Sequential Collaboration Network of Article “Switzerland”in English WikipediaThe number of Nodes = 3998Average path length = 5.468
  • 42. Sequential Collaboration Network of Article “Fondue”in English WikipediaThe number of Nodes = 457Average path length = 10.485
  • 43. n Method: Sequential collaboration network Building a sequential collaboration network, connecting a relation from editor A to editor B, if editor B follows on work done by editor A. order an article 1 Editor A Editor A 2 Editor B 3 4 Editor A Editor B Editor C Editor C 5
  • 44. Our Previous Study: Featured Articles in English Wikipedia each sequential collaboration network Linear graph The average path length of 2,545 articles [Jun 27 2009] The order of each sequential collaboration network (The number of editors in each article)T. Iba, K. Nemoto, B. Peters & P. Gloor, "Analyzing the Creative Editing Behavior of WikipediaEditors Through Dynamical Social Network Analysis", COINs2011, 2009T. Iba and S. Itoh, "Sequential Collaboration Network of Open Collaboration", NetSci09, 2009
  • 45. Editorial Collaboration Networks ofWikipedia Articles in Various Languages n Method: Sequential collaboration network n Analysis 1: Comparison of 12 different languages n Analysis 2: Distribution of account and IP users n Analysis 3: Distribution of Featured Articles
  • 46. nAnalysis 1: Comparison of 12 different languages Target Languages Rank 1: English Rank 2: German Rank 3: French Rank 4: Polish Rank 5: Italian Rank 6: Japanese Rank 7: Spanish Rank 8: Dutch Rank 9: PortugueseAnalyzing ALL articles as of January Rank 10: Russian1st, 2011 in each language. … Rank 15: FinnishThe ranking based on the data as of …January 6th, 2011. Rank 20: Turkish
  • 47. English Rank 1 3,490,325 articles each sequential collaboration network The average path length of The order of each sequential collaboration network (The number of editors in each article)
  • 48. English Rank 1 3,490,325 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 49. English Rank 1 3,490,325 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 50. German Rank 2 1,155,210 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 51. French Rank 3 1,039,251 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 52. Polish Rank 4 752,734 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 53. Italian Rank 5 750,634 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 54. Japanese Rank 6 718,974 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 55. Spanish Rank 7 676,866 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 56. Dutch Rank 8 656,079 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 57. Portuguese Rank 9 638,747 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 58. Russian Rank 10 627,139 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 59. Finnish Rank 15 255,712 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 60. Turkish Rank 20 152,262 articles each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 61. English German French PolishItalian Japanese Spanish DutchPortuguese Russian Finnish Turkish
  • 62. Result of Analysis 1: Comparison of 12 different languages •Scatter plot of all articles exhibits a tilted triangle in all languages. •The height of triangle gets shorter as the number of articles decreases.
  • 63. Editorial Collaboration Networks ofWikipedia Articles in Various Languages n Method: Sequential collaboration network n Analysis 1: Comparison of 12 different languages n Analysis 2: Distribution of account and IP users n Analysis 3: Distribution of Featured Articles
  • 64. nAnalysis 2: Distribution of account and IP users IP users Account users
  • 65. Scatter plot of articles in English Wikipedia each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 66. Scatter plot of articles withnumber of IP users / number of total editors each sequential collaboration network The average path length of 0.0 PIP 1.0 Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 67. Scatter plot of articles withnumber of IP users / number of total editors PIP = 0.0 PIP = 0.1 PIP = 0.2 PIP = 0.3 PIP = 0.4 PIP = 0.5 PIP = 0.6 PIP = 0.7 PIP = 0.8
  • 68. Scatter plot of articles withnumber of IP users / number of total editors each sequential collaboration network The average path length of 0.0 PIP 1.0 Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 69. Result of Analysis 2: Distribution of account and IP users •Top and right area of the “triangle” in scatter plot consist of articles which ratios of users is high. •As a result, both the average path length and order of network can be large in these areas. PIP = 0.0 PIP = 0.6
  • 70. Editorial Collaboration Networks ofWikipedia Articles in Various Languages n Method: Sequential collaboration network n Analysis 1: Comparison of 12 different languages n Analysis 2: Distribution of account and IP users n Analysis 3: Distribution of Featured Articles
  • 71. nAnalysis 3: Distribution of Featured Articles 3,372 featured articles / 3,732,033 articles In English Wikipedia
  • 72. Scatter plot of all articles in English Wikipedia each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 73. Scatter plot of featured articles on the all articlesin English Wikipedia each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 74. Scatter plot of featured articles on the all articlesin English Wikipedia each sequential collaboration network The average path length of Double logarithmic graph The order of each sequential collaboration network (The number of editors in each article)
  • 75. Result of Analysis 3: Distribution of Featured Articles •Features articles are located at a certain area in the scatter plot. •It implies that there would be characteristic patterns of collaboration producing good results.
  • 76. Editorial Collaboration Networks ofWikipedia Articles in Various Languages n Method: Sequential collaboration network n Analysis 1: Comparison of 12 different languages n Analysis 2: Distribution of account and IP users n Analysis 3: Distribution of Featured Articles
  • 77. Editorial Collaboration Networks ofWikipedia Articles in Various Languages •Scatter plot of all articles commonly exhibits a tilted triangle in all languages, but the height of triangle gets shorter as the number of articles decreases. •Top and right area of the “triangle” in scatter plot consist of articles which the ratios of IP users are high. •Features articles are located at a certain area in the scatter plot.
  • 78. Collaborators Natsumi Yotsumoto Former Student, Iba Lab. Satoshi Itoh A Former Graduate Student, Ko Matsuzuka Iba Lab. Student, Iba Lab. Daiki Muramatsu Peter Gloor Student, Iba Lab. Research Scientist, MIT CCI Keiichi Nemoto Bui Hong Ha Visiting Scholar, MIT CCI Former Student, Iba Lab. Fuji Xerox
  • 79. Papers & Presentations S. Itoh and T. Iba, "Analyzing Collaboration Network of Editors in JapaneseWikipedia", International Workshop and Conference on Network Science 09, 2009 S. Itoh, Y. Yamazaki, T. Iba, "Analyzing How Editors Write Articles in Wikipedia",Applications of Physics in Financial Analysis (APFA7), Tokyo, Japan, 2009 T. Iba & S. Itoh, "Sequential Collaboration Network of Open Collaboration",International Workshop and Conference on Network Science 09, 2009 S. Itoh & T, Iba, "Analyzing Collaboration Network of Editors in JapaneseWikipedia", International Workshop and Conference on Network Science 09, 2009 T. Iba, K. Nemoto, B. Peters, & P. A. Gloor, "Analyzing the Creative Editing Behaviorof Wikipedia Editors Through Dynamic Social Network Analysis", Procedia - Socialand Behavioral Sciences, Vol.2, Issue 4, 2010, pp.6441-6456 T. Iba, K. Matsuzuka, D. Muramatsu, "Editorial Collaboration Networks of WikipediaArticles in Various Languages", 3rd Conference on Collaborative Innovation Networks(COINs2011), 2011 伊藤 諭志, 伊藤 貴一, 熊坂 賢次, 井庭 崇, 「マスコラボレーションにおけるコンテンツ形成プロセスの分析」, 人工知能学会 第20回セマンティックウェブとオントロジー研究会, 2009 四元 菜つみ, 井庭 崇, 「Wikipedia におけるコラボレーションネットワークの成長」, 情報処理学会 第7回ネットワーク生態学シンポジウム, 2011
  • 80. Collaboration Networks of Linux Dynamics as Networks the dynamics of system/phenomena are represented as networks
  • 81. Linux-Activists Mailing List& comp.os.minix Newsgroup Takashi Iba (2008)
  • 82. Linux-Activists Mailing List (Nov. 1991) Takashi Iba (2008)
  • 83. Linux-Activists Mailing List (Nov. 1991) Takashi Iba (2008)
  • 84. Linux-Activists Mailing List (Nov. 1 – 14, 1991) Takashi Iba (2008)
  • 85. comp.os.minix Newsgroup (Aug. 1991) Takashi Iba (2008)
  • 86. comp.os.minix Newsgroup (Oct. 1991) Takashi Iba (2008)
  • 87. comp.os.minix Newsgroup (Dec. 1991) Takashi Iba (2008)
  • 88. comp.os.minix Newsgroup (Jan. 1992) Takashi Iba (2008)
  • 89. comp.os.minix Newsgroup (Aug. 1991) Takashi Iba (2008)
  • 90. comp.os.minix Newsgroup (Oct. 1991) Takashi Iba (2008)
  • 91. comp.os.minix Newsgroup (Jan. 1992) Takashi Iba (2008)
  • 92. comp.os.minix Newsgroup (Aug. 1991 - Jan. 1992) Takashi Iba (2008)
  • 93. Co-Purchase Networks of Books, CDs, DVDs Dynamics as Networks the dynamics of system/phenomena are represented as networks
  • 94. Rakuten Books: A famous Japanese Online Store http://books.rakuten.co.jp/ We’ve got POS data of random-sampled 30,000 customers (2005-2006) - Customer ID (masked) - When purchasing - Which book (CD, DVD) purchasing * sample customers of books, CDs, and DVDs are different one another.
  • 95. Building Co-Purchase Network Sequential Connectionfrom data (1) (2)
  • 96. Co-Purchase Network of Books Sequential Connection Number of nodes = 68,701 Number of edges = 113,940 * Target Customers = 30,000 * All Links are Visualized.
  • 97. Co-Purchase Network of CD Sequential Connection Number of nodes = 14,038 Number of edges = 22,727 * Target Customers = 30,000 * All Links are Visualized.
  • 98. Co-Purchase Network of DVD Sequential Connection Number of nodes = 10,875 Number of edges = 24,535 * Target Customers = 30,000 * All Links are Visualized.
  • 99. In-Degree Distribution Sequential Connectionfollow Power Law Book γ=2.56 CD DVD γ=2.33 γ=2.09
  • 100. Out-Degree Distribution Sequential Connectionfollow Power Law Book γ=2.57 CD DVD γ=2.43 γ=1.99
  • 101. Weight Distribution follow Power Law Sequential Connection Books CDs DVDs
  • 102. Mapping Genre in Co-Purchase Sequential ConnectionNetwork of DVDsNode Coloringlow k highSequential ConnectionTarget Customers = 30,000Weight > 1
  • 103. Papers & Presentations T. Iba, M. Mori, "Visualizing and Analyzing Networks of Co-Purchased Books,CDs and DVDs", NetSci’08, June, 2008 Y. Kitayama, M. Yoshida, S. Takami and T. Iba, “Analyzing Co-Purchase Networkof Books in Japanese Online Store”, poster, NetSci’08, June, 2008 R. Nishida, M. Mori and T. Iba, “Analyzing Co-Purchase Network of CDs inJapanese Online Store”, poster, NetSci’08, June, 2008 S. Itoh, S. Takami and T. Iba, “Analyzing Co-Purchase Network of DVDs inJapanese Online Store”, poster, NetSci’08, June, 2008 井庭 崇, 北山 雄樹, 伊藤 諭志, 西田 亮介, 吉田 真理子, 「オンラインストアにおける商品の共購買ネットワークの分析」, 情報処理学会 第5回ネットワーク生態学シンポジウム, 2009 Collaborators Mariko Yoshida Former Student, Iba Lab. Satoshi Itoh Ryosuke Nishida Former Graduate Student, Former Graduate Student, Iba Lab. Iba Lab. Yuki Kitayama Masaya Mori Former Graduate Student, Rakuten Institute of Technology Iba Lab.
  • 104. State Networks of Chaotic Dynamical Systems Dynamics as Networks the dynamics of system/phenomena are represented as networks
  • 105. Drawing a map of system’s behaviorThe networks of state transitions in several discretizedchaotic dynamical systems are scale-free networks.It is found in Logistic map, Sine map, Cubic map,General symmetric map, Gaussian map, Sine-circle map.
  • 106. Fundamentals: Chaos Highly complex behavior is generated, although it’s governed by a very simple rule.
  • 107. Fundamentals: Logistic Map xn+1 = 4µ xn ( 1 - xn ) a simple population growth model (non-overlapping generations) xn ... population (capacity) 0 < xn < 1 (variable) µ ... a rate of growth 0 < µ < 1 (constant) x0 = an initial value n=0 x1 = 4µ x0 ( 1 - x0 ) n=1 x2 = 4µ x1 ( 1 - x1 ) May, R. M. Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645–647 (1974). n=2 x3 = 4µ x2 ( 1 - x2 ) May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).
  • 108. Fundamentals: Logistic Map xn+1 = 4µ xn ( 1 - xn ) The behavior depends on the value of control parameter µ. 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 x x x 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 n n n µ 0 0.25 0.5 0.75 0.89... 1 1 1 0.8 0.8 0.6 0.6 x x 0.4 0.4 0.2 0.2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 n n
  • 109. Fundamentals: Logistic Map xn+1 = 4µ xn ( 1 - xn ) bifurcation diagram 1x 0 µ 0 0.25 0.5 0.75 0.89... 1 This diagram shows long-term values of x.
  • 110. 1 1Fundamentals: Logistic Map 1 0.8 0.8 0.8 0.6 0.6 0.6 x x x 0.4 0.4 0.4 xn+1 = 4µ xn ( 1 - xn ) 0.2 0 0 20 40 n 60 80 100 0.2 0 0 20 40 n 60 80 100 0.2 0 0 20 40 n 60 80 100 bifurcation diagram 1x 0 µ 0 0.25 0.5 0.75 0.89... 1 1 1 This diagram shows long-term values of x. 0.8 0.8 0.6 0.6 x x 0.4 0.4 0.2 0.2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 n n
  • 111. I want a map!for taking an overview of the whole.map1. a. A representation, usually on a plane surface, of a region of the earth or heavens. b. Something that suggests such a representation, as in clarity of representation.2. Mathematics: The correspondence of elements in one set to elements in the same set or another set. - The American Heritage Dictionary of the English Language
  • 112. I want a map of the map!for taking an overview of the whole behavior of the iterated map.map Logistic Map xn+1 = 4 xn ( 1 - xn ) ?1. a. A representation, usually on a plane surface, of a region of the earth or heavens. b. Something that suggests such a representation, as in clarity of representation.2. Mathematics: The correspondence of elements in one set to elements in the same set or another set. - The American Heritage Dictionary of the English Language
  • 113. Existing methods are time series 1NOT for drawing a map 0.8for taking an overview of the whole 0.6 x 0.4 0.2They merely visualize the trajectory of 0 0 20 40 60 80 100an instance of the system’s behavior, ngiving an initial value. cobweb plot 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 bifurcation diagram
  • 114. Visualizing State Transitions of a Systemfor taking an overview of the whole. system state xn+1 = 4µ xn ( 1 - xn ) state: the value of x
  • 115. xn+1 = 4µ xn ( 1 - xn )map x is treated as discrete which implies the finite number of states abstractiontarget xn+1 = 4µ xn ( 1 - xn )world x is a real number which has no limit to smallness (detail)
  • 116. xn+1 = 4µ xn ( 1 - xn ) x is treated as discrete which implies the finite number of statesTransit map State-transition mapNetwork of stations Network of states
  • 117. Discretizing into the finite number of statesTo subdivide a unit interval of variables into a finite number of subintervals x
  • 118. Discretizing into the finite number of statesTo subdivide a unit interval of variables into a finite number of subintervals x h()(ex.) round-up h( ) is the rounding function:d = 1: 0.14 0.2 round-up, round-off, or round-downd = 2: 0.146 0.15The discretization is carried out byrounding the value x to d decimal places.Therefore,
  • 119. Obtaining the set of state transitionsTo apply the map f into all states in order to obtain all state transitions. h()(ex.) round-upd = 1: 0.14 0.2 the set of stated = 2: 0.146 0.15The discretization is carried out byrounding the value x to d decimal places.Therefore,
  • 120. Obtaining the set of state transitionsTo apply the map f into all states in order to obtain all state transitions. h() f the set of state transitionswhere h( ) is the rounding function, and
  • 121. Building state-transition networksTo connect each state into its successive state f the set of state transitions State-transition network is a “map” of the whole behavior of the system, so one can take an overview from bird’s-eye view. building networks
  • 122. State-transition networks The order of the network N = 1/Δ + 1. The in-degree of each node may exhibit various values. The out-degree of every node must be always equal to 1, since it is a deterministic system. Each connected component mustThe whole behavior is often have only one loop or cycle: fixedmapped into more than one point or periodic cycle.connected components, whichrepresent basins of attraction.
  • 123. State-transition networks as a “dried-up river” node: geographical point in the river link: a connection from a point to another The direction of a flow in the river is fixed. The network is dried-up river, where there are no water flows on the riverbed. There are many confluences of two or more tributaries. There are no branches of the flow.
  • 124. Numerical simulation traces an instance Determining a starting point and discharging water, you will see that the water flow downstream on the river network. That is a happening you witness when conducting a numerical simulation of system’s evolution in time. Numerical simulation represents an instance of flow on the state-transition networks.
  • 125. Let’s draw a mapof the logistic map! for taking an overview of the whole xn+1 = 4µ xn ( 1 - xn )
  • 126. The state-transition networks for the logistic mapControl Parameter: µ = 1 Therefore, xn+1 = h( 4 xn ( 1 - xn ) )Round-Up into the decimal place, d = 1 Therefore, 11 states xn xn+1 f h0.0 0.00 0.00.1 0.36 0.40.2 0.64 0.70.3 0.84 0.9 0.2 0.80.4 0.96 1.00.5 1.00 1.0 0.70.6 0.96 1.0 0.9 0.30.7 0.84 0.9 0.40.8 0.64 0.7 0.5 1.00.9 0.36 0.4 0.11.0 0.00 0.0 0.0 0.6
  • 127. The state-transition networks for the logistic mapControl Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, d = 2 Therefore, 101 states xn xn+1 f h0.00 0.0000 0.000.01 0.0396 0.040.02 0.0784 0.080.03 0.1164 0.120.04 0.1536 0.160.05 0.1900 0.190.06 0.2256 0.230.07 0.2604 0.270.08 0.2944 0.300.09 0.3276 0.330.10 0.3600 0.360.11 0.3916 0.400.12 0.4224 0.430.13 0.4524 0.460.14 0.4816 0.490.15 0.5100 0.510.16 0.5376 0.54
  • 128. The state-transition networks for the logistic mapControl Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, d = 3 Therefore, 1001 states xn f h xn+10.000 0.000000 0.0000.001 0.003996 0.0040.002 0.007984 0.0080.003 0.011964 0.0120.004 0.015936 0.0160.005 0.019900 0.0200.006 0.023856 0.0240.007 0.027804 0.0280.008 0.031744 0.0320.009 0.035676 0.0360.010 0.039600 0.0400.011 0.043516 0.0440.012 0.047424 0.0480.013 0.051324 0.0520.014 0.055216 0.0560.015 0.059100 0.0600.016 0.062976 0.063
  • 129. The state-transition networks for the logistic mapControl Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, d = 4 Therefore, 10001 states xn xn+1 f h0.0000 0.00000000 0.00000.0001 0.00039996 0.00040.0002 0.00079984 0.00080.0003 0.00119964 0.00120.0004 0.00159936 0.00160.0005 0.00199900 0.00200.0006 0.00239856 0.00240.0007 0.00279804 0.00280.0008 0.00319744 0.00320.0009 0.00359676 0.00360.0010 0.00399600 0.00400.0011 0.00439516 0.00440.0012 0.00479424 0.00480.0013 0.00519324 0.00520.0014 0.00559216 0.00560.0015 0.00599100 0.00600.0016 0.00638976 0.0064
  • 130. The cumulative in-degree distributions of state-transitionnetworks for the logistic mapwith µ = 1 in the case from d = 1 to d = 7 The dashed line has slope -2. The networks are scale-free networks with the degree exponent γ = 1, regardless of the value of d.
  • 131. The state-transition networks for the logistic mapControl Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, ranging from d = 1 to d = 4 d=1 d=2 d=3 d=4 scale-free networks!
  • 132. Does the scale-free property depend on the control parameter µ ?
  • 133. Does the scale-free property depend on the control parameter µ ? xn+1 = 4µ xn ( 1 - xn ) 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 x x x 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 n n n µ 0 0.25 0.5 0.75 0.89... 1 1 1 0.8 0.8 0.6 0.6 x x 0.4 0.4 0.2 0.2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 n n
  • 134. The state-transition networks for the logistic mapControl Parameter: ranging from µ = 0 to µ = 1 xn+1 = 4µ xn ( 1 - xn )Round-Up into the decimal place, d = 3
  • 135. The cumulative in-degree distributions of state-transitionnetworks for the logistic mapfrom µ = 0.125 to 1.000 by 0.125 in the case d = 7 The dashed line has slope -2. The networks are scale-free networks with the degree exponent γ = 1, regardless of the value of the parameter µ.
  • 136. Does the scale-free property depend on the control parameter µ ? NO.The scale-free property is independent of the control parameter µ.
  • 137. How the scale-free network is emerged? o w? H
  • 138. How the scale-free network is emerged? 1.0 xn+1 = 4µ xn ( 1 - xn ) 0.8 xn+1 0.6 0.4 0.2 xn 0.2 0.4 0.6 0.8 1.0
  • 139. How the scale-free network is emerged?If x is a real number, namely in the mathematically ideal condition, the state 0.84 must have only 2 previous values: 0.30 and 0.70. 1.0 0.84 0.8 xn+1 = 4µ xn ( 1 - xn ) xn+1 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 0.30 0.70 xnIn the condition,the state-transition network cannot be a scale-free network ...
  • 140. How the scale-free network is emerged?
  • 141. How the scale-free network is emerged?The trick is discretization.
  • 142. How the scale-free network is emerged? The relation between a ex. µ = 1, d = 3 subinterval on y-axis and its corresponding range on xn+1 in-degree the x-axis for the logistic 1.000 31 map function. 0.999 14 0.998 10 0.997 8 0.996 8 0.995 6 0.994 6 0.993 6 0.992 6
  • 143. Mathematical Derivation
  • 144. The cumulative in-degree distribution follows a law given by:Summarizing The second term makes a large effect on the distribution, only when k is quite close to k,max because . It means that the “hub” states have higher in- degree than the typical scale-free network whose in-degree distribution follows a strict power law.
  • 145. The cumulative in-degree distribution follows a law given by: Summarizing The second term makes a large effect on the distribution, only when k is quite close to k,max because . It means that the “hub” states have higher in- degree than the typical scale-free network whose in-degree distribution follows a strict power law.taking the limit In the case Δ is extremely small ( d is extremely large), The cumulative in-degree distribution follows a power law: Summarizing
  • 146. The cumulative in-degree distribution follows a law given by: Summarizingtaking the limit In the case Δ is extremely small ( d is extremely large), The cumulative in-degree distribution follows a power law: Summarizing
  • 147. Comparison between the results of numerical computationsand mathematical predictionswith µ = 0.1 and µ = 1.0, in the case d = 6
  • 148. Are there any other maps whosestate-transition networks are scale-free networks? Yes!
  • 149. One-dimensional maps Sine map Cubic map ♯1 Cubic map ♯2
  • 150. The state-transition networks for other one-dimensional maps Logistic map Sine map µ = 1.0, d = 3 µ = 1.0, d = 3 Cubic map ♯1 Cubic map ♯2 µ = 1.0, d = 3 µ = 1.0, d = 3
  • 151. The cumulative in-degree distributions of state-transitionnetworks for other one-dimensional maps d=6 The dashed line has slope -2 The networks are scale-free networks with the degree exponent γ = 1.
  • 152. One-dimensional maps General Symmetric map The parameter α is controlling the flatness around the top. α = 2.0: the logistic map
  • 153. The state-transition networks for the general symmetric mapα = 1.5 α = 2.0 α = 2.5α = 3.0 α = 3.5 α = 4.0 d=3
  • 154. The cumulative in-degree distributions of state-transitionnetworks for the general symmetric map d=6 The parameter α influences the degree exponent γ, while the scale-free property is maintained.
  • 155. Other types of maps Gaussian map exponential Sine-circle map discontinuous Delayed logistic map two-dimensional
  • 156. The state-transition networks for the other types of maps Sine-circle map Gaussian map a = 4.0, b = 0.5. d = 3 a = 1.0, b = −0.3, d = 3 Delayed logistic map a = 2.27, d = 2
  • 157. The cumulative in-degree distributions of state-transitionnetworks for the other types of maps d=6 d=3 The dashed line has slope -2 the networks are scale-free networks with the degree exponent γ = 1.
  • 158. Chaotic mapsthat have scale-free state-transition networks Logistic map one-dimensional Sine map one-dimensional Cubic map one-dimensional Gaussian map one-dimensional, exponential Sine-circle map one-dimensional, discontinuous Delayed logistic map two-dimensional
  • 159. Do all chaotic maps havescale-free state-transition networks? No.
  • 160. NOT scale-free state-transition networks of chaotic maps Tent map Binary Shift map Gingerbreadman map a = 2.0. d = 3 d=3 d=1
  • 161. NOT scale-free state-transition networks of chaotic maps Cusp map Pincher map Henon map a = 2.0. d = 4 a =. d = 4 a = 1.4, b = 0.3, d = 2 Lozi map Henon area-preserving map Standard (Chirikov) map a = 1.7, b = 0.5, d = 2 a = 0.24. d = 2 a = 1.0. d = 1
  • 162. Papers & Presentations T. Iba, "Hidden Order in Chaos: The Network-Analysis Approach To DynamicalSystems", Unifying Themes in Complex Systems Volume VIII: Proceedings of theEighth International Conference on Complex Systems, Sayama, H., Minai, A. A.,Braha, D. and Bar-Yam, Y. eds., NECSI Knowledge Press, Jun., 2011, pp.769-783 T. Iba, "Scale-Free Networks Hidden in Chaotic Dynamical Systems", arXiv:1007.4137v1, 2010! T. Iba with K. Shimonishi, J. Hirose, A. Masumori, "The Chaos Book: New Explorations for Order Hidden in Chaos", 2011
  • 163. Dynamics as Networksthe dynamics of system/phenomena are represented as networks Chord Networks of Music Chord-Transition Networks of Music Collaboration Networks of Wikipedia Collaboration Networks of Linux Co-Purchase Networks of Books, CDs, DVDs State Networks of Chaotic Dynamical Systems
  • 164. Network Analysis for Understanding Dynamicswant to capture the process / dynamics ofphenomena as a whole.introducing network analysis in a new way.to build a directed network by connectingbetween nodes, based on the sequential orderof occurrence.a new viewpoint “dynamics as network”, whichis a distinct from “dynamics of network” and“dynamics on network.”
  • 165. Network Analysis for Understanding Dynamics - Wikipedia, Music & Chaos - 時間発展のネットワーク分析 TAKASHI IBA 井庭  崇 Ph.D. in Media and Governance Associate Professor at Faculty of Policy Management, Keio University, Japan twitter in Japanese: takashiiba twitter in English: taka_iba