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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
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)
   Creativity

Teaching at SFC, Keio University
   Social Systems Theory
   Pattern Language
   Simulation Design                                          Social Systems Theory
   Science of Complex Systems                                 [Reality+ Series] (2011)
                                                              in Japanese
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
Network Analysis for Understanding Dynamics

want to capture the process / dynamics of
phenomena as a whole.

introducing network analysis in a new way.

to build a directed network by connecting
between nodes, based on the sequential order
of occurrence.

a new viewpoint “dynamics as network”, which
is a distinct from “dynamics of network” and
“dynamics on network.”
Dynamics
Dynamics
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
Dynamics of Networks
Structural 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", NetSci'07, 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
Dynamics of Networks
Alliance Networks of Nations




                the evolution of network structure in time
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
Dynamics on Networks
Iterated 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", NetSci'07, 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
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
Dynamics of Networks



Dynamics on Networks



Dynamics as Networks
Dynamics as Networks
the dynamics of system/phenomena are represented as networks
Dynamics as Networks
Sequential 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
Dynamics as Networks
Sequential 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
Dynamics as Networks
Sequential Collaboration Network of Wikipedians



The Sequential Collaboration Network
of an Article on Wikipedia (English)
“Australia”




  the dynamics of system/phenomena are represented as networks
Dynamics as Networks
Sequential 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
Dynamics as Networks
the dynamics of system/phenomena are represented as networks
Dynamics as Networks
the 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
Chord Networks of Music


              Dynamics as Networks
the dynamics of system/phenomena are represented as networks
Sequential Chord Network of Music
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
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’S
YESTERDAY ONCE MORE      THOSE GOOD OLD DREAMS
                                                 YOUR FIRST TIME
Sequential Chord Network of Music [Integrated]




Carpenters
major 19 songs

I NEED TO BE IN LOVE
WE’VE ONLY JUST BEGUN
TOP OF THE WORLD
RAINY DAYS AND MONDAY
GOODBY TO LOVE
SING
ONLY YESTERDAY
I WON’T LAST A DAY WITHOUT YOU
SOLITAIRE
PLEASE MR.POSTMAN
JAMBALAYA (ON THE BAYOU)
HURTING EACH OTHER
THERE’S A KIND OF HUSH (ALL OVER THE WORLD)
FOR ALL WE KNOW
BLESS THE BEASTS AND CHILDREN
(THEY LONG TO BE) CLOSE TO YOU
YESTERDAY ONCE MORE
THOSE GOOD OLD DREAMS
MAKE BELIEVE IT’S YOUR FIRST TIME
Sequential Chord Network
                                                        in-degree distribution
of Music [Integrated]                                    1




Carpenters                                             0.1



major 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
Chord-Transition Networks of Music


                  Dynamics as Networks
    the dynamics of system/phenomena are represented as networks
Sequential Chord-Transition Network of Music
Sequential Chord-Transition Network of Music



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
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’S
YESTERDAY ONCE MORE      THOSE GOOD OLD DREAMS
                                                 YOUR FIRST TIME
Sequential Chord-Transition Network of Music




Carpenters
major 19 songs

I NEED TO BE IN LOVE
WE’VE ONLY JUST BEGUN
TOP OF THE WORLD
RAINY DAYS AND MONDAY
GOODBY TO LOVE
SING
ONLY YESTERDAY
I WON’T LAST A DAY WITHOUT YOU
SOLITAIRE
PLEASE MR.POSTMAN
JAMBALAYA (ON THE BAYOU)
HURTING EACH OTHER
THERE’S A KIND OF HUSH (ALL OVER THE WORLD)
FOR ALL WE KNOW
BLESS THE BEASTS AND CHILDREN
(THEY LONG TO BE) CLOSE TO YOU
YESTERDAY ONCE MORE
THOSE GOOD OLD DREAMS
MAKE BELIEVE IT’S YOUR FIRST TIME
Sequential Chord-Transition
Network of Music [Integrated]                                 in-degree distribution
                                                               1




Carpenters
major 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
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", International
School and Conference on Network Science 2010 (NetSci2010), 2010




     Collaborators


                 Junya Hirose
                 Former Graduate Student
Collaboration Networks of Wikipedia


                   Dynamics as Networks
     the dynamics of system/phenomena are represented as networks
Editorial Collaboration Networks of
Wikipedia 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.
Editorial Collaboration Networks of
Wikipedia 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
Editorial Collaboration Networks of
Wikipedia 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
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
Sequential Collaboration Network of Article
“Collaborative Innovation Networks” in English Wikipedia




The number of Nodes = 51
Average path length = 6.399
Sequential Collaboration Network of Article “Basel”
in English Wikipedia




The number of Nodes = 594
Average path length = 6.577
Sequential Collaboration Network of Article “Switzerland”
in English Wikipedia




The number of Nodes = 3998
Average path length = 5.468
Sequential Collaboration Network of Article “Fondue”
in English Wikipedia




The number of Nodes = 457
Average path length = 10.485
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
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 Wikipedia
Editors Through Dynamical Social Network Analysis", COINs2011, 2009
T. Iba and S. Itoh, "Sequential Collaboration Network of Open Collaboration", NetSci'09, 2009
Editorial Collaboration Networks of
Wikipedia 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
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: Portuguese
Analyzing ALL articles as of January    Rank 10: Russian
1st, 2011 in each language.             …
                                        Rank 15: Finnish
The ranking based on the data as of     …
January 6th, 2011.                      Rank 20: Turkish
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
English      German     French    Polish




Italian      Japanese   Spanish   Dutch




Portuguese   Russian    Finnish   Turkish
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.
Editorial Collaboration Networks of
Wikipedia 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
nAnalysis 2: Distribution of account and IP users




                                                 IP users




                                                Account
                                                 users
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)
Scatter plot of articles with
number 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)
Scatter plot of articles with
number 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
Scatter plot of articles with
number 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)
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
Editorial Collaboration Networks of
Wikipedia 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
nAnalysis 3: Distribution of Featured Articles




         3,372 featured articles / 3,732,033 articles
                   In English Wikipedia
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)
Scatter plot of featured articles on the 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)
Scatter plot of featured articles on the 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)
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.
Editorial Collaboration Networks of
Wikipedia 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
Editorial Collaboration Networks of
Wikipedia 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.
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
Papers & Presentations

 S. Itoh and T. Iba, "Analyzing Collaboration Network of Editors in Japanese
Wikipedia", 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 Japanese
Wikipedia", International Workshop and Conference on Network Science '09, 2009
  T. Iba, K. Nemoto, B. Peters, & P. A. Gloor, "Analyzing the Creative Editing Behavior
of Wikipedia Editors Through Dynamic Social Network Analysis", Procedia - Social
and Behavioral Sciences, Vol.2, Issue 4, 2010, pp.6441-6456
  T. Iba, K. Matsuzuka, D. Muramatsu, "Editorial Collaboration Networks of Wikipedia
Articles 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
Collaboration Networks of Linux


                 Dynamics as Networks
   the dynamics of system/phenomena are represented as networks
Linux-Activists Mailing List
& comp.os.minix Newsgroup




                               Takashi Iba (2008)
Linux-Activists Mailing List (Nov. 1991)




                                           Takashi Iba (2008)
Linux-Activists Mailing List (Nov. 1991)




                                           Takashi Iba (2008)
Linux-Activists Mailing List (Nov. 1 – 14, 1991)




                                                   Takashi Iba (2008)
comp.os.minix Newsgroup (Aug. 1991)




                                      Takashi Iba (2008)
comp.os.minix Newsgroup (Oct. 1991)




                                      Takashi Iba (2008)
comp.os.minix Newsgroup (Dec. 1991)




                                      Takashi Iba (2008)
comp.os.minix Newsgroup (Jan. 1992)




                                      Takashi Iba (2008)
comp.os.minix Newsgroup (Aug. 1991)




                                      Takashi Iba (2008)
comp.os.minix Newsgroup (Oct. 1991)




                                      Takashi Iba (2008)
comp.os.minix Newsgroup (Jan. 1992)




                                      Takashi Iba (2008)
comp.os.minix Newsgroup (Aug. 1991 - Jan. 1992)




                                                  Takashi Iba (2008)
Co-Purchase Networks of Books, CDs, DVDs



                     Dynamics as Networks
       the dynamics of system/phenomena are represented as networks
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.
Building Co-Purchase Network   Sequential Connection
from data




     (1)



           (2)
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.
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.
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.
In-Degree Distribution          Sequential Connection
follow Power Law
                         Book


                                    γ=2.56




 CD                      DVD


             γ=2.33                 γ=2.09
Out-Degree Distribution          Sequential Connection
follow Power Law
                          Book

                                     γ=2.57




 CD                       DVD


            γ=2.43                   γ=1.99
Weight Distribution follow Power Law   Sequential Connection


                         Books




   CDs                    DVDs
Mapping Genre in Co-Purchase   Sequential Connection
Network of DVDs
Node Coloring

low   k   high




Sequential Connection
Target Customers = 30,000
Weight > 1
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 Network
of Books in Japanese Online Store”, poster, NetSci’08, June, 2008
  R. Nishida, M. Mori and T. Iba, “Analyzing Co-Purchase Network of CDs in
Japanese Online Store”, poster, NetSci’08, June, 2008
  S. Itoh, S. Takami and T. Iba, “Analyzing Co-Purchase Network of DVDs in
Japanese 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.
State Networks of Chaotic Dynamical Systems


                      Dynamics as Networks
        the dynamics of system/phenomena are represented as networks
Drawing a map of system’s behavior




The networks of state transitions in several discretized
chaotic dynamical systems are scale-free networks.
It is found in Logistic map, Sine map, Cubic map,
General symmetric map, Gaussian map, Sine-circle map.
Fundamentals: Chaos




       Highly complex behavior is generated,
       although it’s governed by a very simple rule.
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).
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
Fundamentals: Logistic Map
     xn+1 = 4µ xn ( 1 - xn )
     bifurcation diagram
 1




x


 0                                                                    µ
     0              0.25               0.5       0.75   0.89...   1

     This diagram shows long-term values of x.
1                                           1



Fundamentals: 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
 1




x


 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
I want a map!
for taking an overview of the whole.




map
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
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
Existing methods are                          time series
                                               1


NOT for drawing a map                         0.8



for taking an overview of the whole
                                              0.6
                                          x
                                              0.4

                                              0.2

They merely visualize the trajectory of        0
                                                     0   20    40         60    80    100

an instance of the system’s behavior,
                                                                     n




giving 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
Visualizing State Transitions of a System
for taking an overview of the whole.



 system


       state




  xn+1 = 4µ xn ( 1 - xn )
  state: the value of x
xn+1 = 4µ xn ( 1 - xn )
map      x is treated as discrete
         which implies the finite number of states




             abstraction



target   xn+1 = 4µ xn ( 1 - xn )
world    x is a real number
         which has no limit to smallness (detail)
xn+1 = 4µ xn ( 1 - xn )
                      x is treated as discrete
                      which implies the finite number of states




Transit map           State-transition map
Network of stations          Network of states
Discretizing into the finite number of states
To subdivide a unit interval of variables into a finite number of subintervals




          x
Discretizing into the finite number of states
To 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-down
d = 2: 0.146     0.15

The discretization is carried out by
rounding the value x to d decimal places.
Therefore,
Obtaining the set of state transitions
To apply the map f into all states in order to obtain all state transitions.




                                                             h()




(ex.) round-up
d = 1: 0.14       0.2                                                     the set of state
d = 2: 0.146      0.15

The discretization is carried out by
rounding the value x to d decimal places.
Therefore,
Obtaining the set of state transitions
To apply the map f into all states in order to obtain all state transitions.




                                                             h()




                                                                f


                                                                          the set of
                                                                          state transitions




where h( ) is the rounding function, and
Building state-transition networks
To 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
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 must
The whole behavior is often       have only one loop or cycle: fixed
mapped into more than one         point or periodic cycle.
connected components, which
represent basins of attraction.
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.
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.
Let’s draw a map
of the logistic map!
 for taking an overview of the whole


      xn+1 = 4µ xn ( 1 - xn )
The state-transition networks for the logistic map
Control 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          h
0.0       0.00       0.0
0.1       0.36       0.4
0.2       0.64       0.7
0.3       0.84       0.9                             0.2                   0.8
0.4       0.96       1.0
0.5       1.00       1.0                                           0.7
0.6       0.96       1.0                                     0.9                 0.3
0.7       0.84       0.9                                   0.4
0.8       0.64       0.7                     0.5
                                                     1.0
0.9       0.36       0.4                                                 0.1
1.0       0.00       0.0
                                               0.0
                                                           0.6
The state-transition networks for the logistic map
Control 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            h
0.00       0.0000       0.00
0.01       0.0396       0.04
0.02       0.0784       0.08
0.03       0.1164       0.12
0.04       0.1536       0.16
0.05       0.1900       0.19
0.06       0.2256       0.23
0.07       0.2604       0.27
0.08       0.2944       0.30
0.09       0.3276       0.33
0.10       0.3600       0.36
0.11       0.3916       0.40
0.12       0.4224       0.43
0.13       0.4524       0.46
0.14       0.4816       0.49
0.15       0.5100       0.51
0.16       0.5376       0.54
The state-transition networks for the logistic map
Control 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+1
0.000       0.000000       0.000
0.001       0.003996       0.004
0.002       0.007984       0.008
0.003       0.011964       0.012
0.004       0.015936       0.016
0.005       0.019900       0.020
0.006       0.023856       0.024
0.007       0.027804       0.028
0.008       0.031744       0.032
0.009       0.035676       0.036
0.010       0.039600       0.040
0.011       0.043516       0.044
0.012       0.047424       0.048
0.013       0.051324       0.052
0.014       0.055216       0.056
0.015       0.059100       0.060
0.016       0.062976       0.063
The state-transition networks for the logistic map
Control 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                h
0.0000       0.00000000       0.0000
0.0001       0.00039996       0.0004
0.0002       0.00079984       0.0008
0.0003       0.00119964       0.0012
0.0004       0.00159936       0.0016
0.0005       0.00199900       0.0020
0.0006       0.00239856       0.0024
0.0007       0.00279804       0.0028
0.0008       0.00319744       0.0032
0.0009       0.00359676       0.0036
0.0010       0.00399600       0.0040
0.0011       0.00439516       0.0044
0.0012       0.00479424       0.0048
0.0013       0.00519324       0.0052
0.0014       0.00559216       0.0056
0.0015       0.00599100       0.0060
0.0016       0.00638976       0.0064
The cumulative in-degree distributions of state-transition
networks for the logistic map
with 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.
The state-transition networks for the logistic map
Control 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!
Does the scale-free property depend
   on the control parameter µ ?
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
The state-transition networks for the logistic map
Control Parameter: ranging from µ = 0 to µ = 1   xn+1 = 4µ xn ( 1 - xn )
Round-Up into the decimal place, d = 3
The cumulative in-degree distributions of state-transition
networks for the logistic map
from 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.
Does the scale-free property depend
    on the control parameter µ ?

                NO.

The scale-free property is independent
     of the control parameter µ.
How the scale-free network is emerged?


                                    o w?
                                H
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
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
                                             xn
In the condition,
the state-transition network cannot be a scale-free network ...
How the scale-free network is emerged?
How the scale-free network is emerged?

The trick is discretization.
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
Mathematical Derivation
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.
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
The cumulative in-degree
                                  distribution follows a law
                                  given by:
                   Summarizing




taking the limit


                                 In the case Δ is extremely small
                                 ( d is extremely large),
                                 The cumulative in-degree
                                 distribution follows a power law:
                   Summarizing
Comparison between the results of numerical computations
and mathematical predictions
with µ = 0.1 and µ = 1.0, in the case d = 6
Are there any other maps whose
state-transition networks are scale-free networks?



        Yes!
One-dimensional maps



       Sine map



       Cubic map ♯1



       Cubic map ♯2
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
The cumulative in-degree distributions of state-transition
networks 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.
One-dimensional maps



      General Symmetric map




           The parameter α is controlling
           the flatness around the top.

           α = 2.0: the logistic map
The state-transition networks for the general symmetric map

α = 1.5           α = 2.0           α = 2.5




α = 3.0           α = 3.5           α = 4.0




 d=3
The cumulative in-degree distributions of state-transition
networks for the general symmetric map




                                                               d=6




                                The parameter α influences the degree
                                exponent γ, while the scale-free property
                                is maintained.
Other types of maps



    Gaussian map           exponential




    Sine-circle map        discontinuous




    Delayed logistic map   two-dimensional
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
The cumulative in-degree distributions of state-transition
networks 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.
Chaotic maps
that 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
Do all chaotic maps have
scale-free state-transition networks?




     No.
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
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
Network Analysis for Understanding Dynamics: Wikipedia, Music & Chaos
Papers & Presentations

  T. Iba, "Hidden Order in Chaos: The Network-Analysis Approach To Dynamical
Systems", Unifying Themes in Complex Systems Volume VIII: Proceedings of the
Eighth 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
Dynamics as Networks
the 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
Network Analysis for Understanding Dynamics

want to capture the process / dynamics of
phenomena as a whole.

introducing network analysis in a new way.

to build a directed network by connecting
between nodes, based on the sequential order
of occurrence.

a new viewpoint “dynamics as network”, which
is a distinct from “dynamics of network” and
“dynamics on network.”
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

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Network Analysis for Understanding Dynamics: Wikipedia, Music & Chaos

  • 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) Creativity Teaching 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 Dynamics want to capture the process / dynamics of phenomena as a whole. introducing network analysis in a new way. to build a directed network by connecting between nodes, based on the sequential order of occurrence. a new viewpoint “dynamics as network”, which is a distinct from “dynamics of network” and “dynamics on network.”
  • 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 Networks Structural 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", NetSci'07, 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 Networks Alliance 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 Networks Iterated 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", NetSci'07, 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 Networks Dynamics on Networks Dynamics as Networks
  • 14. Dynamics as Networks the dynamics of system/phenomena are represented as networks
  • 15. Dynamics as Networks Sequential 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 Networks Sequential 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 Networks Sequential Collaboration Network of Wikipedians The Sequential Collaboration Network of an Article on Wikipedia (English) “Australia” the dynamics of system/phenomena are represented as networks
  • 18. Dynamics as Networks Sequential 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 Networks the dynamics of system/phenomena are represented as networks
  • 20. Dynamics as Networks the 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 Networks the dynamics of system/phenomena are represented as networks
  • 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’S YESTERDAY ONCE MORE THOSE GOOD OLD DREAMS YOUR FIRST TIME
  • 25. Sequential Chord Network of Music [Integrated] Carpenters major 19 songs I NEED TO BE IN LOVE WE’VE ONLY JUST BEGUN TOP OF THE WORLD RAINY DAYS AND MONDAY GOODBY TO LOVE SING ONLY YESTERDAY I WON’T LAST A DAY WITHOUT YOU SOLITAIRE PLEASE MR.POSTMAN JAMBALAYA (ON THE BAYOU) HURTING EACH OTHER THERE’S A KIND OF HUSH (ALL OVER THE WORLD) FOR ALL WE KNOW BLESS THE BEASTS AND CHILDREN (THEY LONG TO BE) CLOSE TO YOU YESTERDAY ONCE MORE THOSE GOOD OLD DREAMS MAKE BELIEVE IT’S YOUR FIRST TIME
  • 26. Sequential Chord Network in-degree distribution of Music [Integrated] 1 Carpenters 0.1 major 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
  • 29. Sequential Chord-Transition Network of Music 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
  • 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’S YESTERDAY ONCE MORE THOSE GOOD OLD DREAMS YOUR FIRST TIME
  • 31. Sequential Chord-Transition Network of Music Carpenters major 19 songs I NEED TO BE IN LOVE WE’VE ONLY JUST BEGUN TOP OF THE WORLD RAINY DAYS AND MONDAY GOODBY TO LOVE SING ONLY YESTERDAY I WON’T LAST A DAY WITHOUT YOU SOLITAIRE PLEASE MR.POSTMAN JAMBALAYA (ON THE BAYOU) HURTING EACH OTHER THERE’S A KIND OF HUSH (ALL OVER THE WORLD) FOR ALL WE KNOW BLESS THE BEASTS AND CHILDREN (THEY LONG TO BE) CLOSE TO YOU YESTERDAY ONCE MORE THOSE GOOD OLD DREAMS MAKE BELIEVE IT’S YOUR FIRST TIME
  • 32. Sequential Chord-Transition Network of Music [Integrated] in-degree distribution 1 Carpenters major 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", International School 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 of Wikipedia 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 of Wikipedia 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 of Wikipedia 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 Wikipedia The number of Nodes = 51 Average path length = 6.399
  • 40. Sequential Collaboration Network of Article “Basel” in English Wikipedia The number of Nodes = 594 Average path length = 6.577
  • 41. Sequential Collaboration Network of Article “Switzerland” in English Wikipedia The number of Nodes = 3998 Average path length = 5.468
  • 42. Sequential Collaboration Network of Article “Fondue” in English Wikipedia The number of Nodes = 457 Average 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 Wikipedia Editors Through Dynamical Social Network Analysis", COINs2011, 2009 T. Iba and S. Itoh, "Sequential Collaboration Network of Open Collaboration", NetSci'09, 2009
  • 45. Editorial Collaboration Networks of Wikipedia 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: Portuguese Analyzing ALL articles as of January Rank 10: Russian 1st, 2011 in each language. … Rank 15: Finnish The 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 Polish Italian Japanese Spanish Dutch Portuguese 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 of Wikipedia 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 with number 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 with number 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 with number 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 of Wikipedia 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 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)
  • 74. Scatter plot of featured articles on the 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)
  • 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 of Wikipedia 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 of Wikipedia 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 Japanese Wikipedia", 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 Japanese Wikipedia", International Workshop and Conference on Network Science '09, 2009 T. Iba, K. Nemoto, B. Peters, & P. A. Gloor, "Analyzing the Creative Editing Behavior of Wikipedia Editors Through Dynamic Social Network Analysis", Procedia - Social and Behavioral Sciences, Vol.2, Issue 4, 2010, pp.6441-6456 T. Iba, K. Matsuzuka, D. Muramatsu, "Editorial Collaboration Networks of Wikipedia Articles 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 Connection from 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 Connection follow Power Law Book γ=2.56 CD DVD γ=2.33 γ=2.09
  • 100. Out-Degree Distribution Sequential Connection follow 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 Connection Network of DVDs Node Coloring low k high Sequential Connection Target Customers = 30,000 Weight > 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 Network of Books in Japanese Online Store”, poster, NetSci’08, June, 2008 R. Nishida, M. Mori and T. Iba, “Analyzing Co-Purchase Network of CDs in Japanese Online Store”, poster, NetSci’08, June, 2008 S. Itoh, S. Takami and T. Iba, “Analyzing Co-Purchase Network of DVDs in Japanese 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 behavior The networks of state transitions in several discretized chaotic 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 1 x 0 µ 0 0.25 0.5 0.75 0.89... 1 This diagram shows long-term values of x.
  • 110. 1 1 Fundamentals: 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 1 x 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. map 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
  • 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 1 NOT for drawing a map 0.8 for taking an overview of the whole 0.6 x 0.4 0.2 They merely visualize the trajectory of 0 0 20 40 60 80 100 an instance of the system’s behavior, n giving 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 System for 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 abstraction target 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 states Transit map State-transition map Network of stations Network of states
  • 117. Discretizing into the finite number of states To subdivide a unit interval of variables into a finite number of subintervals x
  • 118. Discretizing into the finite number of states To 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-down d = 2: 0.146 0.15 The discretization is carried out by rounding the value x to d decimal places. Therefore,
  • 119. Obtaining the set of state transitions To apply the map f into all states in order to obtain all state transitions. h() (ex.) round-up d = 1: 0.14 0.2 the set of state d = 2: 0.146 0.15 The discretization is carried out by rounding the value x to d decimal places. Therefore,
  • 120. Obtaining the set of state transitions To apply the map f into all states in order to obtain all state transitions. h() f the set of state transitions where h( ) is the rounding function, and
  • 121. Building state-transition networks To 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 must The whole behavior is often have only one loop or cycle: fixed mapped into more than one point or periodic cycle. connected components, which represent 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 map of the logistic map! for taking an overview of the whole xn+1 = 4µ xn ( 1 - xn )
  • 126. The state-transition networks for the logistic map Control 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 h 0.0 0.00 0.0 0.1 0.36 0.4 0.2 0.64 0.7 0.3 0.84 0.9 0.2 0.8 0.4 0.96 1.0 0.5 1.00 1.0 0.7 0.6 0.96 1.0 0.9 0.3 0.7 0.84 0.9 0.4 0.8 0.64 0.7 0.5 1.0 0.9 0.36 0.4 0.1 1.0 0.00 0.0 0.0 0.6
  • 127. The state-transition networks for the logistic map Control 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 h 0.00 0.0000 0.00 0.01 0.0396 0.04 0.02 0.0784 0.08 0.03 0.1164 0.12 0.04 0.1536 0.16 0.05 0.1900 0.19 0.06 0.2256 0.23 0.07 0.2604 0.27 0.08 0.2944 0.30 0.09 0.3276 0.33 0.10 0.3600 0.36 0.11 0.3916 0.40 0.12 0.4224 0.43 0.13 0.4524 0.46 0.14 0.4816 0.49 0.15 0.5100 0.51 0.16 0.5376 0.54
  • 128. The state-transition networks for the logistic map Control 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+1 0.000 0.000000 0.000 0.001 0.003996 0.004 0.002 0.007984 0.008 0.003 0.011964 0.012 0.004 0.015936 0.016 0.005 0.019900 0.020 0.006 0.023856 0.024 0.007 0.027804 0.028 0.008 0.031744 0.032 0.009 0.035676 0.036 0.010 0.039600 0.040 0.011 0.043516 0.044 0.012 0.047424 0.048 0.013 0.051324 0.052 0.014 0.055216 0.056 0.015 0.059100 0.060 0.016 0.062976 0.063
  • 129. The state-transition networks for the logistic map Control 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 h 0.0000 0.00000000 0.0000 0.0001 0.00039996 0.0004 0.0002 0.00079984 0.0008 0.0003 0.00119964 0.0012 0.0004 0.00159936 0.0016 0.0005 0.00199900 0.0020 0.0006 0.00239856 0.0024 0.0007 0.00279804 0.0028 0.0008 0.00319744 0.0032 0.0009 0.00359676 0.0036 0.0010 0.00399600 0.0040 0.0011 0.00439516 0.0044 0.0012 0.00479424 0.0048 0.0013 0.00519324 0.0052 0.0014 0.00559216 0.0056 0.0015 0.00599100 0.0060 0.0016 0.00638976 0.0064
  • 130. The cumulative in-degree distributions of state-transition networks for the logistic map with 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 map Control 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 map Control 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-transition networks for the logistic map from 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 xn In 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
  • 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: Summarizing taking 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 computations and mathematical predictions with µ = 0.1 and µ = 1.0, in the case d = 6
  • 148. Are there any other maps whose state-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-transition networks 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 map α = 1.5 α = 2.0 α = 2.5 α = 3.0 α = 3.5 α = 4.0 d=3
  • 154. The cumulative in-degree distributions of state-transition networks 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-transition networks 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 maps that 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 have scale-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
  • 163. Papers & Presentations T. Iba, "Hidden Order in Chaos: The Network-Analysis Approach To Dynamical Systems", Unifying Themes in Complex Systems Volume VIII: Proceedings of the Eighth 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
  • 164. Dynamics as Networks the 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
  • 165. Network Analysis for Understanding Dynamics want to capture the process / dynamics of phenomena as a whole. introducing network analysis in a new way. to build a directed network by connecting between nodes, based on the sequential order of occurrence. a new viewpoint “dynamics as network”, which is a distinct from “dynamics of network” and “dynamics on network.”
  • 166. 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