Network Analysis for Understanding Dynamics- Wikipedia, Music & Chaos -
TAKASHI IBA
Ph.D. in Media and GovernanceAssociate Professor
at Faculty of Policy Management, Keio University, Japantwitter in Japanese: takashiiba
twitter in English: taka_iba
時間発展のネットワーク分析
井庭 崇
Teaching at SFC, Keio University
Social Systems TheoryPattern LanguageSimulation DesignScience of Complex Systems
Takashi Iba
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)
2004 -Sociology (Autopoietic Systems Theory)Network AnalysisPattern Language (A method to describe tacit knowledge)Creativity
Introduction toComplex Systems(1998) in Japanese
Studying ...
井庭 崇
Social Systems Theory [Reality+ Series] (2011) in Japanese
Network Analysis for Understanding Dynamics- Wikipedia, Music & Chaos -
TAKASHI IBA
Ph.D. in Media and GovernanceAssociate Professor
at Faculty of Policy Management, Keio University, Japantwitter in Japanese: takashiiba
twitter in English: taka_iba
時間発展のネットワーク分析
井庭 崇
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
Dynamics
DynamicsDynamics
Dynamics on Networksthe information spread or interaction on networks
When Network Scientists talk about dynamics...
Dynamics of Networksthe evolution of network structure in time
Dynamics
Dynamics
Structural Change of Alliance Networks of Nations
Dynamics of Networks
the evolution of network structure in time
1850 1946 2000
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
Alliance Networks of Nations
Dynamics of Networks
the evolution of network structure in time
Dynamics on Networksthe information spread or interaction on networks
When Network Scientists talk about dynamics...
Dynamics of Networksthe evolution of network structure in time
Iterated Games on Alliance Network of Nations, as an example
Dynamics on Networks
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
Dynamics on Networksthe information spread or interaction on networks
When Network Scientists talk about dynamics...
Dynamics of Networksthe evolution of network structure in time
Dynamics on Networks
Dynamics as Networks
Dynamics of Networks
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Editor B
Editor C
Editor A
sequentialorderan article
Editor A
Editor B
Editor C
Sequential Collaboration Network of Wikipedians
Dynamics as Networks
the dynamics of system/phenomena are represented as networks
Editor A
Sequential Collaboration Network of Wikipedians
Dynamics as Networks
the dynamics of system/phenomena are represented as networks
The Sequential Collaboration Networkof an Article on Wikipedia (English)“Star Wars Episode IV: A New Hope”
Sequential Collaboration Network of Wikipedians
Dynamics as Networks
The Sequential Collaboration Networkof an Article on Wikipedia (English)“Australia”
the dynamics of system/phenomena are represented as networks
Sequential Collaboration Network of Wikipedians
Dynamics as Networks
The Sequential Collaboration Networkof an Article on Wikipedia (English)“Damien (South Park)”
the dynamics of system/phenomena are represented as networks
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Chord Networks of Music
Chord-Transition Networks of Music
Collaboration Networks of Wikipedia
Collaboration Networks of Linux
Co-Purchase Networks of Books, CDs, DVDs
State Networks of Chaotic Dynamical Systems
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Chord Networks of Music
Sequential Chord Network of Music
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
(Carpenters)Sequential Chord Network of Music
SOLITAIRE PLEASE MR.POSTMAN JAMBALAYA(ON THE BAYOU) HURTING EACH OTHER
THOSE GOOD OLD DREAMS
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 MAKE BELIEVE IT’SYOUR FIRST TIME
I NEED TO BE IN LOVEWE’VE ONLY JUST BEGUNTOP OF THE WORLDRAINY DAYS AND MONDAYGOODBY TO LOVESINGONLY YESTERDAYI WON’T LAST A DAY WITHOUT YOUSOLITAIREPLEASE MR.POSTMANJAMBALAYA (ON THE BAYOU)HURTING EACH OTHERTHERE’S A KIND OF HUSH (ALL OVER THE WORLD)FOR ALL WE KNOWBLESS THE BEASTS AND CHILDREN(THEY LONG TO BE) CLOSE TO YOUYESTERDAY ONCE MORETHOSE GOOD OLD DREAMSMAKE BELIEVE IT’S YOUR FIRST TIME
Carpenters
Sequential Chord Network of Music [Integrated]
major 19 songs
0.001
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1 10 100
p(k)
k 0.001
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1 10 100
0.001
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1 10 100
P(k)
k 0.001
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1 10 100 0.001
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P(w)
w
in-degree distribution
out-degree distributionweight distribution
Carpentersmajor 19 songs
Sequential Chord Network of Music [Integrated]
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Chord-Transition Networks 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
GOODBY TO LOVE SING ONLY YESTERDAY I WON’T LAST A DAYWITHOUT YOU
Sequential Chord-Transition Network of Music
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
THOSE GOOD OLD DREAMSYESTERDAY ONCE MORE MAKE BELIEVE IT’SYOUR FIRST TIME
Sequential Chord-Transition Network of Music
I NEED TO BE IN LOVEWE’VE ONLY JUST BEGUNTOP OF THE WORLDRAINY DAYS AND MONDAYGOODBY TO LOVESINGONLY YESTERDAYI WON’T LAST A DAY WITHOUT YOUSOLITAIREPLEASE MR.POSTMANJAMBALAYA (ON THE BAYOU)HURTING EACH OTHERTHERE’S A KIND OF HUSH (ALL OVER THE WORLD)FOR ALL WE KNOWBLESS THE BEASTS AND CHILDREN(THEY LONG TO BE) CLOSE TO YOUYESTERDAY ONCE MORETHOSE GOOD OLD DREAMSMAKE BELIEVE IT’S YOUR FIRST TIME
Carpentersmajor 19 songs
0.001
0.01
0.1
1
1 10
p(k)
k 0.001
0.01
0.1
1
1 10
0.001
0.01
0.1
1
1 10
p(k)
k 0.001
0.01
0.1
1
1 10 0.001
0.01
0.1
1
1 10 100
P(w)
w 0.001
0.01
0.1
1
1 10 100
in-degree distribution
out-degree distributionweight distribution
Carpentersmajor 19 songs
Sequential Chord-Transition Network of Music [Integrated]
Junya HiroseFormer Graduate Student
Collaborators
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
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Collaboration Networks of Wikipedia
•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
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
Editor A
Editor B
Editor A
Editor C
an article1
2
3
4
5
Editor A
Editor B
Editor C
order
Building a sequential collaboration network, connecting a relation from editor A to editor B, if editor B follows on work done by editor A.
n Method: Sequential collaboration network
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 = 594Average 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
Editor A
Editor B
Editor A
Editor C
an article1
2
3
4
5
Editor A
Editor B
Editor C
order
Building a sequential collaboration network, connecting a relation from editor A to editor B, if editor B follows on work done by editor A.
n Method: Sequential collaboration network
Our Previous Study: Featured Articles in English Wikipedia
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
kLinear graph
T. Iba, K. Nemoto, B. Peters & P. Gloor, "Analyzing the Creative Editing Behavior of Wikipedia Editors Through Dynamical Social Network Analysis", COINs2011, 2009T. Iba and S. Itoh, "Sequential Collaboration Network of Open Collaboration", NetSci'09, 2009
2,545 articles [Jun 27 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
Rank 1: EnglishRank 2: GermanRank 3: FrenchRank 4: PolishRank 5: ItalianRank 6: JapaneseRank 7: SpanishRank 8: DutchRank 9: PortugueseRank 10: Russian…Rank 15: Finnish…Rank 20: Turkish
Target Languages
Analyzing ALL articles as of January 1st, 2011 in each language.
The ranking based on the data as of January 6th, 2011.
nAnalysis 1: Comparison of 12 different languages
English Rank 13,490,325 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
English Rank 13,490,325 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
English Rank 13,490,325 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
German Rank 21,155,210 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
French Rank 31,039,251 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Polish Rank 4752,734 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Italian Rank 5750,634 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Japanese Rank 6718,974 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Spanish Rank 7676,866 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Dutch Rank 8656,079 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Portuguese Rank 9638,747 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Russian Rank 10627,139 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Finnish Rank 15255,712 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
Turkish Rank 20152,262 articles
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Double logarithmic graph
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
nAnalysis 2: Distribution of account and IP users
IP users
Accountusers
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Scatter plot of articles in English Wikipedia
Double logarithmic graph
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Scatter plot of articles with number of IP users / number of total editors
Double logarithmic graph
0.0 PIP 1.0
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
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Scatter plot of articles with number of IP users / number of total editors
Double logarithmic graph
0.0 PIP 1.0
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
nAnalysis 3: Distribution of Featured Articles
3,372 featured articles / 3,732,033 articlesIn English Wikipedia
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Scatter plot of all articles in English Wikipedia
Double logarithmic graph
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Scatter plot of featured articles on the all articles in English Wikipedia
Double logarithmic graph
The order of each sequential collaboration network (The number of editors in each article)
The
aver
age
path
leng
th o
f e
ach
sequ
entia
l col
labo
ratio
n ne
twor
k
Scatter plot of featured articles on the all articles in English Wikipedia
Double logarithmic graph
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
Ko MatsuzukaSatoshi ItohA Former Graduate Student, Iba Lab.
Peter GloorResearch Scientist, MIT CCI
Keiichi NemotoVisiting Scholar, MIT CCIFuji Xerox
Student, Iba Lab.
Daiki MuramatsuStudent, Iba Lab.
Natsumi YotsumotoFormer Student, Iba Lab.
Bui Hong HaFormer Student, Iba Lab.
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
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Collaboration Networks of Linux
Takashi Iba (2008)
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)
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Co-Purchase Networks of Books, CDs, DVDs
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 Networkfrom data
(1)
(2)
Sequential Connection
Co-Purchase Network of Books
* Target Customers = 30,000* All Links are Visualized.
Number of nodes = 68,701Number of edges = 113,940
Sequential Connection
Co-Purchase Network of CD
* Target Customers = 30,000* All Links are Visualized.
Sequential Connection
Number of nodes = 14,038Number of edges = 22,727
Co-Purchase Network of DVD
* Target Customers = 30,000* All Links are Visualized.
Sequential Connection
Number of nodes = 10,875Number of edges = 24,535
In-Degree Distributionfollow Power Law
Book
CD DVD
γ=2.56
γ=2.33 γ=2.09
Sequential Connection
Out-Degree Distribution follow Power Law
Book
CD DVD
γ=2.57
γ=2.43 γ=1.99
Sequential Connection
Weight Distribution follow Power Law
Books
CDs DVDs
Sequential Connection
Mapping Genre in Co-PurchaseNetwork of DVDs
Sequential ConnectionTarget Customers = 30,000Weight > 1
k highlow
Node Coloring
Sequential Connection
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
Satoshi ItohFormer Graduate Student, Iba Lab.
Ryosuke Nishida
Yuki Kitayama
Mariko Yoshida
Former Graduate Student, Iba Lab.
Former Graduate Student, Iba Lab.
Former Student, Iba Lab.
Masaya MoriRakuten Institute of Technology
Collaborators
Papers & Presentations
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
State Networks of Chaotic Dynamical Systems
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.
Highly complex behavior is generated,although it’s governed by a very simple rule.
Fundamentals: Chaos
a simple population growth model (non-overlapping generations)
May, R. M. Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645–647 (1974).May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).
xn+1 = 4µ xn ( 1 - xn )
0 < xn < 10 < µ < 1 (constant)
(variable)xn population (capacity)...
µ a rate of growth...
Fundamentals: Logistic Map
x1 = 4µ x0 ( 1 - x0 ) n = 0
n = 1 x2 = 4µ x1 ( 1 - x1 )
n = 2 x3 = 4µ x2 ( 1 - x2 )
x0 = an initial value
0
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x
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µ0
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x
n
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x
n
The behavior depends on the value of control parameter µ.xn+1 = 4µ xn ( 1 - xn )
Fundamentals: Logistic Map
0.25 0.5 0.89... 10.75
µ0
bifurcation diagram
x
010.25 0.5 0.89...
1
0.75
Fundamentals: Logistic Mapxn+1 = 4µ xn ( 1 - xn )
This diagram shows long-term values of x.
This diagram shows long-term values of x.
xn+1 = 4µ xn ( 1 - xn ) Fundamentals: Logistic Map
bifurcation diagram
µ0 0.25 0.5 0.89... 10.75
x
0
1
0
0.2
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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 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
Logistic Mapxn+1 = 4 xn ( 1 - xn )
Existing methods areNOT for drawing a map
bifurcation diagram
0.2 0.4 0.6 0.8 1.0
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cobweb plot
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time series
They merely visualize the trajectory of an instance of the system’s behavior,giving an initial value.
for taking an overview of the whole
Visualizing State Transitions of a Systemfor taking an overview of the whole.
system
state
state: the value of x xn+1 = 4µ xn ( 1 - xn )
x is a real numberwhich has no limit to smallness (detail)
xn+1 = 4µ xn ( 1 - xn ) targetworld
abstraction
map x is treated as discretewhich implies the finite number of states
xn+1 = 4µ xn ( 1 - xn )
Transit map State-transition mapNetwork of stations Network of states
xn+1 = 4µ xn ( 1 - xn ) x is treated as discretewhich implies the finite number of states
Discretizing into the finite number of statesTo subdivide a unit interval of variables into a finite number of subintervals
x
Discretizing into the finite number of states
h( ) is the rounding function:round-up, round-off, or round-down
To subdivide a unit interval of variables into a finite number of subintervals
h ( )
The discretization is carried out byrounding the value x to d decimal places.Therefore,
x
(ex.) round-upd = 1: 0.14 0.2d = 2: 0.146 0.15
To apply the map f into all states in order to obtain all state transitions.
Obtaining the set of state transitions
the set of state
The discretization is carried out byrounding the value x to d decimal places.Therefore,
(ex.) round-upd = 1: 0.14 0.2d = 2: 0.146 0.15
h ( )
To apply the map f into all states in order to obtain all state transitions.
Obtaining the set of state transitions
f
the set ofstate transitions
h ( )
where h( ) is the rounding function, and
Building state-transition networks
the set of statetransitions
building networks
To connect each state into its successive state
State-transition network is a “map” of the whole behavior of the system, so one can take an overview from bird’s-eye view.
f
State-transition networksThe order of the network N = 1/Δ + 1.
Each connected component must have only one loop or cycle: fixed point or periodic cycle.
The whole behavior is often mapped into more than one connected components, which represent basins of attraction.
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.
State-transition networks as a “dried-up river”
The network is dried-up river, where there are no water flows on the riverbed.
node: geographical point in the riverlink: a connection from a point to another
The direction of a flow in the river is fixed.
There are no branches of the flow.
There are many confluences of two or more tributaries.
Numerical simulation represents an instance of flow on the state-transition networks.
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 traces an instance
Let’s draw a mapof the logistic map!
for taking an overview of the whole
xn+1 = 4µ xn ( 1 - xn )
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Control Parameter: µ = 1 Therefore, xn+1 = h( 4 xn ( 1 - xn ) )Round-Up into the decimal place, d = 1 Therefore, 11 states
The state-transition networks for the logistic map
0.0 0.00 0.00.1 0.36 0.40.2 0.64 0.70.3 0.84 0.90.4 0.96 1.00.5 1.00 1.00.6 0.96 1.00.7 0.84 0.90.8 0.64 0.70.9 0.36 0.41.0 0.00 0.0
xn+1xnf h
Control Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, d = 2 Therefore, 101 states
The state-transition networks for the logistic map
xn+1xn
0.00 0.0000 0.000.01 0.0396 0.040.02 0.0784 0.080.03 0.1164 0.120.04 0.1536 0.160.05 0.1900 0.190.06 0.2256 0.230.07 0.2604 0.270.08 0.2944 0.300.09 0.3276 0.330.10 0.3600 0.360.11 0.3916 0.400.12 0.4224 0.430.13 0.4524 0.460.14 0.4816 0.490.15 0.5100 0.510.16 0.5376 0.54
f h
Control Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, d = 3 Therefore, 1001 states
The state-transition networks for the logistic map
xn+1xn
0.000 0.000000 0.0000.001 0.003996 0.0040.002 0.007984 0.0080.003 0.011964 0.0120.004 0.015936 0.0160.005 0.019900 0.0200.006 0.023856 0.0240.007 0.027804 0.0280.008 0.031744 0.0320.009 0.035676 0.0360.010 0.039600 0.0400.011 0.043516 0.0440.012 0.047424 0.0480.013 0.051324 0.0520.014 0.055216 0.0560.015 0.059100 0.0600.016 0.062976 0.063
f h
Control Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, d = 4 Therefore, 10001 states
The state-transition networks for the logistic map
xn+1xn
0.0000 0.00000000 0.00000.0001 0.00039996 0.00040.0002 0.00079984 0.00080.0003 0.00119964 0.00120.0004 0.00159936 0.00160.0005 0.00199900 0.00200.0006 0.00239856 0.00240.0007 0.00279804 0.00280.0008 0.00319744 0.00320.0009 0.00359676 0.00360.0010 0.00399600 0.00400.0011 0.00439516 0.00440.0012 0.00479424 0.00480.0013 0.00519324 0.00520.0014 0.00559216 0.00560.0015 0.00599100 0.00600.0016 0.00638976 0.0064
f h
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 cumulative in-degree distributions of state-transition networks for the logistic mapwith µ = 1 in the case from d = 1 to d = 7
Control Parameter: µ = 1 Therefore, xn+1 = 4 xn ( 1 - xn )Round-Up into the decimal place, ranging from d = 1 to d = 4
The state-transition networks for the logistic map
d = 1 d = 2
d = 3 d = 4
scale-freenetworks!
Does the scale-free property depend on the control parameter µ ?
Does the scale-free property depend on the control parameter µ ?
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xn+1 = 4µ xn ( 1 - xn )
Control Parameter: ranging from µ = 0 to µ = 1 xn+1 = 4µ xn ( 1 - xn )Round-Up into the decimal place, d = 3
The state-transition networks for the logistic map
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.
from µ = 0.125 to 1.000 by 0.125 in the case d = 7
The cumulative in-degree distributions of state-transition networks for the logistic map
Does the scale-free property depend on the control parameter µ ?
The scale-free property is independent of the control parameter µ.
NO.
How the scale-free network is emerged?
How?
How the scale-free network is emerged?
xn
xn+1
xn+1 = 4µ xn ( 1 - xn )
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How the scale-free network is emerged?
If x is a real number, namely in the mathematically ideal condition,
0.30
0.84
0.70
the state 0.84 must have only 2 previous values: 0.30 and 0.70.
In the condition,the state-transition network cannot be a scale-free network ...
How the scale-free network is emerged?
The trick is discretization.
How the scale-free network is emerged?
How the scale-free network is emerged?
The relation between a subinterval on y-axis and its corresponding range on the x-axis for the logistic map function.
1.000 0.999 0.998 0.997 0.996 0.995 0.994 0.993 0.992
311410886666
in-degreexn+1
ex. µ = 1, d = 3
Mathematical Derivation
The cumulative in-degree distribution follows a law given by:
The second term makes a large effect on the distribution, only when k is quite close to k,because .
max
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.
Summarizing
The cumulative in-degree distribution follows a law given by:
The second term makes a large effect on the distribution, only when k is quite close to k,because .
max
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.
Summarizing
taking the limit
Summarizing
In the case Δ is extremely small ( d is extremely large), The cumulative in-degree distribution follows a power law:
The cumulative in-degree distribution follows a law given by:
Summarizing
taking the limit
Summarizing
In the case Δ is extremely small ( d is extremely large), The cumulative in-degree distribution follows a power law:
with µ = 0.1 and µ = 1.0, in the case d = 6
Comparison between the results of numerical computations and mathematical predictions
Are there any other maps whose state-transition networks are scale-free networks?
Yes!
One-dimensional maps
Sine map
Cubic map ♯2
Cubic map ♯1
Logistic mapµ = 1.0, d = 3
Sine mapµ = 1.0, d = 3
Cubic map ♯2µ = 1.0, d = 3
Cubic map ♯1µ = 1.0, d = 3
The state-transition networks for other one-dimensional maps
The dashed line has slope -2The networks are scale-free networks with the degree exponent γ = 1.
The cumulative in-degree distributions of state-transition networks for other one-dimensional maps
d = 6
One-dimensional maps
General Symmetric map
The parameter α is controlling the flatness around the top.
α = 2.0: the logistic map
α = 1.5 α = 2.0 α = 2.5
α = 3.0 α = 3.5 α = 4.0
d = 3
The state-transition networks for the general symmetric map
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
Delayed logistic map
Sine-circle map
exponential
discontinuous
two-dimensional
The state-transition networks for the other types of maps
Sine-circle mapa = 4.0, b = 0.5. d = 3
Gaussian mapa = 1.0, b = −0.3, d = 3
Delayed logistic mapa = 2.27, d = 2
The dashed line has slope -2the networks are scale-free networks with the degree exponent γ = 1.
d = 6d = 3
The cumulative in-degree distributions of state-transition networks for the other types of maps
Chaotic mapsthat have scale-free state-transition networks
Gaussian map
Delayed logistic map
Sine-circle map
one-dimensional, exponential
one-dimensional, discontinuous
two-dimensional
Sine map
Cubic map
Logistic map
one-dimensional
one-dimensional
one-dimensional
Do all chaotic maps have scale-free state-transition networks?
No.
NOT scale-free state-transition networks of chaotic maps
Tent mapa = 2.0. d = 3
Binary Shift mapd = 3
Gingerbreadman mapd = 1
Cusp mapa = 2.0. d = 4
Henon mapa = 1.4, b = 0.3, d = 2
Henon area-preserving mapa = 0.24. d = 2
Standard (Chirikov) mapa = 1.0. d = 1
Pincher mapa =. d = 4
Lozi mapa = 1.7, b = 0.5, d = 2
NOT scale-free state-transition networks of chaotic maps
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!
Papers & Presentations
T. Iba with K. Shimonishi, J. Hirose, A. Masumori, "The Chaos Book: New Explorations for Order
Hidden in Chaos", 2011
Dynamics as Networksthe dynamics of system/phenomena are represented as networks
Chord Networks of Music
Chord-Transition Networks of Music
Collaboration Networks of Wikipedia
Collaboration Networks of Linux
Co-Purchase Networks of Books, CDs, DVDs
State Networks of Chaotic Dynamical Systems
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
Network Analysis for Understanding Dynamics- Wikipedia, Music & Chaos -
TAKASHI IBA
Ph.D. in Media and GovernanceAssociate Professor
at Faculty of Policy Management, Keio University, Japantwitter in Japanese: takashiiba
twitter in English: taka_iba
時間発展のネットワーク分析
井庭 崇