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約6年前 (2012/05/18)にアップロードin学び

My presentation slides for the Invited Talk at IEICE Technical Committee on Information Network S...

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

- 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 Dy

Dyna

namics

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
- Dyna

Dy m

na ics

c - When Network Scientists talk about dynamics..

Dynam

Dynamics of N

cs

etworks

the evolution of network structure in time

Dyna

Dy m

na i

m cs

c o

s n 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

Edit

Edi or A

r

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

0.1

major 19 songs

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

Editor

4

A

Editor B

Editor C

5

Editor C - 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

Editor

4

A

Editor B

Editor C

5

Editor C - Our Previous Study: Featured Articles in English Wikipedia

Linear graph

2,545 articles [Jun 27 2009]

The average path length of

each sequential collaboration network

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

Rank 10: Russian

Analyzing ALL articles as of January

…

1st, 2011 in each language.

Rank 15: Finnish

…

The ranking based on the data as of

Rank 20: Turkish

January 6th, 2011. - English

Rank 1

3,490,325 articles

The average path length of

each sequential collaboration network

The order of each sequential collaboration network

(The number of editors in each article) - English

Rank 1

3,490,325 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - English

Rank 1

3,490,325 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - German

Rank 2

1,155,210 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - French

Rank 3

1,039,251 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Polish

Rank 4

752,734 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Italian

Rank 5

750,634 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Japanese

Rank 6

718,974 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Spanish

Rank 7

676,866 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Dutch

Rank 8

656,079 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Portuguese

Rank 9

638,747 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Russian

Rank 10

627,139 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Finnish

Rank 15

255,712 articles

The average path length of

each sequential collaboration network

Double logarithmic graph

The order of each sequential collaboration network

(The number of editors in each article) - Turkish

Rank 20

152,262 articles

The average path length of

each sequential collaboration network

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.

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

The average path length of

each sequential collaboration network

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

The average path length of

P

0.0

IP

1.0

each sequential collaboration network

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

The average path length of

P

0.0

IP

1.0

each sequential collaboration network

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

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

The average path length of

each sequential collaboration network

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

The average path length of

each sequential collaboration network

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

The average path length of

each sequential collaboration network

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.

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

Ko Matsuzuka

A Former Graduate Student,

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)

T - 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

Mariko Yoshida

Collaborators

Former Student, Iba Lab.

Satoshi Itoh

Ryosuke Nishida

Former Graduate Student,

Iba Lab.

Former Graduate Student,

Iba Lab.

Yuki Kitayama

Masaya Mori

Former Graduate Student,

Iba Lab.

Rakuten Institute of Technology - 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

n

0

20

40

60

80

100

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. - Fundamentals: Logistic Map

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

x

x

x

0.4

0.4

0.4

x0.2

0.2

0.2

0

n+1 = 4µ xn ( 1 -

0

xn )

0

0

20

40

60

80

100

n

0

20

40

60

80

100

0

20

40

60

80

100

n

n

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

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100

0

20

40

60

80

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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.

Logistic Map

xn+1 = 4 xn ( 1 - xn )

?

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 - Existing methods are

time series

NOT for drawing a map

1

0.8

0.6

x

for taking an overview of the whole

0.4

0.2

They merely visualize the trajectory of

0

0

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40

60

80

100

n

an instance of the system’s behavior,

cobweb plot

giving an initial value.

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 - x

map

n+1 = 4µ xn ( 1 - xn )

x is treated as discrete

which implies the finite number of states

abstraction

xn+1 = 4µ xn ( 1 - xn )

target

x is a real number

world

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

0.5

0.8 0.64 0.7

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

xn+1

f

h

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 )

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n

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n

µ

0

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0.89...

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x

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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?

How? - How the scale-free network is emerged?

1.0

xn+1 = 4µ xn ( 1 - xn )

0.8

x

0.6

n+1

0.4

0.2

0.2

0.4 x 0.6

0.8

1.0

n - 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

xn+1 = 4µ xn ( 1 - xn )

0.8

x

0.6

n+1

0.4

0.2

0.2

0.4

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1.0

0.30

x

0.70

n

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 = 0.24. d = 2

a = 1.0. d = 1

a = 1.7, b = 0.5, d = 2 - 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