Network Analysis for Understanding Dynamics: Wikipedia, Music & Chaos

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


TAKASHI IBA





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


the 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 Networks

Dynamics as 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


the dynamics of system/phenomena are represented as networks

Editor A




The Sequential Collaboration Networkof an Article on Wikipedia (English)“Star Wars Episode IV: A New Hope”



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




The Sequential Collaboration Networkof an Article on Wikipedia (English)“Damien (South Park)”




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



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

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out-degree distributionweight distribution

Carpentersmajor 19 songs

Sequential Chord Network of Music [Integrated]



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


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


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


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out-degree distributionweight distribution


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



•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






Editor A

Editor B

Editor A

Editor C

an article1

2

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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.


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


Sequential Collaboration Network of Article “Fondue”in English Wikipedia


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.


Our Previous Study: Featured Articles in English Wikipedia

The order of each sequential collaboration network (The number of editors in each article)

The

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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]






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


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Double logarithmic graph



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German Rank 21,155,210 articles


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French Rank 31,039,251 articles


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Polish Rank 4752,734 articles


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Italian Rank 5750,634 articles


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Japanese Rank 6718,974 articles


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Spanish Rank 7676,866 articles


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Dutch Rank 8656,079 articles


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Portuguese Rank 9638,747 articles


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Russian Rank 10627,139 articles


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Finnish Rank 15255,712 articles


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Turkish Rank 20152,262 articles


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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.






nAnalysis 2: Distribution of account and IP users

IP users

Accountusers


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Scatter plot of articles in English Wikipedia



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Scatter plot of articles with number of IP users / number of total editors


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



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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






nAnalysis 3: Distribution of Featured Articles

3,372 featured articles / 3,732,033 articlesIn English Wikipedia


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Scatter plot of all articles in English Wikipedia



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Scatter plot of featured articles on the all articles in English Wikipedia



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Scatter plot of featured articles on the all articles in English Wikipedia


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.







•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.


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



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)



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


Co-Purchase Network of CD




Co-Purchase Network of DVD




In-Degree Distributionfollow Power Law

Book

CD DVD

γ=2.56

γ=2.33 γ=2.09


Out-Degree Distribution follow Power Law

Book

CD DVD

γ=2.57

γ=2.43 γ=1.99


Weight Distribution follow Power Law

Books

CDs DVDs


Mapping Genre in Co-PurchaseNetwork of DVDs

Sequential ConnectionTarget Customers = 30,000Weight > 1

k highlow

Node Coloring


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




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

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The behavior depends on the value of control parameter µ.xn+1 = 4µ xn ( 1 - xn )

Fundamentals: Logistic Map

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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

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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

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cobweb plot

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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 )

0.0

1.0

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0.1

0.9 0.3

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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


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



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



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


d = 1 d = 2

d = 3 d = 4

scale-freenetworks!

Does the scale-free property depend on the control parameter µ ?


0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100n

x

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100n

x

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

x

n

µ0

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

x

n

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

x

n

0.25 0.5 0.89... 10.75

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 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


The scale-free property is independent of the control parameter µ.

NO.

How the scale-free network is emerged?

How?


xn

xn+1

xn+1 = 4µ xn ( 1 - xn )

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

xn

xn+1

xn+1 = 4µ xn ( 1 - xn )

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0


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 ...


The trick is discretization.



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 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:


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!


T. Iba with K. Shimonishi, J. Hirose, A. Masumori, "The Chaos Book: New Explorations for Order

Hidden in Chaos", 2011








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


TAKASHI IBA





井庭崇

Education

Network Analysis for Understanding Dynamics: Wikipedia, Music & Chaos