複雜系統與相關研究
蔡憶佳
淡江大學資訊工程系
個人化智慧通訊實驗室
2005 June 24
這篇簡介是集合目前各種有關研究複雜系統的複雜性科學與談論複雜系統的文章並加入作者個人的意見與解
釋。內容的取材當然就反應出個人的喜好與偏見，但同時也儘量加入原始出處，以供讀者自行參考。
所謂複雜系統，目前並沒有標準定義。我們可以找到許多文章嘗試要明確的定義它。一般而言，複雜系統是指
整體系統特性無法完全由其個別組成份子的特性所解釋。也就是說，在集合眾多個體組成一個龐大的系統後，
所呈現的一些特性是組成個體所沒有的。P. W. Anderson [5] 在 1972 年的文章標題《量多就不一樣(more is
different)》數量多就會有不一樣的現象就是形容這種情形。由於很難有一個可以涵蓋全貌的解釋，我們藉由一
些論文中學者對於複雜系統的描述來管窺一二。
早於 1962 年，諾貝爾經濟獎得主 Herbert Simon [1] 形容複雜系統是一個系統可解析為具有多重關係的許多元
件，因為每一元件的行為是被其他所有元件的行為所影響。
首先，在 Advances in Complex Systems 期刊的目標與展望中1，提到“複雜系統是由一些(通常是十分眾多)互動（通
常是強烈的）的個體、單元、或構成體所組成。需要發展、使用新的科學工具、非線性模型、非均衡描述與電
腦模擬來幫助我們進一步瞭解它。”另外在 1999 年《科學 Science》的一期特刊中，由化學、生化訊號系統、神
經系統、生物群聚演化、自然成形圖樣、氣候與經濟等領域探討複雜系統。在這一系列文章中，各方學者談到
複雜系統，我們整理如下：
複雜系統是一初始值或微細擾動對於後續演變有敏銳影響，是具有眾多獨立的互動元件，或是具有許多演
變路徑的系統。(化學，Whitesides and Ismagilov)
複雜系統是難以理解與驗證的，無論是在設計時或功能運作時或同時便是如此。(生化訊號系統，Weng,
Bhalla and Iyengar)
複雜系統是許多元件間具有多重互動的系統。(氣候，D. Rind)
複雜系統是隨著時間漸漸演化與開展的系統。(經濟，W. Brian Arthur).
在同一期的《科學 Science》特刊中，Goldenfeld and Kadanoff [[6] 的文章 Simple Lessons from Complexity，除了
談到在複雜系統的研究領域，與傳統物理研究的不同點，同時也將複雜系統描述成：
複雜系統是一高度結構化系統，展現出具變化性結構的組織。(Goldenfeld and Kadanoff)
1
www.tbi.univie.ac.at/~studla/ACS/AimsScopes.html
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接著，在 2002 年 Proceedings of the National Academy of Sciences 的 Self-Organized Complexity in the Physical,
Biological, and Social Sciences2 特刊中，探討在物理、生物、與社會科學領域裏複雜系統中的自行組織複雜度。
首先由頻率與大小的分佈談起，例如地震在某段時間內出現的次數與其斷裂區域大小的關係符合冪次律。另外
河川流域網路其一固定大小的河流出現次數與其長度也是呈冪次律關係。在時間序列的領域中，全球的平均溫
度、各種股價指數、匯率、心跳間隔時間、混流中某一點的速率等，都是在看似紊亂的表象下呈現某種規律性。
由上面的幾項描述，我們知道複雜系統是遍佈於各個領域，由實體界到人為的網路關係，而探討這些系統的角
度也有許多不同點，有的著重現象與統計的規律性，有的則是嘗試要建立模型。因此所使用的工具也有相當程
度的不同，有的是使用數學與統計，有的則是使用電腦模擬。
複雜系統在科學與科技的領域中扮演必要的角色。科學是探討複雜系統而科技是如何使用複雜系統。科學界想
要探討與理解複雜系統(發現簡單的規則與方程式導致複雜現象)，科技與工程界則是想要控制複雜系統(建立一
簡單的介面來使用複雜的設備或使用簡易的介面隱匿複雜的工具與儀器)。由於其難理解，在科學與工程界的一
項重要議題即是如何預測與控制這些系統的行為。複雜系統難以理解是一些現象與機制所導致。這些現象與機
制也時常在渾沌論、人工生命、演化計算與基因演算等領域使用的一些意念與技術中所探討。另一些嘗試則是
使用整體系統思維藉由整體的考量來研究複雜系統中的現象。
以下所談複雜系統的特徵是基於 Wikipedia3 中所述。
複雜系統的特徵
突現(emergence)
所謂突現是指在複雜系統中導致新的具一致性的結構、圖樣與特質的過程。突現是由於系統中元件互動的模式
歷時而來。突現通常是簡單的元件間看似簡單的互動後所產生的無法預期的，不可忽略的結果。複雜系統與似
複雜而紛亂的系統，其最大的不同點在於構成元件間的互動模式會導致某些行為與樣式突現於系統中。突現行
為通常表現於一群在同一環境中簡單構成體形成整體性的複雜結構。此一複雜行為並不是單一個體的特性，同
時也無法由低階個體的行為所能解釋或預測。一群魚或一群鳥的外廓與整體行動是項容易理解的例子，而且整
群的控制機制往往比單一個體的行為更難掌握。在許多地方都可以觀察到突現的歷程或行為，例如多細胞生物、
車流模式、組織行為或電腦模擬。股票市場是一個龐大的突現範例。整體而言股市忠實的呈現出世界上各別公
司的相對價值，然而每位投資者都只對於其投資組合中的一些公司有相當程度的瞭解，而且必須遵守該市場的
法令規則。由這些投資者在市場中的互動而突現了整體股市的複雜行為。研究突現行為是依照所研究的領域而
有所不同，並不是一視同仁的。
關係是非線性
組成元件間的互動極少是簡單的因果關係的，往往是一個微小的刺激因導致龐大的或者完全無效的結果。所謂
的蝶翼效應即是指此。
關係具有回饋機制
兼具有負向(阻泥)的或正向(放大)的回饋機制是複雜系統的關鍵。代理者的行為結果回饋到本身而這項回饋影響
2
3
www.pnas.org/content/vol99/suppl_1/
en.wikipedia.org/wiki/Complex_system
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此代理者未來的行為。這些時時改變的非線性關係是使得複雜系統如此特殊的主因。
複雜系統是開放式的
複雜系統是開放式的系統，能量與資訊隨時流經其邊界。因此複雜系統通常不屬於平衡系統，雖然呈現出常態
變化而且外觀看起來為穩定的狀態。
部份無法涵蓋全部
在複雜系統中的元件無法得知整體系統的發展。否則，在此元件上我們就可以觀察到整體複雜系統的行為了。
而複雜現象也不單單是由元件間互動關係所造成，由此可知一項結論是無法由單一元件控制整體系統。
複雜系統是有歷史記憶的
複雜系統的歷程是無法忽略的。一個周遭條件極小的改變可能引起未來十分大的變異。這也是所謂的“蝶翼效應”。
複雜系統是層層相疊的
複雜適性系統的另一項特質是系統中的元件(或稱構成體)往往也是複雜適性系統。例如經濟體是由組織機構所構
成，而這些組織機構是由人群所組成。人則是由中樞神經與內分泌系統所控制的各種器官與生命系統構成，而
這些全部由各種細胞所組成。在每一階層都可自成為複雜適應系統。
沒有明確的邊界
複雜系統的邊界是很難劃分清楚的。若要劃分邊界，通常是基於觀察者的需求與偏見而非基於任何內部的特性。
例如一個人的邊界似乎是可以很清楚的劃出來，不過再仔細深入想想，模擬兩可的情形便出現了。比如衣服是
屬於人的界線內或界線外？如果有人從房間的另一角落或由火車上看著你，尤其是使用一種貪婪或惡毒的眼神
注視你，那麼你的邊界有沒被侵入呢？你的邊界是實質上的還是情緒上的？情緒上的影響會導致實質上的改
變。而食物在什麼時刻可以算成是你身體的一部份？
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複雜網路
複雜網路是複雜系統的骨幹，網路節點表示複雜系統的組成份子，個體或構成體或另一複雜系統的部份。連線
則代表這些組成體間的互動。經由分析實體複雜系統中的網路拓樸，複雜網路呈現出一些統計特性是與隨機網
路有顯著差異。其中，網路節點的連接度與出現次數分佈呈現「冪次律(power law)」關係，複雜網路往往具有
短距連通性，也就是說網路中任意兩點的平均最短距離是與整體網路節點個數取對數呈正比。另外，複雜網路
的群聚係數也較隨機網路高許多。這三種統計特性是目前廣為人知的複雜網路特性，不過未來是否還有其他特
性，則有待探討。
處處都可以找到複雜系統的例子，許多更是與生命攸關的：從一個細胞到完整的生物個體，天氣與氣候系統，
生態與經濟，大腦與神經系統。也由於複雜系統如此的普遍，因此在談到複雜系統時，會時常看到許多相關的
研究領域。
複雜論(complexity theory) 是研究在確定非線性系統(deterministic non-linear system)中非穩定無週期性(unstable,
aperiodic)行為的理論。由於複雜論在科學研究的範疇中還是相當新的學說，因此有許多研究還是進行中與無法
明確定義清楚。此理論同時也包括許多相關領域，例如碎形(fractal)、混沌論(chaos theory)、細胞自動機(cellular
automata)、滲透理論(percolation theory)、人工生命(artificial life)、複雜適性系統(complex adaptive system)和非線
性動力(non-linear dynamics)。
碎形(fractal)是一個具有無盡的細節及自我相似的幾何物件。若是使用維度來說明，碎形具有的幾何維度並不是
整數，而是介於整數間的分數。依碎形 (fractal)一詞的創作者 Benoit Mandelbrot 的定義，碎形是一個其
Hausdorff-Besicovitch 維度絕對大於拓樸維度的集合。大陸使用分形來描述 fractal 便是基於其維度是一個分數。
混沌論(chaos theory)是一門研究混沌現象的科學，其肇始於物理學、氣象學，嗣後延伸觸角及於哲學、社會學
與管理學。所謂混沌，是指在描述系統的方程式中並沒有隨機項目，也就是說一切變數是可精確計算得知。然
而在某些參數值下，整體系統呈現出對於微擾動十分敏感的特性，也造成了系統的未來發展是無法精確計算出
來。因此將此種無法精確預測的現象稱為混沌。混沌論與非線性系統是息息相關的。
細胞自動機或晶格自動機(cellular automata)是研究在大自然中如何形成各式各樣圖案的。在一類似晶體格子點中
執行一簡單的自動機，其下一狀態是由周遭其他自動機與自己的上一狀態所決定的。這種簡單的相互影響及自
動機有限的變化卻可以造成許多類似自然界所產生的圖樣。例如某些貝殼上面依生長而慢慢展開的花紋。早期
Conway 所設計的生命遊戲(Game of Life)也是屬於一種晶格自動機。另外也有一些研究顯示晶格自動機可用來呈
現偏微分方程式的解。
滲濾理論(percolation theory)是研究無序系統臨界現象的統計理論，最初用來描述非均相介質或多孔介質中的傳
導特性。當媒介的密度達到一臨界值時，滲透物突然能夠從媒介一端達到另一端。
人工生命(artificial life) 是研究如何模擬真正的生命體或是其某些特質，例如繁衍、有性繁殖、群聚、與共演化
等。由 Chris Langton (1989, 1992)所提出，其中幾項較為眾人所知的有模擬鳥類魚類等的群聚活動，螞蟻的社群
活動。另外 Tom Ray 的 Tierra 程式中包含不同種族的個體，有略食者、寄生蟲與被獵的族群，在電腦中呈現出
複雜的演化生態體系。
「複雜適應系統(complex adaptive system)」是有別於其他研究複雜系統的方式，在於其大量使用電腦模擬同時
強調整體系統的觀點。其所研究的對象也較集中於生態系統或市場等組織，因為這些較傳統的細胞、有機生命、
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公司或機器較為鬆散。一個複雜適應系統包括許多自動化或半自動的構成體，這些構成體藉由演化努力達成極
大化某些標準度量“好”或“適應性”。此名稱是由 John H. Holland 與 Murray Gell-Mann 所創作，用來描述在聖太
菲學院中的一項跨領域研究。John H. Holland 是演化計算與基因演算法的首創者之一。複雜適應系統是組成體
（可能是細胞、物種、個體、工廠或國家）在動態網路中同時互動，時時主動作為或回應動作。複雜適應系統
的控制是分散式的並沒有中央控制單元。群體一致性的行為主要由組成體間的競爭與合作而來。整體系統的行
為是由許多組成體無時無刻所作的無數的決定所產生。與「多組成體系統(multi-agent system)」的最大差異在於
強調高階特性例如自我相似性，複雜性，突現與自我組織等。多組成體系統是由多數的、互動的組成體構成。
而在複雜適應系統中，不僅是組成體會適應並且改變，連系統也會因此而適應並改變。複雜適應系統是複雜的、
自我相似的，互動與適應的組成體集合。一些重要的特性例如適應（或者用穩態）
、通訊、合作、功能特化、時
空組織與繁殖。這些可以在各種層面找到。例如細胞的功能特化、適應與繁殖同樣的也發生在大型的組織中。
通訊、合作也發生於各種層面，由組成體到系統。
非線性動力(non-linear dynamics)是指描述系統動力的方程式本身為非線性的。
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就字源學的觀點來看 complexity，拉丁字 complexus 是由希臘字 pleko 或 plektos 演化而來。其字義為編結或
扭繞。夸克論作者 Murray Gell-Mann 在談到複雜論時，曾考慮使用一個新字 Plectics4 來描述複雜論，其原因即
在於這個領域包含太多而不太容易使人印象深刻，並且可以兼顧這門研究所強調的簡單性與複雜性並存。尤其
是要顯示出這些複雜系統是由簡單的規則互相作用引發的複雜現象。在印歐語系中的字根 plek- 演變為拉丁文
動詞 plicare，表示折疊，另外衍生一字 simplex 表示單次折疊。由這個字演變成現今英文中的 simple。然而 plek同樣的也演變為拉丁文過去分詞 plexus，表示編結或互相糾結。後來演變成 complexus，意思是編結在一起，後
來英文中的 complex 則是由此演變而來的。希臘文與 plexus 同義的是 plektos，產生了數學上的專有名詞
symplectic。此字也有編結一起的意思，不過它的字源是由希臘文而不是由拉丁文演變到英文。
"I think the next century will be the century of complexity."
Stephen Hawking (Complexity Digest 2001.10 March-0502001).
各種複雜系統範例
以下的各項論文標題都是談到將某系統視為複雜系統。首先是將生命現象視為是複雜系統[2]，研究所有生命系
統的通則。在細胞中維持遞迴式生產過程所需的化學物質會滿足某些統計規律。由基因表現所呈現的冪次律
(power law)與化學物質出現的對數常態(log normal)機率分佈。再由物理中的變異消散理論(fluctuation-dissipation
theory)研究顯型基因變動與基因變動間的關係。
甚至有些學者開始思考使用複雜系統模型分析癌症。因為腫瘤細胞為了生存空間與資源相互競爭。由於 somatic
突變會產生許多基因變異種類而有些突變使腫瘤細胞具有繁殖上或生存上優勢。因此腫瘤細胞為演化與天擇的
影響。使用計算模型來模擬腫瘤細胞生長與演化及其接受各種療法後的演變。
Our understanding of what constitutes a "cancer" is becoming more and more refined. Intricate and complex pathways
are involved in the development of an organism. The organism originates as a single cell and progresses through rapid
proliferation and differentiation, followed by steady state repair and repletion, and finally senescence. The genetic
information for every part of the mature organism exists in the genome of each cell, yet the individual differentiated cell
never expresses most of this genetic information. By studying aberrancies in the system, such as the development of
neoplasia, mechanisms that control the process of gene activation and repression are being elucidated at a rapid rate. In
an article entitled “The Hallmarks of Cancer” (Cell 100:57-70, 2000) Hanahan and Weinberg presented a unifying
conceptual model of cancer biology. Based on the past quarter century of intensive research, common themes in cancer
have emerged. The “Hanahan-Weinberg Model” identifies 6 global traits common to all cancers. These 6 factors are:
self-sufficiency in growth signaling, insensitivity to antigrowth signals, evasion of programmed cell death, development
of limitless replicative potential, the capacity for sustained angiogenesis, and upregulation of genes associated with
tissue invasion and metastasis. The acquisition of each of these physical characteristics by the cancer cell represents the
successful breaching of an evolutionarily anticancer strategy developed by the multicellular organism to maintain
physiologic homeostasis. The cancer cell within the living host eukaryote can be considered analogous to a single-celled
organism that is in direct competition for scarce resources. Thus, from the perspective of the deranged cell, acquisition
of these cancer-associated traits is appropriately adaptive to ensure survival in a stressful environment, but from the
perspective of the host the development of a population of cells with these traits is ultimately maladaptive and lethal.
While each cancer is unique, with myriad individual mutations and epigenetic changes manifest, it may be possible in
the future to target specific anticancer strategies to these common cancer traits. Advances in cancer research will be
hastened by the advent of important technological breakthroughs, such as the widespread application of the high
throughput capabilities of genomics, proteomics, and pharmacogenomics. The future of cancer medicine may be
radically different within the next decade because of these advances in our understanding basic cancer biology.
Modeling Cancer as a Complex System
Mendoza, L. & Alvarez-Buylla, E. R. (1998). Dynamics of the genetic regulatory network for Arabidopsis thaliana
4
www.santafe.edu/sfi/People/mgm/plectics.html
6/70
flower
morphogenesis.
Journal
of
Theoretical
Biology
193:
(http://www.santafe.edu/education/international/intlfel03/files/Mendoza-EAB1998.pdf)
(http://192.12.12.14/education/international/intlfel03/files/Mendoza-EAB1998.pdf)
307
-
319.
Pattern of self-organization in tumour systems: complex growth ...( http://cherrypit.princeton.edu/papers/paper-191.pdf)
Modelling,
Simulation
and
Visualisation
of
Stem
(http://www.interdisciplinary.co.uk/content_assets/dinvernoprophetmodelling.pdf)
Cell
Behaviour
免疫系統視為一複雜系統
allows to model and simulate multicellular biological systems on the scale of membrane receptor mediated cellular
interactions. It is capable of including the molecular components of these systems (as, e.g., membrane receptors or
freely diffusing signal molecules) with realistic numbers of molecules, reaction rates, and diffusion coefficients.
Ahmed E. and Hashish A.H. (2004), ”On Modeling the immune system as a complex system”, Theor. BioSci.
(Accepted).
免疫系統
Meaning-Making
in
the
Immune
(http://muse.jhu.edu/journals/perspectives_in_biology_and_medicine/v047/47.3neuman.pdf)
System
A STOCHASTIC MODEL OF THE IMMUNE SYSTEM IN TWO-DIMENSIONAL SHAPE SPACE
(http://www.math.bme.hu/~szabados/bmb7.doc)
The
Immune
System
as
a
Complex
System:
Description
and
Simulation
...
(http://www.physnet.uni-hamburg.de/services/fachinfo/___Volltexte/Martin___Meier-Schellersheim/Martin___Meier-S
chellersheim.ps)
MEIER-SCHELLERSHEIM,
FB Physik, Univ. Hamburg
Martin
The Immune System as a Complex System: Description and
Simulation of the Interactions of its Constituents
Dissertation, 107 p.
(http://www-library.desy.de/preparch/desy/thesis/desy-thesis-01-006.ps.gz)
An
Emergent
Model
of
(http://www.chrisandtrudi.com/Chris/Portfolio/Thesis.pdf)
Immune
Cognition
An Emergent Model of Immune Cognition (http://www.chrisandtrudi.com/Chris/Portfolio/Thesis.pdf)
Perelson AS and TB Kepler (1995) The immune system as a complex system Adaptation by somatic mutation. In Chaos
& Complexity 93, J. Tran Tranh Van, P. Berge, R. Conte and M. Dubois, eds., Editiones Frontieres, Gif-sur-Yvette,
France, pp 97-106.
The general architecture of the immune system as a complex system involving billions of interacting cells and
molecules will be illustrated, paying particular attention to the evolutionary moulded interaction between innate and
clonotypical immunity. It will be illustrated how a specific immune response to a particular epitope emerges from a
structural background characterized by a high degeneracy and promiscuity of receptors and ligands. The hypothesis will
be pursued that the changes that the immune system undergoes on a long temporal scale (immunosenescence) allow us
to understand additional characteristics of the system, and particularly the role that chronic antigenic load and
“immunological noise” can play in maintaining immunological memory. Two mathematical models regarding this last
topic will be illustrated.
噬菌體 T7
THE
EXTENSION,
APPLICATION,
AND
GENERALIZATION
T7 ...( http://www.its.caltech.edu/~you/publications/Thesis_LingchongYou.pdf)
OF
A
將心臟視為一複雜適應系統
The Heart as a Complex Adaptive System (http://www.saplanners.org.za/SAPC/papers/Serfontien-56.pdf)
7/70
PHAGE
The Basal Ganglia as a Complex system
Telomeres
and
Telomerase
Activity
Are
Regulated
as
a
Complex ...( http://content.karger.com/ProdukteDB/produkte.asp?Aktion=ShowPDF&ProduktNr=227088&Ausgabe=2
29884&ArtikelNr=76101&filename=76101.pdf)
Understanding
Neural
Complexity:
A
(http://www.institutnicod.org/Reduction/Bickle_M_M_2001.pdf)
Role
Organization
of
the
neuronal
circuits
in
(http://taylorandfrancis.metapress.com/index/57PH46UR1U5G3CAW.pdf)
the
For
central
Reduction
nervous
...
由數十億個神經細胞所組成的腦，
心智
http://www.goertzel.org/books/spirit/uni5.htm
http://www.schuelers.com/chaos/chaos8.htm
http://www.schuelers.com/ChaosPsyche/table_of_contents.htm
生態系統與生物圈
Ecosystems and the Biosphere as Complex Adaptive Systems (http://www.cs.columbia.edu/~traub/sloan/LevinEcos.pdf)
(http://www.key-inc.com/ecosystems.pdf)
Incorporating
Complexity
in
Ecosystem
(http://journal-ci.csse.monash.edu.au/ci/vol07/lparro01/lparro01.pdf)
A Complex Ecological Framework of Aboriginal Family Resilience
(http://www.cst.ed.ac.uk/2005conference/papers/LaBoucane-Benson_paper.pdf)
The
biosphere
as
a
complex
(http://www.pubs.royalsoc.ac.uk/phil_trans_bio_content/news/biosphere.html)
演化
Evolution
As
A
Complex
System
(http://csmres.jmu.edu/geollab/Fichter/GeoBio405/VPTest-03.pdf)
A
Hands-on
Modeling
Approach
to
(http://www.umich.edu/~icls/proceedings/pdf/Centola.pdf)
Survey
Evolution:
Modelling
adaptive
of
system
Vertebrate
Learning
about
History
the
地球
The Magnetosphere as a Complex System (http://fisica.ciencias.uchile.cl/~jrogan/investigacion/papers/paperJAV05.pdf)
The Earth as a complex system (http://www.igbp.kva.se/congress/abstracts/Ghil_congress_abstract.pdf)
Earth Charter Curriculum Stimulus Material: Science (http://www.earthcharterusa.org/pdfs/curriculum_science.pdf)
組織動力
http://www.new-paradigm.co.uk/complex-od.htm
A Study of Querical Data Networks as “Complex Systems” http://www-scf.usc.edu/~banaeika/papers/NSF.pdf
8/70
SOCIETY AS A COMPLEX ADAPTIVE SYSTEM (http://www.n4bz.org/gst/gst11.htm)
都市發展
Metropolitan
Development
as
a
Complex
(http://edq.sagepub.com/cgi/content/abstract/13/2/141)
System:
A
New
Approach
to
Sustainability
Sustainable Development and Complex Systems (https://fy.chalmers.se/frt/PRT1995-2001.pdf)
TECHNOLOGY AND THE FUTURE OF CITIES (http://users.libero.it/ulbusi/pdfeng/94ocittafi.PDF)
大學
The University Viewed as a Complex System II (http://www.informs.org/Conf/ATL96/TALKS/WC27.html)
REFORM AND REMODELING OF THE UNIVERSITY AS A COMPLEX LIVING
(http://med.muni.cz/biomedjournal/pdf/2004/05/247_254.pdf)
SYSTEM
工廠
The Simple Analytics of the Firm as a Complex System (http://econ.geog.uu.nl/emaee/127_foster.pdf)
The Simple Analytics of the Firm as a Complex System (http://econ.geog.uu.nl/emaee/127_foster.pdf)
The Anatomy of Large Scale Systems (http://esd.mit.edu/WPS/esd-wp-2002-05.pdf)
APEC as a Complex Adaptive System (http://www.apec.org.au/docs/higgs.pdf)
網際網路
The Internet as a Complex System (http://www.cs.purdue.edu/nsl/complex.pdf)
Modeling the Internet as a Complex System (http://www.icir.org/floyd/talks/E2E-Jan03.pdf)
Towards
self-organizing
computer
networks:
system ...(http://www.co.umist.ac.uk/~mcaihak2/papers/esoa03_10c.pdf)
Modelling
Peer-to-Peer
Data
Networks
under
Complex
(http://infolab.usc.edu/DocsDemos/DNIS2005.pdf)
軟體系統
Software
systems
as
complex
and ...(http://www.tc.cornell.edu/~myers/papers/softnet.pdf)
networks:
A
System
Structure,
complex
Theory
function,
工程建設
CONSTRUCTION
AS
A
COMPLEX
SYSTEM
(http://www.bertelsen.org/strategisk_r%E5dgivning_aps/pdf/Construction%20as%20a%20Complex%20System.pdf)
(http://strobos.cee.vt.edu/IGLC11/PDF%20Files/02.pdf)
健保系統
Complexity and Health Workforce Issues (http://www.globalhealthtrust.org/doc/abstracts/WG6/HargadonPAPER.pdf)
(http://www.plexusinstitute.com/services/E-Library/cf_download.cfm?file=Primer%20on%20Complexity%20-%20fro
m%20Edgeware,%20adapted%20for%20website.doc&path=\)
Health
Care
Organizations
as
Complex
Adaptive
Systems
James
W. ...( http://www.change-ability.ca/Complex_Adaptive.pdf)
網路資訊流
Network as a Complex System: Information Flow Analysis (http://arxiv.org/pdf/nlin.CD/0110008)
The Information Society as a Complex System (http://www.jucs.org/jucs_6_3/the_information_society_as)
網路管理
On
the
Regulation
of
Networks
(http://law.bepress.com/cgi/viewcontent.cgi?article=1107&context=alea)
9/70
as
Complex
Systems
公路車輛交通網
Highway
Car
Traffic
as
a
Complex
(http://vmsstreamer1.fnal.gov/VMS_Site_03/Lectures/Colloquium/presentations/050420Boccara.pdf)
System
The Emergence of Hierarchy in Transportation Networks (http://www.ce.umn.edu/~levinson/Papers/Emergence.pdf)
航空控管
Engineering a Complex System: A Study of the AOC
http://www.mitre.org/work/tech_papers/tech_papers_04/norman_aoc/norman_aoc.pdf
將語言視為一種複雜適應系統
Language
as
a
complex
adaptive
system
(http://www3.isrl.uiuc.edu/~junwang4/langev/localcopy/pdf/steels00languageAs.pdf)
Language as a complex adaptive system (http://www.csl.sony.fr/downloads/papers/2000/steels-ppsn2000.pdf)
Language
as
a
complex
system:
the
case
of
phonetic
variability
(http://aune.lpl.univ-aix.fr:16080/~meunier/publi/CLG_2004.PDF)
Language as a complex system: the case of phonetic variability
(http://aune.lpl.univ-aix.fr:16080/~meunier/publi/CLG_2004.PDF)
The Origin and Evolution of Language: A Plausible, Strong-AI ...(http://www.isi.edu/~hobbs/origin-mar05.pdf)
Dynamical approaches to linguistics (http://www.loutzenhiser.com/Serious/papers/ThesisComplete.pdf)
Smith, K., Brighton, H., and Kirby, S. (2003) Complex Systems in Language Evolution: the cultural emergence of
compositional structure. Advances in Complex Systems, 6(4):537--558.
The variation of Zipf’s law in human language (http://www.edpsciences.org/articles/epjb/pdf/2005/06/b04435.pdf)
Language
as
a
complex
system:
the
(http://aune.lpl.univ-aix.fr:16080/~meunier/publi/CLG_2004.PDF)
case
of
phonetic
variability
The Complex Dynamics of Language Use on Tasks (http://www.education.leeds.ac.uk/research/ljc_complang.pdf)
The
Emergence
of
Symbol-Based
Communication
in
of ...(http://www.dca.fee.unicamp.br/projects/artcog/files/loula-kimas2005.pdf)
a
Complex
System
On Selfish Memes: culture as complex adaptive system (http://cogprints.org/3471/01/hokky.pdf)
群體行為
Börner, K, Maru, J. T., & Goldstone, R. L. (2004). The simultaneous evolution of article and author networks in
PNAS.
The
Proceedings
of
the
National
Academy
of
Science,
101,
5266-5273
.
(http://cognitrn.psych.indiana.edu/rgoldsto/pdfs/TARL.pdf)
Dynamics
in
Action:
Intentional
Behavior
(http://www.leaonline.com/doi/abs/10.1207/S15327000EM0202_03)
as
a
Complex
System
Why is Economics not a Complex Systems Science? (http://www.econ.iastate.edu/tesfatsi/MacroCAS.Foster.pdf)
Applications
of
Advanced
Science
in
the
New
(http://www.casact.org/coneduc/specsem/sp2003/papers/Smith-Tossani.pdf)
http://chronicle.com/free/v47/i35/35a01601.htm
10/70
Era
of
Risk
Modeling
[1] Simon, H.A. The architecture of complexity. In Proceedings of the American Philosophical Society 106, (1962), pp.
467--487.
[2] Kaneko, K., Life as a Complex Systems. (http://www.isi.it/files/2004/Kaneko.pdf, 1 p.)
[3] Parrott, L. & Kok, R. (2000), Incorporating Complexity in Ecosystem Modelling, Complexity International, Volume
07, Paper ID: lparro01, (http://www.complexity.org.au/ci/vol07/lparro01/)
[4] Ahmed, E ; Elgazzar, A S ; Hegazi, A S, An Overview of Complex Adaptive Systems. Mansoura J. Math .
( http://arxiv.org/abs/nlin/0506059)
[5] Anderson, P.W., More is Different, Science 177, 393-396, 1972. (http:// www.columbia.edu/ itc/sociology/watts/
w3233/client_edit/ anderson_more.pdf, )
[6] Goldenfeld and Kadanoff, Simple Lessons from Complexity, Science
284, 87-89 (Apr 2, 1999).
(http://guava.physics.uiuc.edu/~nigel/articles/complexity.html)(http://www.physics.uiuc.edu/People/Faculty/profile
s/Goldenfeld/Science_87.pdf)
[7] Reynolds, C. W. (1987) Flocks, Herds, and Schools: A Distributed Behavioral Model, in Computer Graphics, 21(4)
(SIGGRAPH '87 Conference Proceedings) pages 25-34.
參考資料
Leigh Tesfatsion and Dan Ashlock, Complexity at Large, Complexity Volume 3, Issue 4 , Pages 3 – 18, 7 Dec
1998.(http://www3.interscience.wiley.com/cgi-bin/fulltext/38816/PDFSTART)
A special edition of Science about complex systems Science Vol. 284. No. 5411 April 2, 1999.
A special edition of Science about complex systems Science Vol. 284.
(http://www.sciencemag.org/content/vol284/issue5411/).
No.
5411
(1999)
基因調控網路
Hidde de Jong,
Modeling and Simulation of Genetic Regulatory Systems: A Literature Review, Journal of Computational Biology
Jan 2002, Vol. 9, No. 1: 67-103
11/70
"A system that involves numerous interacting agents whose aggregate behaviors are to be understood. Such aggregate
activity is nonlinear, hence it cannot simply be derived from summation of individual components behavior." [Jerome
Singer]
"Computers have made it possible to explore the consequences of of relatively simple interactions of relatively simple
things in a way never before possible ... this new capability for observations makes possible significant insights into
phenomena long felt to be complex for serious analysis." - Insights ...
"Simple things interacting in simple ways can yield surprisingly complex outcomes ... Brains too consist of relatively
simple things interacting in relatively simple ways" - Simple Networks ...
"Each of us now can be seen as the center ... so its worth thinking about what all this means ..." - On Beyond ...
(http://serendip.brynmawr.edu/complexity/)
(http://web.cenet.org.cn/web/complexity/index.php3?file=index.php3)
(http://serendip.brynmawr.edu/complexity/life.html)
http://cognitrn.psych.indiana.edu/rgoldsto/complex/p747description.htm
(http://www.calresco.org/links.htm#age)
(http://mitpress.mit.edu/books/FLAOH/cbnhtml/home.html)
(http://pespmc1.vub.ac.be/DEFAULT.html)
The behavior of a complex sytem can not be understood in terms of a simple extrapolation of the properties of its
components, elements and entities. As P.W. Anderson said in his 1972 science article (Science Vol. 177 No. 4047), more
is different. At the macroscopic level of the system, new properties appear which can not be predicted by the properties
of microscopic components.
Typically, the relationships between elements in a complex system are both short-range and long-range. The direct local
interactions are short-range, that is information is normally received from near neighbours. Through top-down feedback
and small-world connections agents can indirect influence all other agents. The richness of the connections means that
communications will pass across the system but will probably be modified on the way.
Complex Adaptive System http://www.theo-physik.uni-kiel.de/theo-physik/schuster/cas.html
(http://www.red3d.com/cwr/boids/)
(http://math.math.sunysb.edu/%7Escott/ants/)
(http://ivytech7.cc.in.us/mathsci/alife/ants.html)
(http://www.hip.atr.co.jp/~ray/tierra/tierra.html)
(http://www.krl.caltech.edu/avida/)
(http://offis.offis.uni-oldenburg.de/projekte/ecotools/fishsim.htm)
(http://www.aridolan.com/JavaFloys.html#FloysApplet)
Humor (http://ivytech7.cc.in.us/mathsci/alife/humor.html)
Gell-Mann (1994), Holland (1995), Jantsch (1980), Maturna and Varela (1992), and Prigogine and Stengers (1984). The
essential principles of CAS have been taken from each of these works and synthesized into a single description. The
description is purposefully concise. A more lengthy description, with application to business organizations, is contained
in the forthcoming paper "A Complex Adaptive Systems Model of Organizational Change," by myself, to appear in the
new journal Nonlinear Dynamics, Psychology, & Life Science.
A CAS behaves/evolves according to three key principles: order is emergent as opposed to predetermined, the
system's history is irreversible, and the system's future is often unpredictable. The basic building blocks of the CAS are
agents. Agents are semi-autonomous units that seek to maximize some measure of goodness, or fitness, by evolving
12/70
over time. Agents scan their environment and develop schema representing interpretive and action rules. These schema
are often evolved from smaller, more basic schema. These schema are rational bounded: they are potentially
indeterminate because of incomplete and/or biased information; hey are observer dependent because it is often difficult
to separate a phenomenon from its context, thereby identifying contingencies; and they can be contradictory. Schema
exist in multitudes and compete for survival.
Existing schema can undergo three types of change: first order change, where action is taken in order to adapt the
observation to the existing schema; second order change, where there is purposeful change in the schema in order to
better fit observations; and third order change, where a schema survives or dies because of the Darwinian survival or
death of its corresponding CAS. Schema can change through random or purposeful mutation, and/or combination with
other schema. Schema change generally has the effect of making the agent more robust (it can perform in light of
increasing variation or variety), more reliable (it can perform more predictably), or grow in requisite variety (in can
adapt to a wider range of conditions).
The fitness of the agent is a complex aggregate of many factors, both local and global. The general health or fitness
of the agent determines what the probability of change will be. Optimization of local fitness allows differentiation and
novelty/diversity; global optimization enhances the CAS coherence as a system and induces long term memory. In
general the probability of second order schema change is a nonlinear function of the fitness value.
Schema define how a given agent interacts with other agents surrounding it. Actions between agents involve the
exchange of information and/or resources. These flows may be nonlinear. Information and resources can undergo
multiplier effects based on the nature of interconnectedness in the system. Agent tags help identify what other agents are
capable of transaction with a given agent; tags also facilitate the formation of aggregates, or meta-agents. Meta-agents
help distribute and decentralize functionality, allowing diversity to thrive and specialization to occur. Agents or
meta-agents also exist outside the boundaries of the CAS, and schema also determine the rules of interaction concerning
how information and resources flow externally.
* Dooley, K. (1997): "A Complex Adaptive Systems Model of Organization Change," Nonlinear Dynamics,
Psychology, & Life Science, Vol. 1, No. 1, p. 69-97.
* Dooley, K., Johnson, T., and D. Bush (1995): "TQM , Chaos, and Complexity," Human Systems Management,
Vol. 14, p. 1-16.
* Gell-Mann, M. (1994): The Quark and the Jaguar. (New York: Freeman & Co.).
* Holland, J.H. (1995): Hidden Order, (Reading, MA: Addison-Wesley).
* Jantsch, E. (1980): The Self-Organizing Universe, Oxford: Pergaman Press.
* Lewin, R. (1992): Complexity: Life at the Edge of Chaos. (New York: MacMillan).
* Maturana, H. and F. Varela (1992): The Tree of Knowledge (Boston: Shambhala.
* Prigogine, I., & I. Stengers (1984): Order Out of Chaos. (New York: Bantam Books).
* Waldrop, M.M. (1992): Complexity: The Emerging Science at the Edge of Chaos. (New York: Simon and
Schuster).
13/70
複雜論與複雜系統
簡介與定義
ESD-WP-2000-02: Ideas on Complexity in Systems-- Twenty Views
by Professor Joseph Sussman (http://esd.mit.edu/wps/esd-wp-2000-02.pdf)
Scherf, Olaf. COMPLEXITY: A CONCEPTUAL CHALLENGE,
SYNTHETIC
ANALYSIS
OF
COMPLEX
(http://www.usyd.edu.au/su/hps/newevents/Auyang1.html)
SYSTEMS
Shared
concepts
between
complex
systems
and
http://www.math-info.univ-paris5.fr/~bouzy/publications/UES96.article.pdf
I
the
–
game
理論延伸探討
The paradigm of complex systems (http://www.chem.kntu.ac.ir/mahjani/complexity.htm)
Defining
and
detecting
emergence
in
complex
(http://www.per.marine.csiro.au/staff/Fabio.Boschetti/3054CO/papers/emergence_kes_final.pdf)
Dependability
of
complex
open
systems:
A
unifying
for ...(http://www.cert.org/research/isw/isw2000/papers/47.pdf)
14/70
THEORIES
of
Go
networks
concept
冪次律分佈(Powerlaw distribution)與各種統計分佈
Zipf, G.K. (1949).
Human Behavior and the Principle of Least Effort, Addison-Wesley Press, New York.
哈佛大學語言學教授 George Kinsley Zipf(1902-1950) 所歸納出有關語言文字的規律性，在自然語言中第 n 個最
常使用的字其使用頻率與 n 成反比。也就是說第二常用字出現頻率為第一常用字出現頻率的 1/2 倍，第三常用
字出現頻率為第一常用字出現頻率的 1/3 倍。
pk ( s, N ) 
1 ks

N
n 1
1 ns
N 表示全部不同的字數，k 代表其使用頻率排名，s 則是整個分布的冪次。
pk ( s, N ) 
1 ks
H N ,s
H N ,s 為第 N 個通用協調數(generalized Harmonic number)。
(http://www.aicpa.org/pubs/jofa/may1999/nigrini.htm)
(http://mathworld.wolfram.com/BenfordsLaw.html)
1938 年在奇異電子公司(General Electric Company)位於紐約 Schenectady 的實驗室擔任物理學家 Frank Benford
由對數表手冊的污損情形推斷使用者一定時常查詢這些數值的對數值，尤其是以數字 1 開頭的頁數污損的最嚴
重。他認為使用者並不會特意的多查詢這樣的數字，因此他開始收集並統計身邊一些時常出現的數字。一共收
集了 20229 組數字，其範疇由河流的面積，棒球統計數字到美國科學界重要人物前 342 名的住址號碼，在這樣
廣泛的數字統計後，他發現以數字 1 開始的機率遠比其他數字開頭的高，約佔全部數字的百分之三十。
Benford 設計了公式來描述這個現象。數字 d 表示由 1 到 9，那麼第一個數為 d 的數字出現機率約等於(1+1/d)
取基底為 10 的對數。
這個規律性在 1881 年 Simon Newcomb
數字 第一位 第二位 第三位 第四位
0
.11968 .10178 .10018
1 .30103 .11389 .10138 .10014
2 .17609 .10882 .10097 .10010
3 .12494 .10433 .10057 .10006
4 .09691 .10031 .10018 .10002
5 .07918 .09668 .09979 .09998
6 .06695 .09337 .09940 .09994
7 .05799 .09035 .09902 .09990
8 .05115 .08757 .09864 .09986
9 .04576 .08500 .09827 .09982
Nigrini, M. "A Taxpayer Compliance Application of Benford's Law." J. Amer. Tax. Assoc. 18, 72-91, 1996.
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Newcomb Simon (1881) Note on the Frequency of the Use of Digits in Natural Numbers American Journal of
Mathematics 4 39-40.
Benford Frank (1938) The Law of Anomalous Numbers Proceedings of the American Philosophical Society 78,
551-572.
Hill, T. P. "A Statistical Derivation of the Significant-Digit Law." Stat. Sci. 10, 354-363, 1996.
(http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A//www.gatsby.ucl.ac.uk/%7Eturner/Benford%27s%2520l
aw/stat-der.pdf&ei=XZPzQs65LcOsigGi0bXtAg)
The
Second
Digit
Phenomenon
http://www.google.com.tw/url?sa=t&ct=res&cd=29&url=http%3A//econwpa.wustl.edu%3A8089/eps/othr/papers/0507/
0507001.pdf&ei=TnzzQrOgOZqCiAHelrXTAg
(http://auditorymodels.org/jba/BOOKS_Historical/)
A
General
Approach
to
Digital
Analysis
exemplified
by
Stock
Market
...
http://www.google.com.tw/url?sa=t&ct=res&cd=27&url=http%3A//www.mathematik.uni-ulm.de/dof/pnposch/paper/po
sch_stockmarket.pdf&ei=B5DzQoHTHqruRKmv6OAB
Heaps’ law
H. S. Heaps. Information Retrieval - Computational and Theoretical Aspects. Academic Press, 1978.
描述一個文件的大小與上面所包含不同的字彙。
VR (n )  Kn  , n 是文件的大小，K[10,100], [0.4,0.6]。
Lotka’s law
Alfred J. Lotka, The Frequency Distribution of Scientific Productivity, Journal of the Washington Academy of Sciences,
16(12):317-323, 1926.
貢獻 n 篇的作者數大約是只貢獻一篇的 1/na ，n 大約等於 2。
k
x 
Pr[ X  x ]   m  ，k>0,xxm>0
 x 
k
 xm 
Pr[ X  x ]  1   
 x 
其 PDF 為
p X ( x )  kxmk x  ( k 1) ，k>0,xxm>0
由 Zipf’s 排名分布到 Pareto 分布
E[ X r ] C1r  b
Pr[ X  C1r  b ]  C2 r
Pr[ X  y ]
y 1 b
Pr[ X  y ]
y  (11 b )  y 
16/70
在這篇簡介中，我們深入探討三個領域，經濟體與金融市場、生命體與神經網路和網路模型與網際網路。
Self-similar approach to market analysis (http://www.edpsciences.org/articles/epjb/pdf/2001/08/b0629.pdf)
早在 1963 年 Benoit Mandelbrot 觀察支加哥商品交易市場中的棉花價格變動，注意到其極端值出現的機率遠比一
般認為的高斯分佈所預測的大。而在使用不同的時間間距匯整價格變動畫成各曲線時，看起來都十分相似。此
為使用碎形來描述價格變動的濫觴，也開啟了使用有別於傳統經濟方法研究金融市場的方法。視經濟體為一複
雜系統，則是在 1987 年，由 Kenneth J. Arrow 及一些經濟學家與物理、生物與電腦專家聚會討論，並於次年編
輯成 The Economy as an Evolving Complex System 一書。自此開始將相關的思維推廣。於 1996 年輯錄的 The
Economy as an Evolving Complex System II 論文輯則是幾乎集中於經濟領域中。其主要的議題是在過去的十年
中，使用複雜系統的觀點來看經濟有些什麼助益？其中談到傳統經濟思維與數學工具研究在下列幾項領域中遭
到一些困境
分散式互動
在經濟體中的活動主要是由分散的，同質或(可能是)異質性的組成體同時互動所決定的。任一組成體的行動會受
到對其他幾位構成體的預期行動所影響，也會受到整體組成體的集合行動所影響。
沒有全面控制者
沒有任何全面性的控制機制可以掌控所有的互動。所有的控制是經由組成體間競爭與協調的機制達成。經濟行
動是經由法律機構所監督，分派的角色，轉移的關連，沒有全面性的、可利用所有機會的競爭者。
互相參差的階層式組織
經濟體包括有許多層面的組織與互動。在某一階層的單位例如作為、行動、策略與產品，通常為下一階層的基
石。整體組織不僅是階層式，而且包括各種跨越階層的互動、關連、通訊管道。
持續的適應行為
行動、策略與產品隨著單一組成體累積經驗後便修改適應，整體系統也時時調適。
持續創新
新的利基隨時在新市場、新科技、新作為或新機構中產生。佔領新利基的行動本身可能創造新的利基。結果便
是持續的、永不停歇的創新。
非均衡動力
由於新利基、新的潛能、新機會時時產生，經濟體並非在任何最佳點或全域的均衡點運作。改善是一直有可能
的而且也真正隨時進行的。
John Holland 於 1987 年稱具有以上特性的系統為適應非線性網絡(adaptive nonlinear network)。這些系統時常出
現於自然與社會中，例如神經系統、免疫系統、生態系與經濟體。其基本的特性是組成體並不只是單純的針對
刺激來反應，組成體還會預估未來。經濟構成體會形成期望值，他們會建立經濟模型來預測未來，並依此決定
行動。這些預測模型並不一定是外顯的、協調的或是相互一致的。將經濟體視為是一個複雜適應非線性網路—
演化複雜系統—對於經濟學基礎，同時對於如何設定理論問題及如何解答有深厚的影響。
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認知學基礎
新古典經濟理論的認知學基礎是單一的，經濟組成體是理性的最佳化者。這表示組成體以統計機率評估未知並
隨著新資訊而使用貝氏原理修正預測，從而選擇滿意度期望值最高的行動。在這樣的單一化組成體認知能力情
況下，組成體完全相互瞭解並知悉所有共同資訊，對所存在的世界均使用理性期望。相對的，聖太菲學院的方
式則顯得多元化。依照現代認知學理論，認知過程並沒有唯一的或主要的。組成體需要在認知上架構自身的問
題，也就是說他們必須要弄懂問題的意義並解決問題。而這些都在有限的認知資源下完成。要懂問題的意義，
要學習、要適應，組成體必須使用各式各樣的分散式認知過程。組成體將外界資訊轉化成行動的依據是由經驗
而來。而這些經驗或認知解釋並不是互無矛盾才能產生有效的行動。因此組成體是活在自我解讀的世界觀中。
而這個複雜世界是由其他時時變化的組成體與他們的行動所構成。組成體無法進行一般所認知的最佳化推論，
並不是因為受到有限的記憶空間與處理能力，而是無法精確定義所謂的最佳化行動。同時新古典經濟理論中的
演繹理性組成體在指出有效的行動時只是邊緣的角色。任何組成體所擁有的相互共同知識，必需由實際的、特
定的認知過程依照實際經驗而得，因此，無法輕易假設所謂的共同知識是存在的。
結構基礎
在一般均衡理論分析，組成體並不是直接與其他組成體互動，而是透過間接的機制—市場—來達成互動。不同
於博奕理論中組成體直接與所有其他組成體互動，由償賠矩陣得知輸贏多寡。所以其互動是簡單的或者可以說
是極端的一對所有的或所有的對所有的。組成體的內部結構更是抽象化，然而由複雜論的角度看，結構是很重
要的。首先，以網路為基礎的結構是很重要的。所有的經濟活動都牽涉到組成體間的互動。所以經濟功能是局
限於由網路所架構出來的圖樣。這些網路的特色是相當稀疏的連線。接著，經濟活動是由社會角色與社會上支
持的程序也就是說經濟組織。再者，經濟單位都具有遞迴的架構，其本身是由其他經濟單位所構成。然而在這
樣的一級級組織下，個體與其行動並非是階層式。由於一級組成個體可能是另一級個體的部份，而且在不同級
的個體可能會有互動。在不同級的組織的行動的因果關係可能是相反的。一級的組織其行動可以是自主的可是
受到其他級個體的行動或行為的樣式所限制。使用聖太菲角度的基礎組織原則是在一級的個體結合形成另一級
的個體。
聖太菲人工股市模型（Santa Fe Artificial Stock Market Model）]
[8] W. Brian Arthur, John H. Holland, Blake LeBaron, Richard Palmer, and Paul Tayler, "Asset Pricing under Endogenous
Expectations in an Artificial Stock Market" pages 15-44 in W. Brian Arthur, Steven N. Durlauf, and David A. Lane,
The Economy as an Evolving Complex System II, Santa Fe Institute Studies in the Sciences of Complexity, Vol.
XXVII, Addison-Wesley, 1997.
[9] Norman Ehrentreich, "The Santa Fe Artificial Stock Market Re-Examined: Suggested Corrections" (html),
Economics Working Paper Archive at WUSTL, 2002, (pdf ms.,22pp., available for downloading from above html site).
[10] Blake LeBaron, "Agent-Based Computational Finance: Suggested Readings and Early Research" (pdf preprint),
Journal of Economic Dynamics and Control 24:5-7 (2000), 679-702. Published article available at Science Direct.
[11]Blake LeBaron, "A Builder's Guide to Agent-Based Financial Markets" (pdf preprint,18pp,207K), Quantitative
Finance 1 (2001), 254-261.
[12] Blake LeBaron, "Building the Santa Fe Artificial Stock Market" (pdf preprint,19pp,126K), Working Paper,
Brandeis University, June 2002.
LeBaron provides an insider's look at the construction of the Santa Fe Artificial Stock Market model. He considers
the many design questions that went into building the model from the perspective of a decade of experience with
agent-based financial markets. He also provides an assessment of the model's overall strengths and weaknesses.
[13] Blake LeBaron, "Calibrating an Agent-Based Financial Market" (pdf,44pp), Graduate School of International
Economics and Finance, Working Paper, Brandeis University, Revised March 2003.
This paper develops an agent-based computational stock market with market participants who adapt and evolve
their behaviors over time. The market model is calibrated to match the variability and growth of dividend payments in
18/70
U.S. data. The market model generates some features that are remarkably similar to those from actual U.S. data,
including the volatility of the dividend process, the persistence in volatility and volume, and fat-tailed return
distributions.
[14] Blake LeBaron, W. Brian Arthur, and Richard Palmer, "Time Series Properties of an Artificial Stock Market
Model" (pdf preprint,30pp,324K), Journal of Economic Dynamics and Control 23 (1999), 1487-1516.
A rigorous technical discussion of the Santa Fe Artificial Stock Market Model, including implementation details.
Anyone interested in the actual implementation of this model should consult this paper in addition to the Arthur et al.
(1997) paper cited above.
Economy from the Perspective of Complex Systems (http://www.kzoo.edu/physics/ccss/materials/comecon.pdf)
COMPLEXITY IN ECONOMICS(http://cob.jmu.edu/rosserjb/COMPLEXITY%20IN%20ECONOMICS.doc)
W. Brian Arthur, "Complexity and the Economy,"Science, 2 April 1999, 284,107-109
將金融市場
Multiscaling
and
non-universality
in
fluctuations
(http://www.nd.edu/~networks/Publication%20Categories/Eisler_EurophyLtr(2005).pdf)
of
Complex
Dynamics
and
Financial
Fragility
in
an
Model(http://www.ecomod.net/conferences/ecomod2003/ecomod2003_papers/Gallegati.pdf)
driven
Agent
...
Based
[15] W. Brian Arthur, "Complexity in Economic and Financial Markets," Complexity, Volume 1, Number 1, 1995, pp. 2
0-25.
Provides, among other things, a brief summary of the Santa Fe Artificial Stock Market model by Arthur et al.
(1997), below.
David F. Batten, "Coevolving Markets" (Chapter 7, pages 209-245), in Discovering Artificial Economics, Westview
Press, 2000.
W. M. van den Bergh, K. Boer, A. de Bruin, U. Kaymak, and J. Spronk, "On Intelligent Agent-Based Analysis of
Financial Markets" (pdf,9pp,341K), Working Paper, Erasmus University, Rotterdam, 2002.
S.-H. Chen and C.-H. Yeh, "Evolving Traders and the Business School with Genetic Programming: A New Architecture
of the Agent-Based Stock Market," Journal of Economic Dynamics and Control 25 (3-4), March 2001, pages 363-393.
Article available at Science Direct.
# S.-H. Chen, T. Lux, and M. Marchesi, "Testing for Nonlinear Structure in an Artificial Financial Market," Journal of
Economic Behavior and Organization 46 (2001), 327-342. Article available at Science Direct.
# John Duffy, "Learning to Speculate: Experiments with Artificial and Real Agents," Journal of Economic Dynamics
and Control 25(3/4), March 2001, pages 295-319. Article available at Science Direct.
This paper employs parallel experiments with real and computational agents to explore issues originally raised by
Kiyotaki and Wright in their well-known search model of money (JPE, 1989). The primary issue of interest is how
individuals might come to accept or learn to adopt a convention in which the particular commodity functioning as
"money" is dominated in rate of return by other assets, in the sense that it has a higher storage cost. The key offsetting
factor is anticipations ("speculation") concerning the ease with which the "money" good can be turned over in trade for
other goods that agents have a higher desire to consume. The author shows how each type of experiment can contribute
to the experimental design and interpretation of results for the other.
# J. Doyne Farmer and John Geanakoplos (eds.), Beyond Equilibrium and Efficiency, 352pp., Oxford University Press,
2005. ISBN: 0-195-15094-5
19/70
From the publisher: "This book presents recent thought on market efficiency, using a complex systems approach to
move past equilibrium models and quantify the actual efficiency of markets. The older view that markets are perfectly
efficient has come under attack from several different directions, including studies of market anomalies, human
psychology, bounded rationality, agent-based modeling, and evolutionary game theory. This volume brings together
some of the best economists, physicists, and biologists working on quantitative models of complex self-organized
behavior relevant to measuring market efficiency, to stimulate new approaches to understanding financial markets."
J. Doyne Farmer is McKinsey Professor at the Santa Fe Institute, and John Geanakoplos is Professor of Economics
at Yale University.
# J. Doyne Farmer and Andrew W. Lo, "Frontiers of Finance: Evolution and Efficient Markets" (pdf,7pp,105K), SFI
Working Paper, April 1999.
# Jens Grossklags, Carsten Schmidt, and Jonathan Siegel, "Dumb Software Agents on an Experimental Asset Market"
(pdf,22pp.), Working Paper, School of Information and Management Systems, UC Berkeley.
# Cars Hommes, "Heterogeneous Agent Models in Economics and Finance", Chapter 8 in Leigh Tesfatsion and Kenneth
L. Judd (editors), Handbook of Computational Economics, Vol. 2: Agent-Based Computational Economics, Handbooks
in Economics Series, North-Holland, Amsterdam, Spring 2006, to appear.
Abstract: This chapter surveys work on dynamic heterogeneous agent models (HAMs) in economics and finance.
Emphasis is given to simple models that, at least to some extent, are tractable by analytic methods in combination with
computational tools. Most of these models are behavioral models with boundedly rational agents using different
heuristics or rule of thumb strategies that may not be perfect, but perform reasonably well. Typically these models are
highly nonlinear, e.g. due to evolutionary switching between strategies, and exhibit a wide range of dynamical behavior
ranging from a unique stable steady state to complex, chaotic dynamics. Aggregation of simple interactions at the micro
level may generate sophisticated structure at the macro level. Simple HAMs can explain important observed stylized
facts in financial time series, such as excess volatility, high trading volume, temporary bubbles and trend following,
sudden crashes and mean reversion, clustered volatility and fat tails in the returns distribution.
# Paul Johnson, "What I Learned from the Artificial Stock Market" (pdf,20pp,107K), Working Paper, Department of
Political Science, University of Kansas, November 5, 2001.
Abstract:This essay describes some of the changes that were incorporated in the ASM-2.2 revision of the code for
the Santa Fe Artificial Stock Market model. It also presents some important lessons for agent-based modelers that can
be illustrated with the code.
# Deddy P. Koesrindartoto, "Treasury Auctions, Uniform or Discriminatory?: An Agent-Based Approach" (html),
Economics Working Paper No. 04013, Department of Economics, Iowa State University, July 2004.
Abstract: This study develops an agent-based computational economics (ACE) framework to explore
experimentally how a Treasury should auction its securities. Buyers are modeled as profit seekers capable of submitting
strategic bids via reinforcement learning. Two distinct auction pricing rules are considered, uniform and discriminatory.
The author shows that these two rules result in systematically different auction outcomes under different treatment
conditions for relative capacity and for price volatility in a secondary security market. In particular, which auction
pricing rule generates greater Treasury revenues varies systematically with these treatment factor specifications. These
findings help to explain why previous Treasury auction studies attempting to determine "the" best Treasury auction
pricing rule have reached contradictory conclusions.
# Blake LeBaron, "Agent-Based Computational Finance", Chapter 9 in Leigh Tesfatsion and Kenneth L. Judd (editors),
Handbook of Computational Economics, Vol. 2: Agent-Based Computational Economics, Handbooks in Economics
Series, North-Holland, Amsterdam, Spring 2006, to appear.
Abstract: This chapter surveys research on agent-based models used in finance. It concentrates on models where
the use of computational tools is critical for the process of crafting models that give insights into the importance and
dynamics of investor heterogeneity in many financial settings.
# T. Lux and M. Marchesi (guest editors), Special Issue on "Heterogeneous Interacting Agents in Financial Markets,"
Journal of Economic Behavior and Organization 49, No. 1, in press. Article available at Science Direct.
# T. H. Noe, M. J. Rebello, and J. Wang, "Corporate Financing: An Agent-Based Analysis," Journal of Finance, Vol. 58,
943-973, June 2003.
# Nicholas S. P. Tay and Scott C. Linn, "Fuzzy Inductive Reasoning, Expectation Formation, and the Behavior of
Security Prices," Journal of Economic Dynamics and Control 25, March 2001, pages 321-361. Article available at
Science Direct.
20/70
# Leigh Tesfatsion, "Introduction to Financial Markets" (html).
# Leigh Tesfatsion, "Information, Bubbles, and the Efficient Markets Hypothesis" (html).
# Leigh Tesfatsion, "Notes on the Santa Fe Artificial Stock Market Model" (html).
# Edward P. K. Tsang and Serafin Martinez-Jaramillo, Computational Finance (pdf,310K,6pp), Feature Article, IEEE
Computational Intelligence Society, August 2004.
This paper briefly outlines the scope and agenda of computational finance research.
# Frank Westerhoff, "Speculative Markets and the Effectiveness of Price Limits", Journal of Economic Dynamics and
Control 28 (2003), 493-508. Article available at Science Direct.
"The End of Certainty in Economics," Talk delivered at the conference Einstein Meets Magritte, Free University of
Brussels, 1994. Appeared in Einstein Meets Magritte, D. Aerts, J. Broekaert, E. Mathijs, eds. 1999, Kluwer Academic
Publishers, Holland.
The Economy as an Evolving Complex System II.
Edited (with S. Durlauf and D. Lane), Addison-Wesley, 1997.
The
Economy
as
an
Evolving
Complex
Volume ...( http://meritbbs.unimaas.nl/staff/silverberg/review.pdf)
System
II.
Proceedings
A Complex System View of why Stock Markets Crash (http://www.newthesis.org/200401/01-200401.pdf)
Why is Economics not a Complex Systems Science? (http://www.econ.iastate.edu/tesfatsi/MacroCAS.Foster.pdf)
Complex Systems Summer Schools 2005 http://www.globalhealthtrust.org/doc/abstracts/WG6/HargadonPAPER.pdf
Agent-based
Microsimulation
of
Economy
Perspective(http://www.idi.ntnu.no/~xiaomeng/paper/itbm-13-02.pdf)
Agent-Based
Computational
Economics_Modelling
(http://web.cenet.org.cn/upfile/68883.pdf)
Handbook
of
Computational
Economics
(http://web.cenet.org.cn/upfile/68884.pdf)
Vol.
from
Economies
2
as
A
Complex
Agent-Based
Complexity
Adaptive
Computational
Systems
Economics
Microsimulation of Complex System Dynamics (http://www.ub.uni-koeln.de/ediss/archiv/2001/11v4116.pdf)
Holbrook, Morris B.. 2003. "Adventures in Complexity: An Essay on Dynamic Open Complex Adaptive Systems,
Butterfly Effects, Self-Organizing Order, Coevolution, the Ecological Perspective, Fitness Landscapes, Market Spaces,
Emergent Beauty at the Edge of Chaos, and All That Jazz." Academy of Marketing Science Review [Online] 2003 (6)
Available: http://www.amsreview.org/articles/holbrook06-2003.pdf
(http://www.amsreview.org/articles.htm)
Mandelbrot, BB (1963). "The Variation of Certain Speculative Prices,” Journal of Business, 36:. 394-429.
(http://www.santafe.edu/arthur/Papers/ADLIntro.html)
REVISITING
MARKET
EFFICIENCY:
THE
STOCK
COMPLEX ...( http://www.quantuminvesting.net/samples/144maub.pdf)
MARKET
Computability
and
Evolutionary
Complexity:
Complex ...( http://www.essex.ac.uk/economics/discussion-papers/papers-text/dp574.pdf)
Agents and Complex Systems (http://www.jot.fm/issues/issue_2002_07/column3.pdf)
A
GAME
PERSPECTIVE
TO
COMPLEX
(http://lib.tkk.fi/Diss/2005/isbn9512277328/isbn9512277328.pdf)
AGENT-BASED
GENETIC
AND
EMERGENT
21/70
Markets
ADAPTIVE
COMPUTATIONAL
AS
A
As
SYSTEMS
MODELS
OF
COMPLEX ...( http://www.public.asu.edu/~kdooley/papers/compmod.PDF)
Darwinism, probability and complexity: market- based ... (http://staff.um.edu.mt/tsam1/publications/Darwinism.pdf)
The Adaptive Markets Hypothesis: Market Efficiency from an ...http://web.mit.edu/alo/www/Papers/JPM2004.pdf
22/70
經濟物理學或金融物理學(econophysics)
這是由物理學家所主導的領域，是指綜合使用統計物理，非線性動力學與組成體為基礎的模擬技術研究經濟與
金融市場現象的跨領域研究。經濟物理學(econophysics) [] 於 1995 年時出現在印度加爾各達舉行的複雜系統國
際研討會中。在 1997 年布達佩斯的國際工作坊則以經濟物理學命名。雖然有些物理學家使用 phynance 來描述
這領域，然而為了與 biophysics(研究在生物學中的物理現象)和 geophysics(研究地質學中的物理)相對應，所以漸
漸多人採用經濟物理學來稱呼這些相關研究。
[16] Sanley, H. E.; Afanasyev, V.; Amaral, L. A. N.; Buldyrev, S. V.; Goldberger, A. L.; Havlin, S.; Leschhorn, H.; Maass,
P.; Mantegna, R. N.; Peng, C.-K.; Prince, P. A.; Salinger, M. A.; Stanley, M. H. R.; Viswanathan, G. M., Anomalous
fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics, Physica A, Volume
224, Issue 1-2, p. 302-321, 1996.
[17] Mantegna, R. N, H.E. Stanley, An introduction to econophysics: correlations and complexity in finance, Cambridge
University
Press,
2000.
(http://assets.cambridge.org/052162/0082/sample/0521620082ws.pdf)
(http://assets.cambridge.org/052162/0082/frontmatter/0521620082_frontmatter.pdf)
[18] De Liso, Nicola, and Giovanni Filatrella, Econophysics: The emergence of a new field? 2002.
(http://www.dise.unisa.it/PDF/deliso_filatrella.pdf) (http://www.dse.unibo.it/prin/wp/at4_2_2002.pdf)
[19] Bouchaud, J.-P., P. Cizeau, L. Laloux and M. Potters, Mutual attractions: physics and finance, Physics World, January
1999
[20] Amaral, L.A.N., P. Cizeau, P. Gopikrishnan, Y. Liu, M. Meyer, C.-K. Peng, H.E. Stanley, Econophysics: can
statistical physics contribute to the science of economics? Computer Physics Communications 121–122 (1999)
145–152 (http://polymer.bu.edu/hes/articles/acglmps99.pdf)
[21] McCauley, J.L., Dynamics of Markets: Econophysics and Finance from a Physicist’s Standpoint, Cambridge
University Press, 2004. (http://www.cap.ca/news/books/Dynamics-McCauley-Martin.pdf)
[22] Burda, Z., J. Jurkiewicz, M.A. Nowak . Is Econophysics a Solid Science? Acta Physica Polonica B34 (2003) 87.
(http://arxiv.org/pdf/cond-mat/0301096)
[23] Di Matteo, Tiziana, Enrico Scalas, Michele Tumminello, Econophysics: a new tool to investigate financial markets,
Bollettino
della
Comunità
Scientifica
in
Australasia,
Sept.
2004.
(http://www.scientific.ambitalia.org.au/bollettino/sept04/dimatteo_ing.pdf)
[24]
Scalas, Enrico, Five Years of Continuous-time Random
(http://econwpa.wustl.edu:8089/eps/fin/papers/0501/0501005.pdf)
Walks
in
Econophysics,
2005.
[25] Vasconcelos, Giovani L., A Guided Walk Down Wall Street: An Introduction to Econophysics , Brazilian Journal of
Physics, vol. 34, no. 3B, September, 2004. (http://www.scielo.br/pdf/bjp/v34n3b/a02v343b.pdf)
[26]
Econophysics:
statistical
physics
of
interacting
(http://www.fzu.cz/departments/theory/seminars/presentations/sem-present-031127.pdf)
(http://members.jcom.home.ne.jp/ephys/Econophysics%20Stauffer.pdf)
agents
[27] Gordon M.B., Nadal J.P., Phan D., Statistical Mechanics Approaches in Economics: Suggested Reading and
Interpretations, (http://www.cenecc.ens.fr/EcoCog/Livre/Drafts/MBGJPNDPv5.pdf)
[28]
Farmer, J. Doyne, Martin Shubik, and Eric Smith, Economics:
(www.santafe.edu/research/publications/workingpapers/05-06-027.pdf)
(http://cowles.econ.yale.edu/P/cd/d15a/d1520.pdf)
[29]
Yakovenko,
Victor
M.,
Research
in
(http://www2.physics.umd.edu/~yakovenk/papers/condmat-0302270.pdf)
the
next
physical
Econophysics,
science?
2003.
[30] Plerou, Vasiliki, Parameswaran Gopikrishnan, Bernd Rosenow, Luis A.N. Amaral, H. Eugene Stanley, Econophysics:
Financial
Time
Series
From
a
Statistical
Physics
Point
of
View,
(http://polymer.bu.edu/~amaral/Papers/physa00a.pdf)
(http://members.jcom.home.ne.jp/ephys/)
(http://iapetus.phy.umist.ac.uk/Teaching/LitStudies/Essays/Econophysics.html)
一方面由於在金融界使用數位技術進行交易與儲存資料漸漸普遍，且每日產生的金融資料龐大而且可以十分容
23/70
易取得。例如 Trades and Quotes Database: a monthly CD-ROM with every transaction at NYSE, AMEX, and
NASDAQ。另一方面一些統計物理學家認為使用統計力學方法可以由這些資料中找到經濟學家所沒注意到的規
律性且或許可由物理定律提供一些可能解釋。
幾個時常見到的議題：財富與收入分佈，價格波動，買賣簿，探討量尺現象，臨界行為，冪次律分佈與寡勝博
奕。
http://www.bwl.uni-kiel.de/vwlinstitute/gwif/teaching/handouts/tdf/Tutorial%20Fat%20Tails.pdf
Essays
on
Asset
Return
Distributions
http://www.google.com.tw/url?sa=t&ct=res&cd=212&url=http%3A//www.int9.com/download/isbn9512262541.pdf&ei
=TgH4QuXyEInqswGbqqj0DQ
價格波動是廣為研究的議題。
http://www.physics.ubc.ca/~jinshanw/project/ecophys/review/review.html
The
Statistical
Physics
of
http://iapetus.phy.umist.ac.uk/Teaching/LitStudies/Essays/Econophysics.html
financial
markets
假設 pt 表示某一金融商品在時間 t 時的價格， pt 是非靜駐式序列(non-stationary)
也就是說，[pt] 是與時間 t 的函數。序列
價格報酬(price returns)
rt (t )  ln p(t  t )  ln p(t ) ，買賣量(volume) V (t ) ，波動係數 (volatility)  (t ) ，相
關係數(correlations)， corrt [T ]  corr[ rt (t  T ), rt (t )]


corr ,t [T ]  corr[ rt (t  T ) , rt (t ) ]
報酬序列 r ( t ) 非高斯分佈，巨額報酬往往集中，集聚的波動係數。
在高頻資料，高流動市場在大於 15 分鐘其報酬平均幾乎為零。
低頻資料(星期，月)少量的自我正相關係數。
不管Δt 為多少，畫出的分怖圖都很相似。此為長尾巴或稱厚尾巴現象。
Pr( r )
r  如果   2 ,
Loretan & Phillips (1994): Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance,
1, 211-248.
Exchange rates & Stock indices have tail indices in range, 2.4-3.8
Müller, U.A., Dacorogna, M.M., Olsen, R.B., Pictet, O.V. (1998) Heavy Tails in High-Frequency Financial Data. In A
Practical Guide to Heavy Tails: Statistical Techniques and Application. Editors: Adler, R.J., Feldman, R.E. & Taqqu,
M.S., Birkhäuser, US.
Stanley et al. (1999), Scaling of the distribution of price fluctuations of individual companies [Phys. Rev E60,
6519-6529]:
For timescales, 5 min to 16 days, tail index of shares is about 3 (2.8 using Hill).
Stanley et al. (1999), Scaling of the distribution of fluctuations of financial market indices [Phys. Rev E60, 5305-5316]:
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一致性「特徵事實(stylized facts)」
由於金融時間序列，例如股價，匯率，利率等，往往因為金融商品，市場，時間研究
然而有些統計特徵是不因金融商品，市場，時間研究而改變的，我們稱這些特徵為「特徵事實(stylized facts)」。
報酬序列
酬序列
rt (t ) (線性)自我相關係數並不顯著，在短時距t20 分鐘左右由於市場微結構效應而趨於顯著。報
rt (t ) 往往具有冪次律分布。其指數通常大於 2 而小於 5。因此變異量無窮大的穩定分布與常態分布無
法描述。pt 序列具有較多的連續下跌而連續上漲的較少。報酬序列
不管任何時距t 下，報酬序列
rt (t ) 在累積長時距t 下顯示接近常態分布。
rt (t ) 均呈現間歇性的高變異量。同時變異量呈現顯著的自我相關係數，可以延
伸到數天。因此我們會說高變異量往往有群聚現象。在經由類似 GARCH 的變異量模型修改後的報酬序列依然
呈現冪次律尾部分布。然而其指數較沒修改前大。絕對值報酬序列
| rt (t ) | 的自我相關係數隨著時距而呈冪次
律遞減，其指數通常介於 0.2 與 0.4 間。大部份的變異量指標均與報酬值呈負相關，此稱為槓桿效應。成交量與
大部份的變異量指標呈正相關。長時距的變異量指標預測短時距的變異量指標往往比用短時距來預測長時距的
變異量指標來的好。
# Cont, Rama (2001) "Empirical properties of asset returns: stylized facts and statistical issues," Qunatitative Finance, 1,
223-236. http://www-stat.wharton.upenn.edu/~steele/Resources/FTSResources/StylizedFacts/Cont2001.pdf
# Malmsten, H. and Terasvirta, T. (2004) "Stylized Facts of Financial Time Series and Three Popluar Models of
Volatility,
SSE/EFI
working
paper.
http://www-stat.wharton.upenn.edu/~steele/Resources/FTSResources/StylizedFacts/MalmstenTerasvirta04.pdf
# Rydberg, T. (2005) "Realistic Statistical Modeling of Financial Data," Technical Report, Nuffield College, Oxford,
UK. http://www-stat.wharton.upenn.edu/~steele/Resources/FTSResources/StylizedFacts/Rydberg.pdf
THE TAIL-FATNESS OF FX RETURNS RECONSIDERED http://arno.unimaas.nl/show.cgi?fid=3084
Empirical properties of asset returns: stylized facts and ...
http://www.cmap.polytechnique.fr/~rama/papers/empirical.pdf
Stylized facts about the distribution of asset returns has been well documented in Bollerslev, Engle and Nelson (1994,
Handbook of Econometrics IV), Ghysels, Harvey and Renault (1996, Handbook of Statistics 14), and others. They
include:
尖峭態峰度分布(LEPTOKURTOSIS)
1.: It has been long observed that asset returns follow a distribution which is far from normal, in particular one that
exhibits a substantial degree of excess kurtosis --Fama (1965, Journal of Business). Merton (1976, Journal of Financial
Economics), among others, notes that mixtures of normal distributions exhibit fat tails relative to the normal, and
therefore models that result in such distributions can be used in order to improve on the BS option pricing results.
2. CLUSTERING: ARCH and stochastic volatility models have been used in the literature to mimic volatility
clustering --Engle (1982, Econometrica), Ghysels, Harvey and Renault (1996) and the references therein. Financial time
series exhibit periods where the volatility is consistently low that alternate with periods of consistently high volatility.
This variation of volatility can be linked to the arrivals of information, and high trading volume --see Mandelbrot and
Taylor (1967, Operations Research), and Karpoff (1987, Journal of Financial and Quantitative Analysis), inter alia. One
can argue that trading does not take place in a uniform fashion across time: new information will result in a more dense
trading pattern with higher trading volumes, which in turn result in higher volatilities.
3. LEVERAGE EFFECTS: Black (1972, Journal of Business) suggests that volatilities and asset returns are
negatively correlated, naming this phenomenon the leverage effect. Falling stock prices imply an increased leverage on
firms, which is presumed by agents to entail more uncertainty, and therefore volatility. This is also referred to in the
literature as the Fisher-Black effect.
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4. LONG MEMORY: Although volatility seems to follow a cyclical pattern, there seems to be a very high degree of
persistency, usually modeled through an IGARCH or a FIGARCH specification--see Baillie, Bollerslev and Mikkelsen
(1993, Journal of Econometrics).
5. VOLATILITY SMILE: Many of the above stylized facts are visualized in the literature through the volatility
smile. The features of the smile are well documented in the literature and include the following:
1. The $ \cup$-shaped relationship between the implied volatility and the moneyness level, with a minimum
around-the-money, although ''smirks'' and ''frowns'' are also encountered --see for example Marsh and Kobayashi (1998,
Univ of Tokyo). This is usually attributed to the fat tails of the returns of the underlying asset.
2. The volatility smile is often symmetric, although documented asymmetries might exist due to the leverage
effects which result into returns with negative skewness--see for example Heston (1993, Review of Financial Studies),
or liquidity issues since the more expensive contracts --which are far in-the-money-- are documented to be the least
liquid ones.
3. The amplitude of the smile decreases with time to maturity. Short maturity options tend to exhibit more
acute volatility smiles, that tend to die out for long maturity contracts.
尾部現象(tail behavior)
量尺現象(scaling behavior)
自我相關函數(autocorrelation functions)
長時記憶自我相關函數(long memory autocorrelation function)
泰勒效應 報酬絕對值具有序列相關性，間隔值，比報酬平方值的序列相關性還高。
A Multifractal Model of Asset Returns http://finance.sauder.ubc.ca/~fisher/pspaps/onlytf/mmar1t.ps.
與計量經濟不同的是，
傳統計量經濟中的 AR 與 ARMA 模型無法充分描述這種隨機過程，
所以另外一些變異量模型例如 ARCH(autoregressive conditional heteroscedasticity) 與 GARCH 模型才發展
Is
it
really
long
memory
we
see
http://www-stat.wharton.upenn.edu/~steele/HoldingPen/StaricaLongRangeQ.pdf
in
nancial
returns?
http://www.cmap.polytechnique.fr/~rama/papers/empirical.pdf
除了由實際金融資料中發掘一致性事實(stylized facts)外，也有使用模型與模擬來探討這些發生的可能原因。
Agent-based
Financial
Markets:
http://www.econ.ku.dk/okokj/colanderconf/style.pdf
Matching
Stylized
Facts
With
Style
Michael Kirchler, and Jürgen Huber, Testing for stylized facts in experimental financial markets
(http://www.google.com.tw/url?sa=t&ct=res&cd=14&url=http%3A//www.essex.ac.uk/wehia05/Paper/Parallel4/Session
3/KirchlerM.pdf&ei=NHb2QuCEHJfuYJSaxMIJ)
Stylized
models
of
financial
markets
and
the
stylized
facts
http://www.google.com.tw/url?sa=t&ct=res&cd=24&url=http%3A//www.ictp.trieste.it/%7Emarsili/Minority.pdf&ei=33
r2Qu6EHLTOYbfU4LoK
波動係數群聚現象
波動係數為正。
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隨著時距加大而呈冪次衰退。


corr ,t [T ]  corr[ rt (t  T ) , rt (t ) ]

A
,   1, 2,   [0.2,0.4]
T
在任何的時距尺度，報酬序列呈現不規則的暴起暴落。
成交量與波動係數相關性強。
A Theory of Large Fluctuations in Stock Market Activity (http://econ-www.mit.edu/faculty/download_pdf.php?id=237)
此種研究方法又可視為市場現象學(market phenomenology)。
“Social scientists for the most part don’t seem to have learned that the theory is always required to fit the data, and that
it is an incorrect procedure that data should be made fit the theory…As a class social scientists have never caught on to
this. As a result they very often won’t even undertake an investigation and collect data unless they have some sort of a
theory or model to fit the data to.”
Osborne, M.F.M. (1977), The Stock Market and Finance from a Physicist’s Viewpoint, p. 19.
“It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of
theories to suit facts.”
Sherlock Holmes (or A. Conan Doyle) , A Scandal in Bohemia.
(http://love.theponytail.net/index.php?option=com_content&task=category&sectionid=3&id=69&Itemid=47)
Levy processes driven by stochastic volatility
Option pricing using the fractional FFT
Non-affine option pricing
Volatility persistence, regime switches, jumps and option pricing
Stochastic volatility and jumps driven by continuous time Markov chains
Bachelier, Louis, 1900, Theory of speculation, reprinted in P. Cootner (ed.), The Random Character of Stock Market
Prices, MIT Press: Cambridge, MA
財富與收入的分佈
在 1897 年 Vilfredo Pareto 在研究十九世紀英國財富與收入分布時
發現
他也注意到這現象並不局限於某段時間或某一國家，而是具有普遍性的。
發現歐洲的收入與財富分佈呈現冪次律現象。
 x 
Pr[ X  x ]   
 xm 
x  xm , k>0
k
27/70
p( x )  k
pdf
xmk
for x  xm
x k 1
p( x | k , xm ) 
kxmk
x k 1
x 
cdf F ( x | k , xm )  1   m 
 x 
k
Pareto 分布是連續值
離散值 Zipf 或 zeta 分布
[x]
期望值 為
xm k
k 1
如果 k1, 期望值為無窮大。
中間值為
xm k 2
標準差為
xm
k
k 1 k  2
如果 k2, 標準差為無窮大。
k>3 時，偏差量為
2(1  k ) k  2
k 3
k
k>4 時，kurtosis
6(k 3  k 2  6k  2)
k (k  3)(k  4)
Entropy
 k  1
ln     1
 xm  k
當 k>n 時，其各次矩
n 
kxmn
k n
特徵函數為
 (t )  k ( ixmt )k ( k , ixmt )
(a,x)為非完備珈瑪函數(incomplete Gamma function)
另外，與 exponential distribution f(x|k)的關係為
p( x | k , xm )  f (ln( x / xm ) | k )
而 Dirac delta function 為 Pareto distribution 的極限
lim p( x | k , xm )   ( x  xm )
k 
Lorenz curve
Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical
Association. 9: 209-219.
28/70
羅倫茲曲線(Lorenz curve)是由 pdf(p(x)) 或 cdf(F(x))
L(F)=
x( F )

L( F ) 

xm

xm
xp( x )dx



F
0
1
xp( x )dx
0
x ( F )dF 
x ( F )dF 
x(F)為 CDF 的反函數
Pareto distribution
x( F ) 
xm
(1  F )1 k
因此羅倫茲曲線為
L( F )  1  (1  F )11 k , k1
Gini 係數量測羅倫茲曲線與 k=平均分布的差異，
1
G  1  2  F ( F )dF 
0
1
2k  1
如果有一組樣本(x1,x2,x3,…,xn)，如何估計 Pareto 分布的參數
疑似度函數(Likelihood function)
n
L(k , xm )   k
i 1
n
xmk
1
n nk

k
x
m  k 1
k 1
xi
i 1 xi
因此，取對數後的疑似度函數
n
( k , xm )  n ln k  nk ln xm  ( k  1)  ln xi
i 1
由於
(k , xm ) 為 xm 的單調遞增函數，因此，xm 越大，對數疑似度越高。
所以我們取 xˆm  min xi
i
要找到 k 的估計值，對
(k , xm ) 取偏微分並將結果設為零。
n
 ( k , xm ) n
  n ln xm   ln xi  0
k
k
i 1
所以得到
xˆm  min xi ， kˆ 
i
n

n
i 1
(ln xi  ln xˆm )
Statistical
Mechanics
of
Money,
http://online.itp.ucsb.edu/online/colloq/yakovenko1/pdf/Yakovenko.pdf
Income,
and
Wealth
Statistical Mechanics of Money, Income, and Wealth: A Short Survey
http://www.google.com.tw/url?sa=t&ct=res&cd=186&url=http%3A//www.nda.ac.jp/cs/AI/wehia04/papers/25VictorYak
ovenko/25StatMech-money.pdf&ei=IQnzQv3oEcmsYfK5ifIP
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Statistical mechanics of money
http://www2.physics.umd.edu/~yakovenk/papers/EPJB-17-723-2000.pdf
The New Theories of Economics
Vilfredo Pareto The Journal of Political Economy, Vol. 5, No. 4. (Sep., 1897), pp. 485-502. Stable URL:
Division
Rules,
Network
Formation,
and
the
Evolution
of
Wealth
http://www.google.com.tw/url?sa=t&ct=res&cd=304&url=http%3A//www.fiu.edu/orgs/economics/wp2005/05-05.pdf&
ei=XArzQp-JHsa6YPiTxfsP
Pareto
law
in
a
Minority
Game
TODA,
Koji
and
http://www.essex.ac.uk/wehia05/Abstract/Parallel6/Session3/NakamuraY.pdf
NAKAMURA,
Yasuyuki
...
Wealth condensation in a simple model of economy http://hussonet.free.fr/wealth.pdf
http://www.ucd.ie/statdept/shanewhelan/phynance.ppt
金融市場模型
自從 1900 年 Louis Bachelier 的 Theory of Speculation
1929 年 Charles Douglas Statistical Groundwork for Investment Policy.
1930 年 Alastair Murray The Compilation of Index Numbers and Yield Statistics relative to Stock Exchange Securities
1944 年 John von Neumann & Oskar Morgenstern: Theory of Games and Economic Behaviour.
1953 年 Maurice Kendall: The Analysis of Time Series, Part I: Prices.
1959 年 M.F.M Osborne Brownian Motion in the Stock Market
1963 年 Benoit Mandelbrot The Variation of certain Speculative Prices
1973 年 Fischer Black & Myron Scholes: The Pricing of Option Contracts and Corporate Liabilities.
Robert Merton: Theory of Rational Option Pricing.
金融時間序列有一些表徵事實是為人所周知的。
高斯的世界
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李維的世界
1963 年 Mandelbrot 建議使用 Lévy 穩定分布來描述
An introduction to stable distributions http://academic2.american.edu/~jpnolan/stable/chap1.pdf
mle.ps http://academic2.american.edu/~jpnolan/stable/mle.ps
http://141.20.100.9/papers/pdf/SFB649DP-2005-008.pdf
穩定分布是指若 IID 隨機變數 X，Y 是屬於一類的機率分布，其和 X+Y 也是屬於同一類的機率分布。具有此封
閉性的機率分布稱為穩定機率分布。
法國數學家 Paul Pierre Levy 提出機率分布成為穩定分布的條件。
這些通常稱為 Lévy 穩定分布，或稱穩定分布。
矩生成函數(moment generating function)
http://www2.sjsu.edu/faculty/watkins/genito.htm
一機率分布的密度函數(probability density function) 為 f ( x ) 時，其矩生成函數
M f ( z )   exp(izx) f ( x )dx
矩生成函數也稱為特徵函數(characteristic function)，其實為機率密度函數經傅立葉轉換(Fourier transformation)後
的函數。
f ( x) 
1
2



 (t )eitx dt
ln  (t )  it   | t | {1  i  sgn(t ) tan( 2 )} ,
2
當=1 時 tan( 
2 ) 用   log | t | 取代。
均值為變異量為2 的常態分布，
ln M ( z )  i  z  12  2
如果隨機變數 X，Y 的矩生成函數為 MX(z)，MY(z)，那麼 X+Y 的矩生成函數為 MX(z) MY(z)。
(http://www.xplore-stat.de/tutorials/stfhtmlnode6.html)
Levy 穩定分布的矩生成函數為
 i z    | z | {1  i  sgn( z ) tan( 2 )}   1
ln M ( z )  
2
i z   | z |{1  i  sgn( z )  ln( | z |)}   1
Nolan (1997)提出另一公式
i z    | z | {1  i  sgn( z ) tan( 2 )[( | z |)1  1]}   1
ln M ( z )  
i z   | z |{1  i  sgn( z ) 2 ln( | z |)}
 1

參數是機率分布的峰值指標，也是尾部指數，通常大於 0 小於等於 2，(0,2) 。常態分布的值為=2。參數
是機率分布的偏度指標，通常介於-1 與 1 之間， [-1,1] 。如果=1，參數就必須為 0。
參數表示尺度，其值必為正。位置參數或位移參數R。
常態分布=2，=0，=2/2，均值為。如果 <2，其變異量為無窮大。
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lim x Pr[ X  x ]  C (1   ) 
x 
lim x Pr[ X   x ]  C (1   ) 
x 


C  2 x  sin( x )dx
0

1
 1 ( )sin 2
至於非常態分布的 Lévy 穩定分布，舉柯西分布(Cauchy distribution)為例，=1，=0，
f ( x) 
1
[ (1  x 2 )]
t 個 IID，均屬於 Lévy 穩定分布，參數為，，， 的隨機變數，
其和也是 Lévy 穩定分布，而參數成為，，t1/，均值則為 t1/。
P. Bak, M. Paczuski and M. Shubik (1997), Price variations in a stock market with many agents. Physica A 246,
430–453.
Per Bak, Simon F. Nørrelykke and Martin Shubik (1999), Dynamics of money. Phys. Rev. E 60, 2528–2532.
J. Doyne Farmer (1998/2002), Market force, ecology and evolution. Santa Fe Institute Working Paper 98-12-117 / Ind.
Corp. Change 11, 895–953.
George Soros on theoretical economics
“Existing theories about the behavior of stock prices are remarkably inadequate. They are of so little value to the
practitioner that I am not even fully familiar with them. The fact that I could get by without them speaks for itself.”
G. Soros, “Alchemy of Finance” 1994
Applying
Physics
Methods
to
Economics
(http://www.cmth.bnl.gov/~maslov/new_econophysics_colloquium.ppt)
James Feigenbaum (2003), Financial physics. Rep. Prog. Phys. 66, 1611–1649.
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and
Finance
寡勝博奕(minority game)
寡勝博奕是一具有 N(通常是奇數)參與者反複的博奕。每一次每位參與者要選擇是兩個狀態的其中之一，可以用
0/1 表示。選擇成為少數的那一方贏。這個博奕是由 El-Farol’s 酒吧問題抽象化而來。由於酒吧容量只有 60 人
而有 100 人想去，因此每天每人依照過去酒吧出現的人數決定要不要去酒吧。
這是每位參與者只有部份資訊與有限理性的博奕。
首先介紹與定義寡勝博奕
[] Damien Challet and Yi-Cheng Zhang (1997), Emergence of cooperation and organization in an evolutionary game.
Physica A 246, 407–418.
Damien Challet, Matteo Marsili and Yi-Cheng Zhang (2000), Modeling market mechanism with minority game.
Physica A 276, 284–315.
[] El-Farol’s bar problem (BrianW. Arthur, Am. Econ. Assoc. Papers and Proc 84, 406, (1994)):
W. Brian Arthur (1994), Inductive Reasoning and Bounded Rationality. Am. Econ. Rev. 84,
406–411.( http://www.santafe.edu/arthur/Papers/Pdf_files/El_Farol.pdf)
Damien Challet, M. Marsili and Gabriele Ottino (2004), Shedding light on El Farol. Physica A 332, 469–482.
http://www.ictp.trieste.it/~marsili/Minority.pdf
http://www.unifr.ch/phystheo/3emecycleMatteo12.pdf
Econophysics:
Unified
Game
Theory
Approach
http://www.google.com.tw/url?sa=t&ct=res&cd=137&url=http%3A//www.todo1services.com/pdfs/WP-CRM19.pdf&ei
=MFv3QusevoJh6M-M-Ak
Credit Networks and Agent Games
Theory
of
Minority
Games
http://www.google.com.tw/url?sa=U&start=163&q=http://www.unifr.ch/phystheo/3emecycleMatteo12.pdf&e=747
http://lagash.dft.unipa.it/~micciche/PAPERPA1.ps
Introduction: 100 years of Brownian motion http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/387.pdf
International Finance, Lévy Distributions, and the Econophysics of ...
http://www.angelfire.com/id/SergioDaSilva/intfinance.pdf
Research in Econophysics http://www2.physics.umd.edu/~yakovenk/papers/condmat-0302270.pdf
(http://www.moneyscience.org/tiki/tiki-index.php?page=Econophysics+Hub)
(http://www.econ.iastate.edu/tesfatsi/CompFinance.Tsang.pdf)
Detection
of
financial
crisis
by
methods
of
multifractal
(http://wwwmayr.informatik.tu-muenchen.de/konferenzen/Jass04/courses/2/Talks/Agaev.ppt)
analysis
r.N. Mantegna and H.E. Stanley: "Scaling Approach to Finance", Cambridge University Press, Cambridge, UK, in press
# L.A.N. Amaral, S.V. Buldyrev, S. Havlin, M.A. Salinger, and H.E. Stanley: "Power Law for a System of Interacting
Units with Complex Internal Structure", Phys. Rev. Lett. 80 (1998) 1385-1388
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(http://www.javeriana.edu.co/universitas_scientiarum/vol6n2/ART5.htm)
The word “econophysics” was introduced by H.E.Stanley to describe the large number of papers written by
physicists in the last ten years on problem of (stock) market, the growth of companies and and related economic
questions(J.P.Bouchaud and M.Potters, 2000; R.N.Mantegna and H.E.Stanley,2000; H.Levy et al.,2000). The first
econophysics model published by physicists in
physics journals
were those published by Mantegna
(R.N.Mantegna,1991), but clearly physicists did not bring the physical methods to the economic science. For example, a
Monte Carlo simulation of a market was already published in 1964 by Stigler (G.J.Stigler,1964) from the Chicago
economics school and economy Nobel laureate Markowitz (G.W.Kim and H.M.Markowitz,1989) published with Kim a
model for the 1987 crash on Wall Street with two types of investors similar to many later models of physicists;
economist Lux (T.Lux,1996) has cited the work of physicists like Haken; and some other articles were published in
between.Thus, the question in the title of the section can be answered with a clear “no”. The field is not new. What
physicists did was to enlarge the number of people using these methods, to get better data, or to use very specific
physics results less known in economics.
The econophysicist D.Stauffer compares in one of his papers “econophysics” with the “discovery” of
America by Columbus half
a millennium ago: Other people came thousands of years earlier from Asia, and the
Normans settled for some time in Vinland, now in Northern Newfoundland. But none of them informed about these
findings a more widespread medium, while the voyage of Columbus really changed life in both America and Europe. In
this sense
the econophysicists are like Columbus, not really knowing they are doing but nevertheless doing
something important.
G.Soros pointed out in 1994 in his “Alchemy of Finance” the inadequacy and the inefficiency of the
existing theories about the behavior of stock prices. Until the last decade the theoretical economics was dominated by
pure mathematics characterized by ridiculous lemma/theorem style, little effort to compare theoretical predictions to
“experiment” (say, prices from real stock markets) and the fact that bulk of papers are inaccessible and of no interest
to “experimentalists” - practitioners of the field. However the pure mathematics has contributed to economics through
the Game Theory approach (the concept of Nash equilibrium where no player can improve on his/her strategy and the
supposition of the perfect rationality of all players) and the phenomenology of stock price fluctuations that are
postuled to be Gaussian and subject to the Efficient Market Hypothesis: all correlations are arbitraged away. There
are many observations in disagreement with these suppositions: First, short term fluctuations are non-Gaussian; The
second, price increments are correlated (the magnitude of price fluctuations has long temporal correlations); The third,
strategies used by trades are correlated as manifested by herd effect.
From this point physics is called to bring its contributions. The physicists can work with empirical data and construct
phenomenological theories. Also, statistical physics field has useful approaches to deal with collective dynamics
composed of many interacting parts.
As the traditional physics, econophysics can be divided into experimental and theoretical, the first trying to analyze
real data from real markets and to make sense of them, the second trying to find microscopic models which give for
some quantities good agreement with the experimental facts. In the last years econophysics has matured enough to
allow some applications, their field being called econo-engineering: the financial applications want to advise banks and
brokers how to estimate risks and demand proper fees to balance these risks.
From the large field of econophysics, we have selected for
present paper only the problem of financial crashes
modelling. In the next section we describe briefly the analogies between these crashes and the thermodynamic phase
transitions.
Starting from here, in Section 3 we propose a simple and suggestive model to explain the critical points arising in the
stock market behavior. The last section draws some conclusions.
D. Heymann ; R. Perazzo ; A. Schuschny;
"Transitions Between Regimes in variants of the Bar Attendance Model", Preprint, , pp. , [2001].
(http://www.nld.df.uba.ar//paperRP/Bam_jee.pdf)
D. Heymann ; R. Perazzo ; A. Schuschny;
"Price Setting in a Schematic Model of Inductive Learning", Preprint, , pp. , [2001].
(http://www.nld.df.uba.ar//paperRP/Proveed.PDF)
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BLACK-SCHOLES… WHAT’S NEXT?( http://www.mathfin.com/nicolas/Q35.pdf)
Frank Hahn (1991), The Next Hundred Years. Econ. J. 101, 47–50. ()
Moshe Levy, Haim Levy and Sorin Solomon (1995), Microscopic Simulation of the Stock Market: the Effect of
Microscopic Diversity. J. Phys. I France 5, 1087–1107.
An Investigation into the Dynamics of Correlation Networks in the ... http://www.maths.ox.ac.uk/~mcdonal4/Thesis.pdf
http://sip.clarku.edu/3e/
Essays on Asset Return Distributions http://www.int9.com/download/isbn9512262541.pdf
Credit Networks and Agent Games http://163.1.148.210/~buttle/thesis.pdf
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複雜網絡
龐雜網路所謂複雜網絡，
(http://complex.upf.es/~ricard/complexnets.html)
(http://cscs.umich.edu/~crshalizi/notebooks/complex-networks.html)
(http://www.nd.edu/~networks/)
http://www.phys.psu.edu/~ralbert/phys597_05/chapter4.pdf
傳統網路模型主要分成兩大類，具有規律性的規則網路與看似雜亂的隨機網路。前者描述晶體格子般的網路架
構，通常用於連繫平行電腦中的微處理機。後者則是由所探討，而在早期使用於網際網路。
Defining
and
detecting
emergence
in
complex
http://www.per.marine.csiro.au/staff/Fabio.Boschetti/3054CO/papers/emergence_kes_final.pdf
Communication
and
Self-Organization
in
Basic ...(http://summa.physik.hu-berlin.de/~frank/download/web-wien.pdf)
Complex
networks
Systems:
Aldous,
D.J.,
2005.
A
STOCHASTIC
COMPLEX
NETWORK
(http://www.ams.org/era/2003-09-19/S1079-6762-03-00123-9/S1079-6762-03-00123-9.pdf)
A
MODEL
Costa, L. da F., F.A. Rodrigues, G. Travieso, and P.R. Villas Boas, 2005. Characterization of Complex Networks: A
Survey of Measurements. (http://arxiv.org/PS_cache/cond-mat/pdf/0505/0505185.pdf, 39 pp.)
Is
all
the
world
a
complex
network?
(http://www.open.ac.uk/socialsciences/staff/gthompson/a%20complex%20network.pdf)
[http://reality.media.mit.edu/dataset.php]
http://complexityworkshop.com/sun/PLaw/
首先探討在小世界網路模型中流行病傳播現象。
M. Kuperman, G. Abramson. "Small World Effect in an Epidemiological Model". Physical Review Letters 86, 13,
2909-2912. (2001)
資訊流
Emergence
of
complex
dynamics
in
a
simple
model
of
signaling
networks
(http://amaral.chem-eng.northwestern.edu/Publications/Papers/Amaral-2004-Proc.Natl.Acad.Sci.U.S.A.-101-15551.pdf)
複雜動態網路中的同步現象
Factors
that
predict
better
synchronizability
on
complex
networks
(http://nlsc.ustc.edu.cn/BJKim/PAPER/PhysRevE_69_067105%20Factors%20that%20predict%20better%20synchroniz
ability%20on%20complex%20networks.pdf)
Neural
Synchrony
and
the
Unity
of
Neurophenomenological ...( http://www.yorku.ca/evant/FV&ETNeuralSynchrony.pdf)
Mind:
A
Immunization of complex networks (http://www-fen.upc.es/~romu/Papers/immuno.pdf)
Modeling
Terrorist
Networks
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Complex
Systems
and
First
Principles ...(http://www.snhu.edu/img/assets/3655/Modeling_Terrorist_Networks_Fellman_Sawyer_and_Wright.doc)
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使用七個項目來看複雜系統
基本個體構成單位組成體(agent)、異質性(heterogeneity) 、組織(organization) 、適應(adaptation) 、回饋(feedback)
動態(dynamics) 、突現(emergence)
在經濟體人群時疫流行病學免疫學金融生態健保
與複雜網絡分析有關的有 瀰(memetics) 是指研究瀰的社會與文化效應。
Propagation of Innovations in Networked Groups http://cognitrn.psych.indiana.edu/rgoldsto/pdfs/mason05.pdf
Weighted networks: analysis, modeling A. Barrat, LPT, Université ...
http://qcd.th.u-psud.fr/page_perso/Barrat/Intro_Complex_Networks.ppt
The architecture of complex weighted networks, Proc Natl Acad Sci U S A > v.101(11); Mar 16, 2004
(http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=374315&blobtype=pdf)
Papers adopting a complex system approach to analyze observed system
Gell-Mann, M. (1992) Complexity and Complex Adaptive Systems. In Hawkins, John A. and Murray Gell-Mann,
editors, The Evolution of Human Languages. Reading, MA: Addison-Wesley.
Modelling
Peer-to-Peer
Data
Networks
(http://infolab.usc.edu/DocsDemos/DNIS2005.pdf)
under
Complex
System
Theory.
Joshua R. Tyler (HP Labs) and John C. Tang, When Can I Expect an Email Response? A Study of Rhythms in Email
Usage, (http://www.hpl.hp.com/research/idl/papers/rhythms/ECSCWFinal.pdf)
Josh Tyler, Dennis Wilkinson, and Bernardo A. HubermanEmail as Spectroscopy: Automated Discovery of Community
Structure within Organizations (http://www.hpl.hp.com/research/idl/papers/email/index.html)
Vladimir Gudkov, Joseph E. Johnson, Network as a Complex System: Information Flow Analysis
(http://arxiv.org/pdf/nlin.CD/0110008)
Modeling the Internet as a Complex System(http://www.icir.org/floyd/talks/E2E-Jan03.pdf)
http://www.hakank.org/webblogg/archives/cat_social_network_analysiscomplex_networks.html
Life as a Complex Systems (http://www.isi.it/files/2004/Kaneko.pdf)
The Earth as a complex system(http://www.igbp.kva.se/congress/abstracts/Ghil_congress_abstract.pdf)
The Magnetosphere as a Complex System (http://fisica.ciencias.uchile.cl/~jrogan/investigacion/papers/paperJAV05.pdf)
The Emergency Department as a Complex System (http://necsi.org/projects/yaneer/emergencydeptcx.pdf)
Highway
Car
Traffic
as
a
Complex
(http://vmsstreamer1.fnal.gov/VMS_Site_03/Lectures/Colloquium/presentations/050420Boccara.pdf)
System
CONSTRUCTION AS A COMPLEX SYSTEM(http://strobos.cee.vt.edu/IGLC11/PDF%20Files/02.pdf)
Verification Methodology for a Complex System-on-a-Chip (http://magazine.fujitsu.com/us/vol36-1/paper05.pdf)
D. C. Mikulecky (http://www.people.vcu.edu/~mikuleck/ON%20COMPLEXITY.html)
An Introduction to Complexity in Social Science
Bernard Pavard and Julie Dugdale (http://www.irit.fr/COSI/training/complexity-tutorial/complexity-tutorial.htm)
The Complexity & Artificial Life Research Concept
for Self-Organizing Systems(http://www.calresco.org/index.htm)
PNAS May 14, 2002; 99 (Suppl. 3) Adaptive Agents, Intelligence, and Emergent Human Organization: Capturing
Complexity through Agent-Based Modeling (http://www.pnas.org/content/vol99/suppl_3/)
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http://www3.interscience.wiley.com/cgi-bin/jhome/38804
http://www.complexity.org.au/
J. P. Crutchfield, The Calculi of Emergence: Computation, Dynamics, and Induction, Physica D 75 (1994) 11-54. (Santa
Fe Institute Working Paper 94-03-016.) (http://www.santafe.edu/projects/CompMech/papers/CalcEmergTitlePage.html)
J.P. Crutchfield and D. P. Feldman Regularities Unseen, Randomness Observed: Levels of Entropy Convergence.
Submitted to Chaos, 2003. 15: 25-54. 2003. (http://arxiv.org/abs/cond-mat/0102181)
(http://hornacek.coa.edu/dave/Publications/ruro.pdf)
Analyzing Molecular Networks(http://www.thep.lu.se/pub/Preprints/04/lu_tp_04_15.pdf)
When information is viewed on a two-dimensional document, it is considered a projection from multi-dimensional
space to this viewing space.
How many web pages are there now?
What is the average distance between two web pages?
http://cyvision.if.sc.usp.br/~francisco/networks/theses.htm
The traditional algorithm computes
What are the common properties between those networks?
Power Laws
http://www-personal.umich.edu/~mejn/courses/2004/cscs535/
http://web.media.mit.edu/~tanzeem/cohn/CoHN.htm
Barabási, Albert-László (2002) Linked: The New Science
Cambridge, MA: Perseus Publishing.
Buchanan, Mark (2002) Small World: Uncovering Nature’s Hidden
London: Weidenfeld & Nicolson.
Taylor, Mark C. (2001) The Moment of Complexity: Emerging Network
Chicago, IL, and London: University of Chicago Press.
Urry, John (2003) Global Complexity, Cambridge: Polity Press.
Yaneer Bar-Yam, Dynamics of Complex Systems, ISBN 0-201-55748-7
The Advanced Book Studies in Nonlinearity series[http://necsi.org/publications/dcs/index.html]
網路流量的尺度現象
Scaling Phenomena in Network Traffic
Week 1
W. Leland, M. Taqqu, W. Willinger, D. Wilson. "On the Self-Similar Nature of Ethernet Traffic (Extended Version),"
IEEE/ACM Transactions on Networking, 2(1):1-15, February 1994. (Presenter D. Towsley.)
V. Paxson, S. Floyd. "Wide-Area Traffic: The Failure of Poisson Modeling," IEEE/ACM Transactions on Networking,
3(3):226-244, June 1995. (Presenter T. Bu)
Week 2
M.E. Crovella, A. Bestavros. "Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes," IEEE/ACM
Transactions on Networking, 5(6):835--846, December 1997. (presenter B. Liu)
A. Erramilli, O. Narayan, W. Willinger. "Experimental Queuing Analysis with Long-Range Dependent Packet Traffic,"
IEEE/ACM Transactions on Networking, 4(2):209-223, April 1996. (Presenter J. Padhye)
Some other papers that might be of interest related to the Crovella, Bestavros paper include.
P. Barford, A. Bestavros, A. Bradley, and M. E. Crovella, "Changes in Web Client Access Patterns: Characteristics and
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Caching Implications," in World Wide Web, Special Issue on Characterization and Performance Evaluation, Vol. 2, pp.
15-28, 1999. This paper updates the study by examining web traffic characteristics in 1998. Briefly, distributions of
object sizes remain heavy-tailed but with different parameter values.
K. Park, G. Kim, M. E. Crovella, "On the Effect of Traffic Self-similarity on Network Performance," Proceedings of the
SPIE International Conference on Performance and Control of Network Systems, November, 1997. This paper
examines the effects that different transport protocols can have on traffic characteristics when fetching WWW objects
whose lengths are characterized by a heavy tail distribution. In the case of TCP, the congestion control algorithm has the
effect of reducing the long range dependence. Nevertheless, simulations show the resulting network traffic to still be
LRD.
Week 3
B.K. Ryu, A. Elwalid. "The Importance of Long-range Dependence of VBR Video Traffic in ATM Traffic Engineering:
Myths and realities," ACM Computer Communication Review, 26:3-14, Oct. 1996. (presenter Z. Koren)
M. Grossglauser, J. Bolot. "On the Relevance of Long Range Dependence in Network Traffic," IEEE/ACM
Transactions on Networking, 1998. (Presenter D. Figueiredo.)
Week 4
P. Abry, D. Veitch. "Wavelet Analysis of Long-Range Dependence Traffic," IEEE Transactions on Information Theory,
44(1):2-15, January, 1998. (Presenter V. Misra.)
V. Misra, W. Gong. "A Hierarchical Model for Teletraffic", Proceedings of 37th IEEE Conference of Decision and
Control, December 1998, Tempa, Fl. (Presenter V. Misra)
Week 5
A. Feldmann, A. C. Gilbert, W. Willinger. "Dynamics of IP Traffic: A Study of the Role of Variability and the Impact of
Control,"ACM Computer Communication Review, Sept. 1999. (Presenter J. Shapiro)
T. Tuan, K. Park. "Multiple timescale congestion control for selfsimilar network traffic," Perfoormance Evalaution,
1999. (presenter B. Liu.)
Week 6
M. Krunz, A. Makowski. "Modeling video traffic using M/G/infinity input processes: A compromise between
Markovian and LRD models," IEEE JSAC 16(5):733-748, June 1888. (presenter P. Ji)
Z. Liu, P. Nain, D. Towsley, Z.-L. Zhang. "Asymptotic behavior of a multiplexer fed by a long-range dependent
process," (Presenter D. Towsley.)
Power Laws in Control
Week 7
J.M. Carlson, J. Doyle. "Highly optimized tolerance: A mechanism for power laws in designed systems,"Phys. Rev. E
60, 1412, (1999) (Presenter W. Gong.)
J.M. Carlson, J. Doyle. "Highly optimized tolerance and generalized source coding", to appear in Phys. Rev. Letters.
(presenter M. Adler)
J.M. Carlson, J. Doyle. "Highly optimized tolerance: Robustness and design" submitted to Phys. Rev. Letters.
Power Laws in Networks
Week 8
J. Chuang, M. Sirbu. "Pricing multicast communications: A cost-based approach," ( .ps.gz, .pdf) Proc. INET'98, 1998.
(Presenter J. Shapiro)
G. Phillips, S. Shenker, H. Tangmunarunkit. Scaling of multicast trees: comments on the Chuang-Sirbu scaling law,"
Proc. SIGCOMM'99, Sept. 1999.
Week 9
M. Faloutsos, P. Faloutsos, C. Faloutsos. "On power law relationships of the Internet topology," Proc. SIGCOMM'99,
Sept. 1999. (presenter D. Figueiredo.)
E. Zegura, K. Calvert, M.J. Donahoo. "A quantitative comparison of graph-based models for Internet
topology,"IEEE/ACM Trans. on Networking, 5(6), Dec. 1997. (presenter B. Wang.)
Small World Phenomena
Week 10
D. Watts, S. Strogatz. "Collective dynamics of small-world networks," Nature, 393:440-442, 1998. (Presenter R. Datta)
J. Kleinberg , R. Kumar, P. Raghavan, S. Rajagopalan, A.S. Tomkins. "The WEB as a graph: measurements, models,
and methods," Intntl. Conf. on Combinatorics and Computing, 1999. (presenter B. Wang.)
Week 11
D. Gibson, J. Kleinberg. "Inferring Web Communities from link topology," Proc. 9th ACM Conf. on Hypertext and
Hypermedia, 1998. (Presenter X. Zhang)
Some web sites that may be of interest in the context of small worlds, power laws in graphs, and graph models of the
web include
The IBM Clever project. http://www.almaden.ibm.com/cs/k53/clever.html
The Xerox Parc Internet Ecologies Area. http://www.parc.xerox.com/istl/groups/iea/
40/70
Complex Network
Luciano da F. Costa, Luis Diambra,
(http://arxiv.org/abs/cond-mat/0306530)
A Complex
Network
Approach to Topographical
Connections
(http://lsc.amss.ac.cn/~ljh/04LC.pdf)
Complex Network Phenomena in Telecommunication Systems (http://www.tinbergen.nl/discussionpapers/04118.pdf)
Pinning
a
Complex
Dynamical
Network
to
Its
Equilibrium
(http://www.ee.cityu.edu.hk/~gchen/pdf/T-CAS%20Pinning.pdf)
Coarse-graining
and
self-dissimilarity
networks(http://www.weizmann.ac.il/mcb/UriAlon/Papers/CoarseGraining.pdf)
of
complex
Enhancing complex-network synchronization (http://www.edpsciences.org/articles/epl/pdf/2005/03/epl8572.pdf)
Modular
Interdependency
in
Complex
(http://eprints.ecs.soton.ac.uk/10621/01/Watson_ALSIDH_micds_preprint.pdf)
Dynamical
Systems
Statics and Dynamics of Complex Network Systems: Supply Chain ...
(http://supernet.som.umass.edu/visuals/kekedissertationpresentation.pdf)
THE EMERGENCE OF COMPLEX NETWORK PATTERNS IN MUSIC ARTIST NETWORKS,
http://www.iua.upf.edu/mtg/ismir2004/review/CRFILES/paper253-d75171ad2cf28b117aa318ac816a3b49.pdf
Complex Network Metrology (http://www.liafa.jussieu.fr/~latapy/Publis/1_3000_Exploration/paper.pdf
A
STOCHASTIC
COMPLEX
NETWORK
MODEL
Introduction(http://www.ams.org/era/2003-09-19/S1079-6762-03-00123-9/S1079-6762-03-00123-9.pdf)
1.
Distributed Resource Brokering in Complex Network Environments
(http://www.mitre.org/news/events/tech03/briefings/intelligent_information/silvey.pdf)
Dynamics of Social Networks (http://www.itp.uni-bremen.de/complex/cplx10066.pdf)
Book purchase networks (http://www.orgnet.com/divided.html)
Social Software (Tracing the Evolution of Social Software)
http://www-106.ibm.com/developerworks/xml/library/x-wxxm23/
http://www.hakank.org/webblogg/archives/cat_social_network_analysiscomplex_networks.html
Functional
cartography
of
complex
metabolic
networks
(http://amaral.chem-eng.northwestern.edu/Publications/Papers/Guimera-2005-Nature-433-895.pdf)
A Hypergraph Model for the Yeast Protein Complex Network (http://www.hicomb.org/papers/HICOMB2004-04.pdf)
(http://reality.media.mit.edu/publications.php)
A
CRITIQUE
OF
DARWIN'S
(http://www.vision.net.au/~apaterson/science/darwin_critique1.htm)
(http://www.vision.net.au/~apaterson/science/darwin_critique2.htm)
THEORY
OF
EVOLUTION
Self-Organization and Irreducibly Complex Systems:
A Reply to Shanks and Joplin (http://www.arn.org/docs/behe/mb_selforganizationreplytoshanksandjoplin.htm)
Controversies in Meme Theory Nick Rose (http://jom-emit.cfpm.org/1998/vol2/rose_n.html)
(http://www.pandasthumb.org/pt-archives/000062.html)
HOW ANTI-EVOLUTIONISTS ABUSE MATHEMATICS
(http://www.math.jmu.edu/~rosenhjd/sewell.pdf)
(http://www.geocities.com/evolvedthinking/Behe.htm)
Chaos,
Complexity
and
Conflict
Major
Michael
(http://www.airpower.maxwell.af.mil/airchronicles/cc/Weeks.html)
R.
Irreducible Complexity and Multiscale Reductionism (http://shum.huji.ac.il/~sorin/rector/ired-complx.htm)
41/70
Weeks
The ecology of the connecticon by Frank Rennie and Robin Mason
(http://www.firstmonday.org/issues/issue8_8/rennie/)
(http://www.nationmaster.com/encyclopedia/List-of-important-publications-in-computer-science#Complexity)
The biochemistry: hemoglobin
Hemoglobin is a complex molecular machine. The elucidation of the detailed molecular mechanism of reversible
oxygen binding and its relationship with respiratory physiology is one of the few triumphs of reductionistic
biochemistry. Hemoglobin is made of 4 protein subunits, each one having a heme group and an oxygen binding site. 2
of the 4 subunits are identical and are denoted alpha and the other 2 are identical and are denoted beta. Alpha subunits
and beta subunits are very similar in three dimensional structure and also similar in amino acid sequence. An alpha
subunit and a beta subunit assemble together to form an alpha/beta dimer. Two alpha/beta dimers make up the intact
hemoglobin tetramer. There are two forms of hemoglobin: the deoxy form with no oxygen bound and the oxy form with
oxygen bound. A key difference between the deoxy and the oxy form appears to be a rotation of about 15° of one
alpha/beta dimer relative to the the other alpha/beta dimer. In the detailed structure it is as if there were two interlocking
gears and in going from one form to the other they slipped a cog. The physiological observations are explained very
nicely by this model. Oxygen does not bind very readily to the deoxy form of hemoglobin, so at low oxygen
concentrations the fraction of possible oxygen molecules bound is very low. This corresponds to the condition of the
blood in the tissues away from the lungs. However, when 2 of the four possible oxygen binding sites have oxygen
bound, the oxy form becomes the most stable energetically and the whole tetramer switches over to the oxy form. Now
the remaining two sites have a high oxygen affinity and bind very readily. So at the critical oxygen concentration there
is a cooperative oxygen binding. This is the condition of the blood in the lungs. As the blood returns to the tissues, the
oxygen concentration drops, the release of the first 2 molecules of oxygen results in a switch back to the deoxy form,
which then makes the release of the remaining oxygen molecules easier, resulting in the dumping of oxygen to the
tissues.
The irreducible complexity argument questions how such a complex molecular machine functioning in such a complex
physiology involving the circulatory and respiratory systems could possibly have evolved step by step. I can say very
little about the evolution of the circulatory and respiratory systems, however, as a result of protein sequence
comparisons and the analysis of the structure of the globin coding regions of the genome, it is possible to construct a
very plausible picture of the origin of a complex machine.
Another structural detail must be noted before we proceed with the evolutionary story. The alpha/beta dimer
self-assembles as a consequence of greasy patches on the surface of each protein. This principle of assembly is due to
the same principle that causes oil drops in water spontaneously coalesce. Interestingly, myoglobin, the oxygen storage
protein found in muscle, has a very similar structure to the hemoglobin monomers, but it does not have the surface
greasy patches, and so does not form multiple subunit assemblies. The two alpha/beta dimers self assemble by the same
principles to form the tetramer.
While no doubt wrong in many of the details, here is a plausible evolutionary scenario that describes the origin of this
molecular machine. This description is largely taken from Hemoglobin: Structure, Function, Evolution & Pathology by
Richard Dickerson and Irving Geis. We'll start with a monomeric oxygen binding globin such as those found in some
insects, annelids, and molluscs and even in plants. [The origin of the first globin-like structure will not concern us,
although there is some speculation that it may have arisen from the cytochrome a which binds to the same kind of heme
group.] These groups do not have globins that have differentiated oxygen storage functions (myoglobin-like) and
oxygen tranport functions (hemoglobin-like). These globins do not have the complex oxygen binding behavior that
hemoglobin has but are similar in their oxygen binding properties to myoglobin.
A key first step is a gene duplication event that allows the preservation of the original functional protein, but provides a
second copy of the gene that can be altered by mutation, providing a source of new material on which selection can
operate. There are multiple versions of globin genes that differ by only a few amino acids in insects and molluscs of the
organisms mentioned above. In humans alpha and beta hemoglobin exist in gene clusters containing multiple copies of
each type gene. In the alpha cluster there are two identical copies of the alpha gene, two copies of the alpha-like zeta
globin (found in fetal hemoglobin), and one alpha-like pseudogene, that appears not to be expressed. In the beta cluster
there five different beta-like genes and one beta-like pseudogene. It appears from various lines of argument that these
have arisen by gene duplication followed by mutations. Some of the mutated copies appear to be functionless, whereas
some of them appear to have new functions, i.e. in fetal hemoglobin with altered oxygen binding. These gene
duplications are pre-adaptations, i.e. changes that occur for other reasons, but once they have occurred they provide the
necessary conditions for some selectable function.
Sea lamprey globin evidences what might be the next intermediate stage. Sea lampreys have a separate myoglobin for
42/70
oxygen storage and a hemoglobin-like molecule for oxygen transport. Lamprey hemoglobin is dimeric rather than
tetrameric. It does display cooperative oxygen binding, though. Lamprey deoxyglobin forms dimers which dissociate
upon oxygen binding. The dimer contacts are in exactly the region of the molecule where one alpha-beta dimer interacts
with the other alpha-beta dimer. This is the region that modulates the 15° rotation and the cog-slipping effect that was
described above. Murray Goodman and co-workers cite evidence from their sequence comparisons that suggest that
mutations accumulated in this region of the molecule at four times the rate for the molecule as a whole during the
evolution of this new function. Clearly, cooperativity of oxygen binding is a consequence of dimerization. But dimer
formation is the result of greasy patches on the surface of the protein, which could well have arisen by a few amino acid
substitutions (or even one as is the case in deoxyhemoglobin S fibers in sickle-cell anemia). Dimer formation would
have been a Darwinian pre-adaptation to the evolution of cooperative oxygen binding.
The next step in hemoglobin evolution is the result of a gene duplication of the ancestral hemoglobin-like gene into the
modern alpha and beta globin genes. Again, the original oxygen transporting function could be preserved, while
mutations acted upon the second copy of the gene. The very similar but slightly different version of the globin allowed
for the formation of the alpha beta dimer which upon interaction with another alpha-beta dimer allowed the preservation
of the tetramer structure even upon oxygenation. Again Goodman's group believe that their sequence comparison data
suggests that the alpha-beta dimer interface accumulated mutations at nearly twice the rate for the whole molecule
during the evolution of this new function. Again, the gene duplication event and the alpha-beta dimer formation are
pre-adaptations to the formation of the complex tetramer.
In the 450 million years since the origin of the hemoglobin tetramer there has been ample time to finely tune the
primitive transport function. And there does appear to be additional evolution, especially related to the rise of
warm-blooded creatures and, as I already mentioned in my discussion of the gene structure of the human alpha and beta
gene clusters, the rise of placental mammals and special adaptations utilized in oxygen transport there.
Hemoglobin is a marvelous molecular machine. When we look at it in all its complex features, many of which we
haven't even discussed today, we might imagine that it's like a mousetrap: you take one part away and it doesn't work
any more. That may, in fact, be true with the modern finely tuned version. But I have shown a plausible evolutionary
scenario based on the structural, mechanistic, genetic, and sequence data concerning globins from all parts of the animal
kingdom.
Biology
&
Philosophy
(http://www.springerlink.com/app/home/journal.asp?wasp=559618bf62a043ccae46a5a1582bea70&referrer=backto&ba
ckto=linkingpublicationresults,1:102856,1;&absoluteposition=17#A17)
Philosophical
Transactions:
Mathematical,
Physical
(http://www.journals.royalsoc.ac.uk/app/home/contribution.asp)
and
Engineering
Functional
cartography
of
complex
metabolic
(http://amaral.chem-eng.northwestern.edu/Publications/Papers/Guimera-2005-Nature-433-895.pdf)
Sciences
networks
數學與統計基礎
圖論
Degree, Weights and Dynamics In Complex Networks (http://www.iis.ee.ic.ac.uk/~v.shen/Report.pdf)
Complex
Behavior
at
Scale:
An
Experimental
Study
of
Low-Power
(http://lecs.cs.ucla.edu/~deepak/PAPERS/empirical.pdf)
Randomized
Algorithms
for
Stability
Analysis
Large-Scale ...(http://decision.csl.uiuc.edu/~alpcan/papers/ComplexSystemSymposium.pdf)
...
of
1. COMPLEX SYSTEMS: CONCEPTUAL INTRODUCTION
Topics: What are the characteristics of simple and complex systems?
Structural, functional, dynamic and algorithmic complexity.
Complexity in physics, biology, economics, and sociology.
Readings:
+ Waldrop, MM: Complexity: the Emerging Science At the Edge of Order and Chaos. New York : Simon & Schuster,
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1992. KCollege Library (KCL): Q175 .W258 1992
+ Bar-Yam, Yaneer Dynamics of Complex System. New England Complex Systems Institut. 1998.
2. HISTORY of COMPLEX SYSTEM RESEARCH
Topics: Some fundamental theories of the 20th centuries are reviewed:
System theory, Cybernetics, Theory of Dissipatice Structures, Synergetics and Catastrophe Theory.
Readings:
+ Bertalanffy, L: General system theory; foundations, development, applications. New York, G. Braziller 1968. KCL:
Q295 .B4 1968
+ Wiener N: Cybernetics; or, Control and communication in the animal and the machine 2nd edition, New York, M.I.T.
Press, 1961 KCL: QA276 .W48 1961
+ Nicolis G. and Prigogine Y.: Exploring Complexity : An Introduction.1989. KCL: Q175 .N417 1989
+ Haken H: The Science of Structure : Synergetics. New York : Van Nostrand Reinhold, c1984. KCL: QH331 .H3413
1984
+ Thom R: Structural stability and morphogenesis; an outline of a general theory of models. Reading, Mass., W. A.
Benjamin, 1975 KCL: QH323.5 .T4813
3. COMPETITION and COOPERATION; the VOLTERRA - LOTKA WORLDS and BEYOND
Topics: The Volterra-Lotka model has its roots in chemistry and ecology. It can be used , however, as a general
paradigm of systems with competitive and cooperative interactions. The mathematics of oscillation. Chemical,
ecological and socioeconomic applications.
Readings:
Scudo FM and Ziegler JR (eds.): The Golden age of theoretical ecology, 1923-1940 : a collection of works by V.
Volterra, V. A. Kostitzin, A. J. Lotka, and A. N. Kolmogoroff. Berlin ; New York : Springer-Verlag, 1978. KCL:
QH541.145 .G64
4. CHAOS and FRACTALS in NATURE and SOCIETY
Topics: Chaos and fractals proved to be very efficent mathematical concepts to understand temporal and spatial
complexity. Elementary mathematical explanation. Chaos in chemistry, population dynamics, brain and economics.
Fractals in physiology. The fractal nature of organizations.
Readings:
+ Gleick, James Chaos: Making a New Science. New York, N.Y., U.S.A. Viking, 1987. KCL: Q172.5.C45 G54 1987
+ Mandelbrot, B: Fractals : form, chance & dimension. San Francisco : W. H. Freeman, c1977. KCL: QA447 .M3613
+ Barnsley, M.: Fractals everywhere. Boston : Academic Press Professional, c1993. KCL: QA614.86 .B37 1993
+PickoverA. (ed.): Fractal horizons : the future use of fractals. KCL: QA614.86 .F6845 1996
+ Gould H. and Tobochnik J: An introduction to computer simulation methods : applications to physical systems.
Reading, Mass. : Addison-Wesley, 1995. KCL: QC21.2 .G67 1995.
5. SELF-ORGANIZATION and SELF_ORGANIZED CRITICALITY
Topics: Self-organization is a vague concept in many respects, still a powerfull notion of modern science. Specifiacally
and counterintuitivly, noise proved to have beneficial (seomtiems indispensable) role in constructing macroscopically
ordered structures. Elementary mathematical models of noise-induced ordering.
In physics, a critical point is a point at which a system radically changes its behavior or structure, for instance, from
solid to liquid. In standard critical phenomena, there is a control parameter which an experimenter can vary to obtain
this radical change in behavior. In the case of melting, the control parameter is temperature. Self-organized critical
phenomena, by contrast, is exhibited by driven systems which reach a critical state by their intrinsic dynamics,
independently of the value of any control parameter. The archetype of a self-organized critical system is a sand pile.
Sand is slowly dropped onto a surface, forming a pile. As the pile grows, avalanches occur which carry sand from the
top to the bottom of the pile. At least in model systems, the slope of the pile becomes independent of the rate at which
the system is driven by dropping sand. Self-organized crtiticality is a useful concept and was used to explain statistical
44/70
features for a wide variety of open systems with many components, ranging from geology to biology and economics. A
few illustrative example will be given
Readings:
+ Kauffman S: At home in the universe : the search for laws of self-organization and complexity. New York : Oxford
University Press, 1995. KCL: QH325 .K388 1995
+ Bak P: How nature works : the science of self-organized criticality. New York, NY, USA : Copernicus, 1999. KCL:
QC173.4.C74 B34 1996
6. GAME THEORY, EVOLUTION, ECONOMICS
Topics: Game theory emerged as an important tool for treating the problem of necessary cooperation to avoid (nuclear
and other) catastrophes. The most famous game is the Prisonner Dilemma. The fundamental types of games will be
discussed. Illustrative examples of applications for evolutionary theory and economics will be given.
Readings:
+ Rapoport, A: The 2 X 2 game. Ann Arbor : University of Michigan Press, 1976. KCL: QA269 .R36 1976
+ Axelrod, R. The evolution of cooperation. New York: Basic Books. New York : Basic Books, 1984. KCL:
HM131 .A89 1984| + Maynard Smith, J: Evolution and the theory of games. Cambridge ; New York : Cambridge
University Press, 1982 KCL: QH371 .M325 1982| + Gardner R: Games for business and economics. New York : Wiley,
c1995. KCL: HD30.22 .G37 1995
+ Hodgson GM: Economics and evolution : bringing life back into economics. Ann Arbor : University of Michigan
Press, 1993. KCL: HB97.3 .H63 1993.
7. NETWORKS EVERYWHERE: FROM MOLECULAR to SOCIAL
Topics: Real world systems in many cases can be represented by networks. Networks can be seen everywhere (neural
networks of the brain, food webs and ecosystems, electric power networks, system of social connections, global
financial network, the world-wide web). Since the famous social psychological experiment of Stanley Milgram, it is
known that from a certain point of view we live in a 'small world.' However, the relationships between the structure of
large networks and their dynamical properties generally are not well known. The performance of many biological,
ecological, economical, sociological, communication and other networks can be illuminated by using new approaches
coming from graph theory, statistical physics and nonlinear dynamics. Examples will be given to illustrate the power of
the new approaches in the understanding of the organization of social structures. Specifically, scientific collaboration
networks will be analized.
Readings:
+ Hayes B: Graph Theory in Practice: Part I, II American Scientist 88(1) and 88(2)2000
+ Newman MEJ: Scientific collaboration networks: I. Network construction and fundamental results, Phys. Rev. E 64,
016131 (2001).
+ Newman MEJ: Scientific collaboration networks: II. Shortest paths, weighted networks, and centrality, Phys. Rev. E
64, 016132 (2001).
8. COMPLEXITY of the BRAIN
Topics: It is often said in a colloquial sense that the brain is a prototype of complex system. Several different notions of
complexity may be more formally related to neural systems. First, structural complexity appears (i) in the arborization
of the nerve terminals at the single neuron level, (ii) in the complexity of the graph structure at the network level, and
(iii) in the systems of networks forming closed loops of closed loops. Second, functional complexity is associated with
the set of tasks performed by the neural system. Third, dynamic complexity can be identified with the different
attractors of dynamic processes, such as point attractors, closed curves related to periodic orbits, and strange attractors
expressing the presence of chaotic behaviour.
Readings:
+ Arbib MA, Érdi P and Szentágothai J: Neural Organization Structure, Function, and Dynamics, The MIT Press, 1998
9. SOCIODYNAMICS: HOW TO BUILD MODELS TO UNDERSTAND EPIDEMICS, ARM RACES, WARS, AND
EPIDEMICS?
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Topics: Simple models can illuminate essential dynamics of complex, and crucially important social systems. Models of
war and arm races can be constructed within the framework of the Volterra-Lotka model
Readings:
+Epstein JM: The calculus of conventional war : dynamic analysis without Lanchester theory. Washington, D.C. :
Brookings Institution, 1985. KCL: U21.2 .E65 1985
+ Epstein JM: Nonlinear Dynamics, Mathematical Biology, and Social Science, Addison-Wesley/Santa Fe Institute;
1997.
Allegory, Allegoresis, and the Hermeneutics of Social Networks (http://www.danielfried.com/dissertation.pdf)
Internetics:
Technologies,
Applications
and
Academic
(http://grids.ucs.indiana.edu/ptliupages/publications/Internetics1.pdf)( http://www.new-npac.org/users/fox/pdftotal/sccs0813.pdf)
46/70
圖論(graph theory)
最早有關圖論的學術論文是由 Leonhard Euler (1707-1783) 所著的 Solutio problematis ad geometriam situs
pertinentis, Commetarii Academiae Scientiarum Imperialis Petropolitanae 8(1736), 128-140。尤拉討論是否有路線可
以穿越 Konigsberg(後稱 Kaliningrad)城中橫跨 Pergel(後稱 Pergolya)河上的橋各剛好一次。這便是有名的七橋問
題。尤拉提出這種路線存在的充要條件。
另外，Gustav Kirchhoff 於 1845 年發表了電路中有關電流與電壓的守恆定律，流進電路中任一點的電流與流出
量必須相同。在電路中的迴路其電壓降必須與提供的電源相同。接著是於 1852 年由 Francis Guthrie 所提出四色
問題，在任何的地圖上，只要使用四種顏色，就能將所有國家著色而且任意相鄰國家屬於不同的顏色。這個問
題直到 1976 年才由 Kenneth Appel 與 Wolfgang Haken 解出。與尤拉稍有不同，在 1856 年 Thomas Pennyngton
Kirkman 與 William Rowan Hamilton 研究的議題則是如何找出經過某些地點正好一次的路線。
最早談論圖頻譜的的學術論文則是在量子化學方面
E. Hückel, Quantentheoretische Beitrage zum Benzolproblem, Z. Phys. 70(1931), 204-286
最早談圖頻譜的數學文獻
L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg, 21(1957), 63--77
基礎定義
圖是由兩個集合所組成，用 G={V,E}表示。其中 V 是節點的集合，E 是連線所成集合。我們可以用 V={1,2,3,…,N}
來表示圖的節點所成集合。使用 E={(v1,v2)|v1,v2 V, v1,v2 間有一連線 }.代表圖中所有連線的集合。如果 (v1,v2)
與(v2,v1)代表不同連線，則稱為有向圖，否則只簡稱圖。節點集合 V 的元素個數使用 |V| 表示，連線集合的元
素多寡則是用 |E|。圖 G 中任一節點 v 的連線個數用 deg[v]表示。與節點 v 有連線的所有節點所成的集合稱為 v
的鄰集，用β[v]表示。根據定義，我們知道 deg[v] = |β[v] |。
漫步(walk)是由一系列節點與連線交互形成，連線接連前後兩節點間。軌跡(trail)是沒有重複連線的漫步。路徑
(path)則是沒有重複節點的漫步。具有相同的開始與結束節點的漫步為閉合。迴路是具有至少一條連線的閉合軌
跡，除了開始與結束節點相同外，沒有重複的節點。
迴圈(loop)與多重連線
連線的兩節點為同一點時，我們稱此連線為迴圈。兩節點間有多條連線時，稱為多重連線圖(multigraph)。
沒有迴圈與多重連線的圖稱為簡單圖，或是稱為圖。
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連通圖(connected graph)
圖中任意兩節點間可以找到一條路徑連接，稱為連通圖。
樹(tree)
連通而且沒有迴路的圖稱為樹。
子圖(subgraph)
圖 G1={V1,E1}, G2={V2,E2}，如果 V1 為 V2 的子集且 E1 為 E2 的子集，則稱 G1 為 G2 的子圖。
支架樹(spanning tree)
圖 G 為連通，其支架樹是包括全部 G 節點同時是樹的子圖。
同構圖(isomorphic graph)
圖 G 與 H 為同構如果可以由 G 重新標示節點而得到 H。
完整圖(complete graph)
圖中任意兩節點間都有一連線，稱為完整圖。用 Kn 表示有 n 個節點的完整圖。
尤拉軌跡(Eulerian trail)
在連通圖中若可以找到一條閉合的、通過所有連線的軌跡，稱為尤拉圖，而軌跡稱為尤拉軌跡。
漢彌爾敦迴路(Hamiltonian cycle)
在連通圖中若可以找到一條包括所有節點的迴路，稱為漢彌爾敦圖，而迴路稱為漢彌爾敦迴路。
Lemma 任意圖的所有節點的連線個數和，等於兩倍連線數目。
定理： 圖 G 為連通。G 為尤拉圖的充要條件為所有節點具有偶數條連線。
證明： 必要條件 若 G 為尤拉圖，則所有節點具有偶數條連線。
充分條件 若所有節點具有偶數條連線，則 G 為尤拉圖。
漢彌爾敦圖的充要條件目前尚為研究議題。
充分條件
Dirac’s 定理： 圖 G 為具有 n 節點的簡單圖，n≧3。若每一節點的連線數 deg[v] ≧n/2，則 G 為漢彌爾敦圖。
Ore’s 定理：圖 G 為具有 n 節點的簡單圖，n≧2。若每一對非相鄰節點 v, w 其連線數和 deg[v]+deg[w] ≧ n ，則
G 為漢彌爾敦圖。
柏拉圖的多面體
四面體(tetrahedron) 由四個三角形構成。
六面體(hexahedron) 由六個四角形構成。
八面體(octahedron) 由兩個四面體上下交疊而成。
十二面體(dodecahedron)由十二個五角形構成。分兩層，最上層有五個五角形，其邊線構成頂部一個平躺的五角
形。下層五個五角形與上層對應，構成底部一平躺五角形。
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二十面體(icosahedron) 由二十個三角形構成。分三層，最上層有五個三角形，中間分兩段，上下段各有五個三
角形，最下層與最上層對稱，有五個三角形。
在柏拉圖的 Timaeus 中提到這五個多面體。
四面體(tetrahedron)
六面體(hexahedron)
十二面體(dodecahedron)
八面體(octahedron)
二十面體(icosahedron)
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圖演算法
表示圖的方式
使用鄰接串列
使用 adjacency lists
陣列 A 有 |V| × 1 個元素，每個元素是一個串列。矩陣元素 A[u] 是一個指標串列指到每一個與 u 相鄰的節點，。
圖 G={V,E} 中的每個節點 u
Adjacency matrix
矩陣 A 有|V| × |V| 個元素，第 i 行第 j 列的元素值表示節點 i 到 j 是否有直接連線，用 1 表示有連線，0 表示
沒有。A 的對角線元素值均設為 0。對角線矩陣 D 的大小為|V| × |V|，每一元素表示其節點的連線數。
Incidence matrix
陣列有|V|× |E|個元素，第 i 行第 j 列的元素值表示節點 I 與連線 j 是否相鄰，
目前在代數圖論中，表示圖的矩陣
矩陣 A 為對角線為 0 的相鄰矩陣。矩陣 D 是每一節點連線數目所形成的對角矩陣。假設矩陣 I 為對角線為 1 的
單元矩陣，而 1 為所有項目均為 1 的矩陣。
相鄰矩陣 A
補相鄰矩陣 A* = 1 – (A+I)
Laplacian L = D – A
signless Laplacian |L| = D + A
Combinatorial Laplacian
Seidel matrix S = A*– A = 1 – (2A + I)
Chapter 7 Algebraic Graph Theory (http://www.math.ksu.edu/~jasonr/book7.pdf)
先廣搜尋法
Problem: For a given graph G, and a specified vertice s in the graph, find all the vertices v that are reachable from s, and
determine the shortest path in G from s to v.
Breadth-first search does this by constructing a breadth-first tree - a tree whose root is s, and the path on that tree from
s to v is the shortest path in G from s to v.
During the execution, BFS algorithm assigns a color to every vertex: white if the vertex has not been reached yet, gray
if the vertex is in the set of vertices currently being processed (a BFS frontier), and black if the vertice and ALL of its
neibhors have been processed.
BFS also computes d[v] for every vertex v, which is the shortest distance from s to v in G.
It also computes p[v] for every vertex v, which is the predecessor of v in the breadth-first tree.
for each vertex v in V
color[v] = white
d[v] = INFINITY
p[v] = NULL
color[s] = gray
d[s] = 0
Queue.clear()
Queue.put(s)
while (! Queue.empty() ) {
u = Queue.get()
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for each v adjacent to u
if(color[v] == white) {
color[v] = gray
d[v] = d[u] +1
p[v] = u
Queue.put(v)
color[u] = black
}
Complexity of BFS
Each vertex is never whitened, so the test at line 12 ensures each vertex is enqueued exactly once, thus dequeued
exactly once. Total queue operations are O(V).
Adjacency lists are scanned only when the vertex is dequeued, thus each adjacency list is scanned exactly once. Total
time for scanning adjacency lists is therefore O(E).
Initialization is O(V). Total running time of the algorithm is O(E+V).
Correctness of BFS
Definition 1. b(s,v) is the minimum number of edges in any path from vertex s to vertex v. If there is no path from s to v,
b(s,v)=INFINITY. B(s,v) is the shortest-path distance.
Lemma 1. Let G = (V, E) be a graph, and sÎV a vertex. The, for any edge (u,v)ÎE:
b(s,v) <= b(s,u)+1
Proof. If u is reachable from s, so is v. The shortest path from s to v cannot be more than the shortest path from s to u
plus the edge from u to v, thus the inequality holds. If u is not reachable from s, then b(s,u) = INFINITY, so the
inequality holds.
Lemma 2. Upon termination, the BFS algorithm computes d[v] for every vertex, and d[v] >= b(s,v)
Proof. By induction on the number i of enqueue operations. For i =1 (after s is enqueued), d[s] = [0] = b(s,s), and d[v] =
INFINITY >= b(s,v) for all v <> s.
For i = n, let us look at a white vertex v discovered at that step from vertex u. By induction, d[u] >= b(s,u). Since d[v]
= d[u]+1 >= b(s,u) + 1 >= b(s,v). QED.
Lemma 3. At all times during the execution of BFS, the queue contains vertices (v1, v2, … vr) such that d[v1] <= d[v2]
<= d[v3] … <= d[vr] AND d[vr] <= d[v1] + 1.
Proof. By induction on the number i of queue operations. For i = 1, queue only has s, so the hypothesis holds.
For i = n:
After dequeuing v1: since d[vr]<=d[v1]+1 and d[v1] <= d[v2], then d[vr] <= d[v2]+1, so the hypothesis holds.
After enqueuing vr+1: d[vr+1] = d[v1] + 1 (line 14) >= d[vr]. Also, d[vr+1] = d[v1] + 1 <= d[v2] + 1, since
d[v1]<=d[v2]. Since v2 is the new head of the queue, the hypothesis holds.
Corollary 4. If vertices u and v are enqueued during execution of BFS, and u is enqueued before v, then d[u] <= d[v].
Theorem 5. Given G=(V,E) and source vertex s, BFS algorithm discovers every vertex v reachable from s, and upon
termination, d[v] = b(s,v). Moreover, for any vertex v reachable from s, one of the shortest paths from s to v is a path
from s to p[v], followed by edge (p[v],v).
Proof. By contradiction. Assume some vertex gets assigned d[v] <> b(s,v). Let v be such a vertex with minimum b(s,v).
By lemma 2, d[v]>=b(s,v), so it must be that d[v]>b(s,v). Vertex v must be reachable from s, for if it’s not, b(s,v) =
INFINITY >=d[v]. Let u be the vertex on a shortest path from s to v, then it must be b(s,v) = b(s,u)+1 = d[u] + 1 (Since
d[u] = b(s,u), otherwise we would have chosen u in our proof). Thus:
d[v] > d[u] + 1 …….. (1)
Let’s prove that this can never happen. Let’s look at the time when BFS dequeues u. At that point, v is either white,
black or gray.
If v is white, line 14 sets d[v] = d[u] + 1
If v is black, than it was already removed from the queue, and by Corollary 4, d[v]<=d[u].
If v is gray, it was painted gray when some other vertex w was dequeued, which was earlier than now (when we are
dequeuing u), so d[v] = d[w] + 1 <= d[u] + 1 (Corollary 4).
So, d[v] = b(s,v) for all v in V. All reachable vertices must be discovered, otherwise they will have d = INFINITY. If p[v]
= u, then d[v] = d[u] + 1, so one of the shortest paths from s to v can be obtained by taking a shortest path from s to u
and then taking the edge (u,v).
先深搜尋法
DFS(G)
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for each vertex u in V do
color[u] = white
parent[u] = NIL
time = 0
for each vertex u in V do
if color[u] == white
DFS-VISIT[u]
DFS-VISIT(u)
color[u] = grey
discover[u] = time
time = time + 1
for each v in ADJ[u]
if color[v] == white then
parent[v] = u
DFS-VISIT(v)
color[u] = black
finish[u] = time
time = time+1
最短路徑
Experimental Evaluation of a New Shortest Path Algorithm ... http://www.mpi-sb.mpg.de/~pettie/papers/alenex02.ps
最小支架樹
Minimum Spanning Trees http://roso.epfl.ch/3emecycle/goemans/zinal3.pdf
http://roso.epfl.ch/3emecycle/goemans/zinal3.pdf
R.C. Prim. Shortest connection networks and some generalizations. Bell System Technical Journal, Volume 36, pp.
1389-1401, 1957.
Prim-MST(G)
select an arbitrary vertex to start
while (there are still non-tree vertices) {
select the edge of minimum weight between the tree and non-tree vertices
add the selected edge and vertices to the tree
}
Kruskal’s 演算法
J.B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the
American Mathematical Society, Volume 7, pp. 48-50, 1956.
Kruskal-MST(G,w)
T={V, }
sort E in order of increasing costs
for (I = I; I<= |V|-1, I++) {
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select the next smallest cost edge e
if (e connects two different connected components) {
add e to the edge of T
}
}
void kruskal (vertex-set V; edge-set E; edge-set T)
int ncomp; /* current number of components */
priority-queue edges /* partially ordered tree */
mfset components; /* merge-find set data structure */
vertex u, v; edge e;
int nextcomp; /* name for new component */
int ucomp, vcomp; /* component names */
{
makenull (T); makenull (edges);
nextcomp = 0; ncomp = n;
for (v$ \in$V) /* initialize a component to have one vertex of V*/
{ nextcomp++ ;
initial (nextcomp, v, components);
}
for (e$ \in$E)
insert (e, edges); /* initialize priority queue of edges */
while (ncomp > 1)
{
e = deletemin (edges);
let e = (u, v);
ucomp = find(u, components);
vcomp = find(v, components);
if (ucomp! = vcomp)
{
merge (ucomp, vcomp, components);
ncomp = ncomp - 1;
}
}
}
* Choose a partially ordered tree for representing the sorted set of edges
* To represent connected components and interconnecting them, we need to implement:
1.MERGE (A, B, C) . . . merge components A and B in C and call the result A or B arbitrarily.
2.FIND (v, C) . . . returns the name of the component of C of which vertex v is a member. This operation will be
used to determine whether the two vertices of an edge are in the same or in different components.
3.INITIAL (A, v, C) . . . makes A the name of the component in C containing only one vertex, namely v
* The above data structure is called an MFSET
void topsort {
Queue Q = new Queue();
for (int i = 0; i < n; i++) {
if(d[i] == 0)
Q.Enqueue(i);
}
while ( !Q.isEmpty() ) {
v = Q.Dequeue();
print(v);
for each w adjacent to v do {
if ( --d[w] == 0 )
Q.Enqueue(w);
}
}
}
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隨機圖(random graph)
談到隨機圖，一般都是談到 Paul Erdös 與 Albert Rényi 在 1960 年所發表的
On the evolution of random graphs. Magyar Tud. Akad. Mat. Kut. Int. Kozl. 5, pp. 17-61, 1960.
Random Graph Models with Hidden Color http://www.thep.lu.se/pub/Preprints/03/lu_tp_03_34.pdf
Random Oxford Graphs http://math.berkeley.edu/~jblasiak/rndgr0604.pdf
目前有三大類隨機圖模型，第一類用  (N,M) 表示，第二類則是用 (N,p)，第三類用 (N,D)表示。 (N,M)表
示由所有 N 個節點 M 條連線的圖所成的集合。(N,p)則是由所有 N 個節點，每條連線是否存在的機率由 p 參數
決定的所有圖所成的集合。第三類 (N,D) 是由 N 個節點，連線數序列 D，組成的所有圖所成集合。
圖 G是指在此類隨機圖中均勻隨機的任取一圖。
因此，指定參數 N,M 後
|  (N,M) | =
 N ( N  1) 2 


M


假設 G   (N,p)有 m 條連線，
Pr[  (N,p) = G ] =
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p m (1  p)
N ( N 1)
m
2
通常圖的某一特性用 Q 表示，例如 Q 代表連通性、或是包括完整圖 K5、或直徑為 log N。
如果增加連線不會使特性 Q 消失，我們稱特性 Q 具有單調遞增性。
我們可以用符合某一特性的所有圖所成集合來表示圖集。
問題 Pr[Q]
如果是 (N,p)，那麼 Pr[Q] 會是 N 與 p 的函數。
有興趣想知道的議題是 “幾乎都是” Q，當 N 
時 Pr[Q]  1 。
可以找到一臨界值

0
lim PrN , p [Q ]  
N 

1
p
pc
0
p
pc

假設 G   (N,p)有 m 條連線，那麼 m 的期望值為 [m] = pN(N-1)/2
證明：
由馬可夫不等式(Markov’s inequality)
X 為隨機變數且 X0，[X] = 
所有 t>0，Pr[Xt]  /t，Pr[Xt]  1/t
由轍諾夫上限(Chernoff’s bound)
 t2
X 為 B(N,p)隨機變數，期望值 [X] =  = np, Pr[ |X-|>t ]  2e 3  , 0 < t  
隨機圖 Gp={V,E} ，假設|V| = N，
 N  1 k
Pr(deg[ x ]  k )  
p (1  p ) N 1k

 k 
使用生成函數

G0 ( x )   pk x k
k 0
z   kpk  G0 (1)
k
Notice that
G0 (1)  1
網路中為巨大元件的比率為 S
S  1  G0 (u)
u 為 zu  G0 (u) 的最小非負值實數解。
除了巨元件本身外，網路中平均元件大小為
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S  1
z 2u 2
G0 (u )[ z  G0(u )]
波以松分布 Poisson distribution is
e   k
k!
 ( x 1)
生成函數 G( x )  e
Pr(deg[ x ]  k ) 
任選兩點，為關鍵連線的機率為 p(1-p2)N-2，為群聚連線的機率為 p(1-(1-p2)N-2)。
為分離狀態的機率為(1-p)(1-p2)N-2，為群聚分離狀態的機率為(1-p)(1-(1-p2)N-2)。
 N  2  2a
Pr( Aij  a )  
p (1  p 2 ) N 2a , a  1, 2,

 a 
,N 2
(To study models of for large graphs like: http://www.lsi.upc.es/~diaz/Gnp.ppt)
(http://www.renesys.com/projects/leiden2001/Presentations/Chung-Graham.ppt)
Graph Clustering for Very Large Topic Maps http://www.gca.org/papers/xmleurope2001/papers/html/s23-2.html
(Chapter 2 Arbitrary Degree Distributions http://www.math.cornell.edu/~durrett/math777/c2s2.pdf)
隨機圖 Gp={V,E} ，假設|V| = N，隨著 p 的改變而有不同。
當 p=0 時，整體是 N 個獨立節點。
當 p=c/N，0<c<1 時，隨機圖 Gp 會有任意大小的迴路。
當 p=1/N 時，隨機圖 Gp 會有雙峰的情形。
當 p=c/N 時，c>1 時，隨機圖 Gp 會有一個大小為Θ(N)的連通元件，其他則為 o(N)的樹。
當 p=log N / N 時，隨機圖 Gp 會是連通。
當 p=w log N / N 時，隨機圖 Gp 會是連通且接近規律圖，每一節點的連線數期望值為 w log N。
由另一種角度看隨機圖。GD={V,E}, D={deg[v1],deg[v2],…,deg[vN]}。
D 為連線數序列。因此我們可以視符合某一連線數序列的圖為一整個圖的集合。
滿足冪次律連線數序列的圖
log y = α- β log x 或 y = eα/ xβ
用 G(α,β)表示。max[deg[vi]] = eα/β
N

1 x  e

e
  (  )e ,  (  ) 為 Riemann Zeta function。

x
e
 x  1   ( 21) e
| E |  (   1)
因此其特性可以由β參數決定。

N
 ( )
|E| =
| E |
1
2
當β=0 時為連通圖。
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β<1 時為連通圖。
1<β<2 時有一個巨型的大小為Θ(N)的連通元件。
2<β<3.4785 時有一個巨型的大小為Θ(N)的連通元件，其他則為 O(log N)的元件。
β>3.4785 時，幾乎沒有連通元件，所有元件大小為 o(N)
β>4 時，所有連通元件大小為冪次律。
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Clique Percolation in Random Networkshttp://angel.elte.hu/clustering/papers/cliquePercolation.pdf
Creating a Giant Component http://www.andrew.cmu.edu/user/kravitz2/papers/bohman_kravitz_giant.pdf
The Optimal Path in an Erd˝os-Renyi Random Graphhttp://polymer.bu.edu/~hes/networks/bbschs04.pdf
(A Random Graph Model for Power Law Graphs, http://www.math.ucsd.edu/~llu/papers/power.pdf)
Efficient and simple generation of random simple connected graphs ...
http://www.liafa.jussieu.fr/~fabien/docs/viger05generation.pdf
Creating a Giant Component http://www.andrew.cmu.edu/user/kravitz2/papers/creatinggiant_may04.pdf
Zipf-Pareto-Yule and Power laws.
Distributions with an inverse polynomial tail have been observed in a number of
contexts. The earliest observations are due to Pareto [Pareto 1897] in the context of economic models. Subsequently,
these statistical behaviors have been observed in the context of literary vocabulary [Yule 44], sociological models [Zipf
49], and even oligonucleotide sequences [Martindale and Konopka 96] among others. Our focus is on the closely
related power law distributions, defined on the positive integers, with the probability of the value i being proportional
to 1/ik for a small positive number k. Perhaps the first rigorous effort to define and analyze a model for power law
distributions is due to Herbert Simon [Simon 55].
Discrete Pareto distribution
Pr[ X  x ] 
1
 ( )  x

1
is known as the Riemann Zeta function.

i 1 i
 ( )  

G ( x )   ( )  i  x i
i 1
Heavy tail property
Not all moments
E[ X k ] are defined
Expected value exists if and only if α>2
Variance and
E[ X 2 ] exist if and only ifα>3
E[ X k ] defined if and only if α>k+1
生成函數
Discrete probability distribution of random variable X  {0,1, 2,

G ( x )   pk x k
k 0
Probability value
58/70
, }
Pr[ X  k ] 
G ( k ) (0)
k!
1 = G(1), E[ X ]  G '(1) ,
所有 Xi 為 IID 離散隨機變數，其生成函數各別為 G X i
n
Sn   ai X i
i 1
n
GSn ( x )   G X i ( x ai )
i 1
因此，當 S = X1 – X2
GS ( x )  G X1 ( x )G X 2 (1 x )
如果 N 是一隨機變數，而 X1,X2,…都是 IID 的隨機變數，
那麼 G X N ( x )  GN (G X ( x ))
因此，
Equilibrium Statistical Mechanics of Network Structures http://angel.elte.hu/~derenyi/publ/LNP_networks.pdf
http://homepages.inf.ed.ac.uk/mrj/ETHbook/
Random graphs with arbitrary iid degrees (http://www.ma.hw.ac.uk/ams/seminars/papers/hooghiemstra.pdf)
http://www.renesys.com/projects/leiden2001/Presentations/Chung-Graham.ppt
http://www.lsi.upc.es/~diaz/Gnp.ppt
http://www.economia.unimi.it/marray/2005/material/L11.pdf
http://www.bioconductor.org/workshops/Bressanone/Friday/L9/L9a.pdf
http://www.cis.upenn.edu/~mkearns/teaching/NetworkedLife/snt.ppt
http://www.stats.ox.ac.uk/~taylor/Presentations/JTNetworks_Chapter%203.ppt
http://compdiag.molgen.mpg.de/ngfn/docs/2004/nov/huber-graphs.pdf
http://gepard.bioinformatik.uni-saarland.de/html/BioinformatikIIIWS0405/V2-ScalefreeNet.ppt
http://www.stats.ox.ac.uk/~taylor/Presentations/Evolution_of_Networks.ppt
http://www.ima.umn.edu/talks/workshops/1-11.2004/misra/misra.ppt
http://www.research.att.com/~volinsky/Graphs/slides/sridhar.ppt
隨機圖
Evolution of Random Networks http://www.cmth.ph.ic.ac.uk/people/k.christensen/papers/published/prl81_1998.pdf
How
to
calculate
the
main
characteristics
http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:cond-mat/0308629
of
random
RANDOM GRAPHS Svante Janson ( http://www.math.uu.se/~svante/papers/sjb4.ps)
如何計算隨機圖的節點連線數目分怖
Generating Function http://www.ece.drexel.edu/telecomm/Talks/Vilas.pdf
Topological phase transitions of random networks http://angel.elte.hu/~fij/homepage/articles/phasetransPhysA.pdf
Are randomly grown graphs really random? http://www.math.cornell.edu/~durrett/math777/CHKNS.pdf
Rigorous Result for the CHKNS Random Graph Model http://www.math.cornell.edu/~durrett/CHKNS/chkns.pdf
http://www.ee.technion.ac.il/courses/049011/lectures/lecture5.pdf
http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/GraphTheoryIII.pdf
59/70
...
Random graphs as models of networks http://192.12.12.14/research/publications/workingpapers/02-02-005.pdf
Models of random regular graphs (http://www.math.uwaterloo.ca/~nwormald/papers/regsurvey.pdf)
產生隨機圖
http://www.matematik.su.se/~mia/generating_random_graphs.pdf
連通元件
THE SIZE OF THE GIANT COMPONENT
http://www.cs.toronto.edu/~molloy/webpapers/size.ps
OF
A
RANDOM
GRAPH
WITH
A
GIVEN
...
Connected components in random graphs with given expected degree ... http://www.math.ucsd.edu/~fan/wp/conn.pdf
GROWTH
OF
COMPONENTS
IN
RANDOM
GRAPHS
1.
Introduction
We
consider
...
http://www.math.uu.se/~svante/papers/sj130.pdf
The Diameter of Random Massive Graphs *
http://digg.cs.tufts.edu/readings/pdf/011.pdf
http://link.aps.org/doi/10.1103/PhysRevLett.85.5468
Rigorous Result for the CHKNS Random Graph Model http://www.math.cornell.edu/~durrett/CHKNS/chkns.pdf
http://www.dmtcs.org/proceedings/html/pdfpapers/dmAC0109.pdf
The volume of the giant component of a random graph with given ...
http://www.math.sc.edu/~lu/papers/conn2.pdf
平均距離
The
Average
Distance
in
a
http://www.math.sc.edu/~lu/papers/ave_full.pdf
Random
Graph
with
Given
Expected
Degrees
On large random graphs with infinite variance Pareto degree ... http://www.vtt.fi/tte/tte21/cost279/TD-02-020.pdf
Distances in random graphs with infinite mean degrees (http://www.win.tue.nl/~rhofstad/infmeanfin.pdf)
Distances in random graphs with finite variance degrees http://ssor.twi.tudelft.nl/~gerardh/onderz/finstubRSArevrev.pdf
Distances
in
random
graphs
with
http://euridice.tue.nl/~dznamens/Math/infvaraap.pdf
finite
mean
and
infinite
variance
...
Distances in random graphs with finite variance degrees http://ssor.twi.tudelft.nl/~gerardh/onderz/finstubRSArevrev.pdf
A General Formalism for Inhomogeneous Random Graphs http://www.thep.lu.se/pub/Preprints/02/lu_tp_02_29.pdf
THE
PHASE
TRANSITION
IN
INHOMOGENEOUS
RANDOM
GRAPHS
http://www.math.uu.se/~svante/papers/sj178.pdf
Kinetic Theory of Random Graphs http://cnls.lanl.gov/~ebn/pubs/rgrev/rgrev.pdf
60/70
冪次律分布(power-law distribution)
1926 Lotka’s Law (distribution of authors in chemical abstracts 1907-1916)
字出現的頻率
Population of City
Yule (later Mandelbrot) statistical study of the literary vocabulary.[Yule, 1944].
Citation analysis [Lotka, 1926].
Zipf human behavior and the principle of least effort. [Zipf, 1947].
Pareto Cours d’economie politique. [Pareto,1897].
Network graph. [Faloutsos-Faloutsos-Faloutsos, 1999].
Oligonucleotide sequences [Martindale-Konopka, 1996].
Access statistics for web pages. (From server logs) [Glassman, 1997].
User behavior (instrument browsers and proxies) [Lukose-Huberman, 1998, Crovella and others,1997-99].
http://www.ljplus.ru/img2/muzyka_sfer/russianlj.html
無尺度網路
Towards a Theory of Scale-Free Graphs: Definition, Properties, and ...
http://www.cds.caltech.edu/~alderd/papers/cit-cds-04-006.pdf
THE
STRUCTURE
OF
RANDOM
GRAPH
ORDERS
http://epubs.siam.org/sam-bin/getfile/SIDMA/articles/28121.pdf
1.
Introduction.
The
random
...
無尺度網路的結構特性
Structural Properties of Scale-Free Networks http://polymer.bu.edu/~hes/networks/reuven5.pdf
Evolution
of
scale-free
random
graphs:
Potts
http://www.physics.iisc.ernet.in/~statphys22/Invitedabstracts/topic12/Dkim.pdf
model
formulation
計算無尺度圖的直徑
The diameter of a scale-free random graph http://www.dpmms.cam.ac.uk/~omr10/diam/diam.pdf
The small giant component in scale-free random graphshttp://www.dpmms.cam.ac.uk/~omr10/giant3/giant3.pdf
ON THE MEAN DISTANCE IN SCALE FREE GRAPHS 1 Introduction
http://ssor.twi.tudelft.nl/~gerardh/onderz/eramcapfin1.pdf
Experimental Evaluation of a New Shortest Path Algorithm \Lambda ...
http://www.cs.utexas.edu/~vlr/papers/alenex02.ps
Scaling exponents and clustering coefficients of a growing random ...
http://www.mpikg-golm.mpg.de/theory/people/zhou/works/PREv66p016125.pdf
Analyzing and Characterizing Small-World Graphs http://www.cs.ucdavis.edu/~martel/main/submitted.pdf
Synchronization
transition
in
scale-free
http://www.uni-saarland.de/fak7/rieger/dslee/paper/paper2_pre.ps
61/70
networks:
Clusters
of
...
小世界網路
Classes of small-world networks http://polymer.bu.edu/~amaral/Papers/pnas00a.pdf
Dynamics of rumor propagation on small-world networks http://cabfst28.cnea.gov.ar/~zanette/s07.pdf
固有值與頻譜
Eigenvalue Bounds (http://www.sfu.ca/~richards/Pages/insna99x.pdf)
The
Principal
Components
Analysis
of
its ...( http://www2.info.ucl.ac.be/people/pdupont/pdupont/pdf/ecml04.pdf)
a
Graph,
and
The Largest Eigenvalue of Sparse Random Graphs http://www.math.princeton.edu/~bsudakov/sparse-eigen.pdf
圖的頻譜
http://www.math.kth.se/~tatiana/Teach/absGraz99.ps
冪次律圖的頻譜
Spectra of random power law graphs http://www.aladdin.cs.cmu.edu/workshops/wsa/papers/spectrum.pdf
The degree sequences and spectra of scale-free random graphs http://www.shef.ac.uk/jhj/scalefree4.pdf
Eigenvalues of Random Power law Graphs http://www.math.ucsd.edu/~fan/wp/eigen.pdf
無尺度圖的頻譜
Spectra and eigenvectors of scale-free networks http://phya.snu.ac.kr/~kahng/spectra.pdf
實際網路的頻譜
http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/NetworksLife_Physica%2
0A%20314,%2025-34%20(2002).pdf
稀疏隨機圖的最大固有值
The Largest Eigenvalue of Sparse Random Graphs http://www.math.princeton.edu/~bsudakov/sparse-eigen.pdf
Spectral Techniques in Graph Algorithms http://www.math.tau.ac.il/~nogaa/PDFS/spectalg.pdf
Finding local community structure in networks http://www.cs.unm.edu/~moore/tr/05-02/local_communities.pdf
THE
COVER
TIME
OF
RANDOM
REGULAR
GRAPHS
(http://epubs.siam.org/sam-bin/getfile/SIDMA/articles/42847.pdf)
Sampling
regular
graphs
and
a
peer-to-peer
network
(http://www.cc.gatech.edu/~mihail/D.8802readings/sampleregular.pdf)
On
the
Limiting
Distribution
of
Eigenvalues
of
Large
Random ...(http://www.math.princeton.edu/mathlab/book/papers/leog.pdf)
Pseudo-random
graphs
Michael
Krivelevich*
Benny
Sudakov
#
1 ...(http://www.math.princeton.edu/~bsudakov/pseudo-random-survey.ps)
On Reliability of Large-Scale Random Networks (http://web.mit.edu/minkyu/www/doc/TON2004.pdf)
62/70
Improving
Network
Robustness
by
Edge
Modification
(http://www.research.ibm.com/people/r/rish/papers/PhysicaA_3_30.pdf)
Chapter 13 Graph-based Evolutionary Algorithms (http://orion.math.iastate.edu/danwell/ma378/chapter13.pdf)
Distances
in
random
graphs
with
finite
variance
degrees
(http://ssor.twi.tudelft.nl/~gerardh/onderz/finstubRSArevrev.pdf)
Minimum connected dominating sets of random cubic graphs(http://www.combinatorics.org/Volume_9/PDF/v9i1r7.pdf)
Small k -Dominating Sets of Regular Graphs (http://www.springerlink.com/index/Q5JFDAMCUE8046JT.pdf)
Spectral analysis of cell-graphs of cancer (http://www.cs.rpi.edu/research/pdf/04-17.pdf)
Learning the topological properties of brain tumors (http://www.cs.rpi.edu/research/pdf/04-14.pdf)
Characterizing
distance-regularity
of
graphs
by
the
spectrum
ER ...( http://com2mac.postech.ac.kr/papers/2005/05-02.ps)
Which graphs are determined by their spectrum? (http://center.uvt.nl/staff/dam/ds.pdf)
Survey
of
Spectra
of
Laplacians
on
Finite
Audrey ...( http://www.expmath.org/restricted/5/5.1/terras.ps)
Spectral
methods
for
analyzing
an ...( http://www.sfu.ca/~richards/Pages/NAS.AJS-WDR.pdf)
and
Spectra
of
Random
Graphs
Project
of
(http://lthiwww.epfl.ch/~leveque/Matrix/report_etienne_rmt.pdf)
the
Eigenvalues
and
Expansion
of
Kahale\Lambda ...( http://dimacs.rutgers.edu/techps/1993/93-70.ps)
Symmetric
Spaces
visualizing
class
networks:
“Random
Regular
Matrices”
Graphs
Nabil
TOPOLOGIES OF SOCIAL INTERACTIONS (http://www.hecer.fi/Seminars/Papers/ioannides.pdf)
Some
results
on
graph
spectra,
to ...( http://www.maths.otago.ac.nz/home/downloads/richard_martin/graphs.pdf)
with
applications
EIGENVALUE SPACINGS FOR REGULAR GRAPHS http://www.math.rutgers.edu/~sdmiller/ima.pdf
Eigenvalues
of
Graphs
0.
Preliminaries.
Let
http://www.maths.nottingham.ac.uk/personal/drw/PG/eigen.pdf
A
be
an
n
×
Spectral
analysis
of
protein-protein
interactions
in
http://www.sst.ph.ic.ac.uk/people/c.kamp/PhysRevE_71_041911.pdf
Synchronizability of degree correlated networks http://xxx.lanl.gov/pdf/cond-mat/0504335/
Pseudo-random
graphs
Michael
Krivelevich*
1 ...( http://www.math.princeton.edu/~bsudakov/pseudo-random-survey.ps)
Benny
n
matrix
...
Drosophila
...
Sudakov
#
http://www-math.mit.edu/~spielman/eigs/
http://www.cs.wpi.edu/~dobrush/cs507/presentation/2001/Project10/ppframe.htm
http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/defEx.htm
http://www.cs.usask.ca/resources/tutorials/csconcepts/1999_8/
RANDOM
GRAPHS
AND
SOCIAL
NETWORKS:
(http://www.iui.se/Networks/Random_Graph_Soc_Net2_IUI.pdf)
An
Economics
algebraic graph theory http://www.math.mcgill.ca/~vetta/CS760.dir/chrisl.pdf
The degree sequences and spectra of scale-free random graphs (http://www.shef.ac.uk/jhj/scalefree4.pdf)
Spectra of ‘‘real-world’’ graphs: Beyond the semicircle law (http://angel.elte.hu/lanczos/pdf/spectra.pdf)
探討網路導致容易同步的結構
Synchronizability of degree correlated networks (http://xxx.lanl.gov/pdf/cond-mat/0504335/)
63/70
Perspective
SYNCHRONIZATION AND GRAPH TOPOLOGY
(http://lanoswww.epfl.ch/personal/ibelykh/MyFiles/synchrony_belykh_with_figures.pdf)
The relationship between and cluster structure
(http://www.library.uu.nl/digiarchief/dip/diss/1895620/c8.pdf)
Learning Eigenfunctions of Similarity: Linking Spectral Clustering ...
http://www.iro.umontreal.ca/~lisa/pointeurs/TR1232.pdf
Graph Spectra and Modal Dynamics of Oscillatory Networks
(http://tableau.stanford.edu/~lall/projects/architectures/papers/ayazifar.pdf)
http://tableau.stanford.edu/~lall/projects/architectures/papers/ayazifar.pdf
Learning Eigenfunctions of Similarity: Linking Spectral Clustering ...
http://www.iro.umontreal.ca/~lisa/pointeurs/TR1232.pdf
Self-Synchronization in Networks of Autonomous Agents
(http://www.seas.upenn.edu/~nima/index_files/MIT%20robotics-final.pdf)
Synchronization in networks of nonlinear dynamical systems coupled ...
(http://www.iop.org/EJ/article/0951-7715/18/3/007/non5_3_007.pdf)
Synchronization in Small–world Systems
(http://chaos-mac.nrl.navy.mil/Section_Stuff/Papers/SWprl4.pdf)
Decentralized Synchronization Protocols with Nearest Neighbor ...
(http://www.cs.jhu.edu/~ijwang/sensys04.pdf)
On the Stability of the Kuramoto Model of Coupled Nonlinear ...
http://www.grasp.upenn.edu/~motee/kuramoto_final.pdf
極大化第二小固有值
On Maximizing the Second Smallest Eigenvalue of a State-dependent ...
(http://www.aa.washington.edu/faculty/mesbahi/papers/max_lambda2.pdf)
Eigenvalue interlacing and the Laplacian (http://www.math.ucsd.edu/~sbutler/interlace.pdf)
Clustering with spectral methods
http://www.ub.uni-konstanz.de/kops/volltexte/2002/886/pdf/thesis.pdf
程式碼 Spectral Gap
http://www.eleves.ens.fr/home/ollivier/specgraph/specgraph.tar.gz
http://mapage.noos.fr/echolalie/graph/gb.zip
http://mapage.noos.fr/echolalie/graph/gbsample.zip
The Generic Graph Component Library (http://www.osl.iu.edu/publications/prints/2000/siek00:_ggcl.pdf)
GDR: A VISUALIZATION TOOL FOR GRAPH ALGORITHMS 1. Introduction.
(http://www.csc.ncsu.edu/faculty/stallmann/GDR_tech_report.pdf)
GRAPH STRUCTURES
http://pil.phys.uniroma1.it/~gcalda/book/Chapter2.pdf
Detecting network communities: a new systematic and efficient algorithm
(http://www.iop.org/EJ/article/1742-5468/2004/10/P10012/jstat4_10_p10012.html)
Optimal and Cooperative Control of Vehicle Formations
(http://cc.ee.ntu.edu.tw/~ncslab/StudyGroup/MultiAgent/Diss2002FaxAVehFormation.pdf)
群聚活動
Flocking in Teams of Nonholonomic Agents
http://www.eece.unm.edu/ifis/papers/Block_Island.pdf
Flocks of Autonomous Mobile Agents
(http://www.unm.edu/~tanner/Papers/AINS_short_paper.pdf)
Random
Deletion
in
a
ScaleFree
Random
Graph
(http://www.internetmathematics.org/volumes/1/4/Cooper.pdf)
http://www.americanscientist.org/template/AssetDetail/assetid/14708/page/6;jsessionid=aaa46IpG52vO-5
Graph Laplacians and Stabilization of Vehicle Formations
64/70
Process
(http://caltechcdstr.library.caltech.edu/2/00/fm01-cds.pdf)
The Principal Components Analysis of a Graph, and its ...
(http://www2.info.ucl.ac.be/people/pdupont/pdupont/pdf/ecml04.pdf)
Mathematica 的 http://www.cs.uiowa.edu/~sriram/Combinatorica/NewCombinatorica.m
http://www.cs.uiowa.edu/~sriram/Combinatorica/
Graph Theoretic Methods in the Stability of Vehicle Formations
(http://www.mth.pdx.edu/~gerardo/papers/ACC2004.pdf)
Spectral
Generalizations
of
Line
(http://www.cambridge.org/catalogue/catalogue_frontmatter.asp?isbn=0521836638)
Graphs
Spectral
characterizations
of
some
distance-regular
graphs
(http://greywww.kub.nl:2080/greyfiles/few/2000/doc/793.pdf)
Characterizing distance-regularity of graphs by the spectrum ER ...(http://com2mac.postech.ac.kr/papers/2005/05-02.ps)
Equivalence
between
isospectrality
and
isolength
(http://www.iop.org/EJ/article/0305-4470/34/30/303/a13003.pdf)
spectrality
for
a
...
Symmetric Powers of Strongly Regular Graphs (http://quoll.uwaterloo.ca/co444/bhpow.pdf)
A
GAME
BASED
ON
SPECTRAL
(http://matematika.etf.bg.ac.yu/publikacije/pub/P16(05)/radd505.pdf)
GRAPH
THEORY
Pseudosimilarity in Graphs-- A Survey Josef Lauri University of ...(http://staff.um.edu.mt/jlau/research/SURV_ARS.ps)
Statistical Learning with Similarity and Dissimilarity Functions
(http://edocs.tu-berlin.de/diss/2004/luxburg_ulrike.pdf)
Switching
of
edges
in
strongly
regular
graphs.
I.
A
family
of
...
(http://www.combinatorics.org/Volume_10/PDF/v10i1r17.pdf)
Finding
Groups
of
Graphs
in
Databases
A
Thesis
Submitted
to
the ...(http://www.binaryshift.us/work/publications/ms-thesis.pdf)
On Models of Coordination Activity and its Disruption:
(http://www.math.nmsu.edu/~jlakey/complexsystems/PING-REP2.pdf)
Metabolic Networks: Activity Detection and Inference (http://www-helix.inrialpes.fr/IMG/pdf/inria.pdf)
群聚
Flocking in Fixed and Switching Networks
http://www.cis.upenn.edu/~lee/04cis640/readingList/hy/boids_automatica5.pdf
使用頻譜幫助畫出圖
Spectral Graph Drawing http://www.ub.uni-konstanz.de/kops/volltexte/2005/1533/pdf/thesis.pdf
A Study of the Relationship between SVM and Gabriel Graph
(http://www.cse.cuhk.edu.hk/~king/PUB/wcci2002-ZhangWang.ppt)
Ultrafast Consensus in Small-World Networks http://www.cds.caltech.edu/~olfati/olfatisaber_acc05.pdf
Statistical properties of Corporate Board and Director Networks
http://www.lps.ens.fr/~battiston/Battis_BoardTopo_03.pdf
Discovering
Hidden
Groups
in
http://www.cs.rpi.edu/~goldberg/publications/hidden-graph.pdf
Uncovering
the
overlapping
community
http://angel.elte.hu/clustering/papers/PallaSuppMat.pdf
structure
Self-similar community structure in a network of human interactions
http://complex.ffn.ub.es/papers/gddga-scso-03.pdf
65/70
Communication
of
complex
Networks
networks
...
The Evolution of the Mathematical Research Collaboration Graph http://www.oakland.edu/enp/eddie.pdf
The corporate boards networks http://pil.phys.uniroma1.it/~gcalda/doc/052.pdf
Analysis of SIGMOD’s Co-Authorship Graph http://database.cs.ualberta.ca/coauthorship/coauthorship.pdf
66/70
Epidemics on Scale-free Networkshttp://www.aims.ac.za/resources/archive/2003/zakEssay.pdf
分子生物學中的網路
http://www.bioconductor.org/workshops/Bressanone/Friday/L9/L9a.pdf
http://compdiag.molgen.mpg.de/ngfn/docs/2004/nov/huber-graphs.pdf
Complex Biological Networks, Random Graphs and Amorphous Computation
http://www.comp.leeds.ac.uk/seth/cluster/May/rg.pdf
Complex Networks in Genomics and Proteomics http://complex.upf.es/~ricard/PROTEOreview.pdf
Computational
Biology
–
http://gepard.bioinformatik.uni-saarland.de/html/BioinformatikIIIWS0405/V2-ScalefreeNet.ppt
Modeling
Genetic
Networks
and
Their
Evolution:
A
Complex
http://www.theo-physik.uni-kiel.de/~bornhol/382_1289.pdf
Protein
complexes
and
functional
modules
http://www.mit.edu/people/leonid/publications/Spirin_Mirny_ProtComplex.pdf
in
Bioinformatik
Dynamical
molecular
...
networks
Evolution in Silico and in Vitro : The RNA Model http://www.degruyter.de/journals/bc/2001/pdf/382_1301.pdf
Structure
and
Function
of
Complex
Networkshttp://www.imedea.uib.es/~dchialvo/Home/Publications/Submitted/TICS_2004inpress.pdf
Brain
新陳代謝網路
Graph Modeling of Metabolism (http://recomb2000.ims.u-tokyo.ac.jp/Posters/pdf/84.pdf)
Functional
cartography
of
complex
metabolic
http://amaral.chem-eng.northwestern.edu/Publications/Papers/Guimera-2005-Nature-433-895.pdf
networks
Neuropercolation:
A
Random
Cellular
Automata
Approach
to
Spatio
...
http://cnd.memphis.edu/neuropercolation/paper/lectureNotes.pdf
Genome Rearrangement: Recent Progress and Open Problems http://www.proba.jussieu.fr/bulletin/ArticleDurrett.pdf
Reconstructing Pathways in Large Genetic Networks from Genetic ...
http://samba.unm.edu/~wagnera/JCompBio04_recon.pdf
Inferring
Network
Mechanisms:
http://phys.columbia.edu/~mjm/droso_supp.pdf
The
Drosophila
melanogaster
Protein
Protein complexes and functional modules in molecular networks
http://www.mit.edu/people/leonid/publications/Spirin_Mirny_ProtComplex.pdf
The Yeast Protein Interaction Network Evolves Rapidly and Contains ...
http://samba.unm.edu/~wagnera/MBE01.pdf
An Introduction to Latent Semantic Analysis
http://lsa.colorado.edu/papers/dp1.LSAintro.pdf
An Introduction to Latent Semantic Analysis by Patrick Kellogg
http://www.patrickkellogg.com/school/papers/LSA.htm
http://www.lri.fr/~roche/CoursDEA/TransparentsCoursDEA/LSA_DEA2.pdf
http://www.lri.fr/~roche/MasterPro/Transparents/LSA_MasterPro2004-2005.pdf
Probabilistic Latent Semantic Indexing http://www.cs.brown.edu/people/th/papers/Hofmann-SIGIR99.pdf
http://www.int.tu-darmstadt.de/publications/Hofmann-UAI99.pdf
Latent Semantic Indexing: a Probabilistic Analysis
http://www.cs.berkeley.edu/~christos/ir.ps
67/70
...
Indexing by Latent Semantic Analysis Scott Deerwester Graduate ...
http://www.aifb.uni-karlsruhe.de/Lehrangebot/Sommer2001/SemanticWeb/papers/deerwester90indexing.pdf
Applications of Latent Semantic Analysis
http://www.k-a-t.com/papers/Cog-Sci-03.pdf
Unsupervised Learning by Probabilistic Latent Semantic Analysis
http://www.cs.helsinki.fi/u/vmakinen/stringology-k04/hofmann-unsupervised_learning_by_probabilistic_latent_semanti
c_analysis.pdf
Improving Text Classification using Local Latent Semantic Indexing
http://research.microsoft.com/users/byzhang/publications/ICDM2004-LLSI.pdf
TOPICAL CLUSTERING OF BIOMEDICAL ABSTRACT by SELF ORGANIZING MAPS
http://biocomp.ge.ismac.cnr.it/contFiles/pdf/BGRS04_FattoreM_ArrigoP_submitted.pdf
Temporal-Semantic Clustering of Newspaper Articles for Event Detection
http://www3.uji.es/~berlanga/Chronology/pris02.pdf
Getting Better Results With Latent Semantic Indexing
http://www.cs.berkeley.edu/~nakov/selected_papers_list/nakov_esslli00.pdf
Indexing by Latent Semantic Analysis Scott Deerwester Graduate ...
http://www.aifb.uni-karlsruhe.de/Lehrangebot/Sommer2001/SemanticWeb/papers/deerwester90indexing.pdf
Intranet indexing using semantic document clustering http://w3.informatik.gu.se/~dixi/reports/lsi.doc
Machine Perception and Learning of Complex Social Systems http://reality.media.mit.edu/pdfs/thesis.pdf
Network thinking in ecology and evolution http://darkwing.uoregon.edu/~pphil/pubs/proulx_tree2005.pdf
Skeleton of Complex Networks http://conf.kias.re.kr/statphys/2004/sp22/new-talk/STATPHYS2004.pdf
Graph Theory Approaches to Protein Interaction Data Analysishttp://www.cs.utoronto.ca/~natasha/GT_PPI.pdf
Dynamics of Cerebral Cortical Networks http://www.genesis-sim.org/GENESIS/iBoG/iBoGpdf/chapt9.pdf
Schrodinger’s Legacy: Systems and Life http://www.hamilton.ie/SystemsBiology/files/2005/schroedinger.pdf
基因
Models of Genome Evolution http://www.cs.nyu.edu/cs/faculty/mishra/PUBLICATIONS/03.evbydup.pdf
蛋白質
Expanding protein universe and its origin from the biological Big Bang
http://dokhlab.unc.edu/papers/dss_pnas02.pdf
生物網路模型
Structure of biological networks (http://insilico.mit.edu/Structure.pdf)
Quantitative Biology for the 21 http://www.maa.org/mtc/Quant-Bio-report.pdf
stochiometry matrices
68/70
...
在描述生物化學的反應時
節點為蛋白質而連線為反應
multigraph
節點為蛋白質而重複連線為不同的反應例如結合或類似
hypergraph
MATLAB 语言是当今国际上科学界 (尤其是自动控制领域) 最具影响力、也是最有活力的软件。它起源于矩阵
运算，并已经发展成一种高度集成的计算机语言。它提供了强大的科学运算、灵活的程序设计流程、高质量的
图形可视化与界面设计、便捷的与其他程序和语言接口的功能。MATLAB 语言在各国高校与研究单位起着重大
的作用。
MATLAB 语言由美国 The MathWorks 开发，2003 年推出了其全新的 MATLAB 6.5.1 正式版。目前最新
版本 Release 14 (MATLAB 7.0) 的 Service Pack 1，2004 年 9 月正式推出。
Matlab 7 (R14) 注 册 码 1 ： 14-58204-39252-07634-11570-16849-09455-22809-05445-13616-2905808276-06885-12215-41987-21894-60423-57622-18647-58411-24238-20443-59027-07209-27706-28292-14609-1539348293-13036-12293-43713-57876-43362
Matlab 7 (R14) 注 册 码 2 ： 14-44889-04614-04275-46147-23559-43066-41714-23083-65272-0499717469-27919-17226-59862-27901-53983-56217-20094-53460-62647-58166-24499-35558-19511-44882-53016-2565861109-03776-34505-00776-15813-07183
Matlab 7 (R14) 注 册 码 3 ： 14-02863-32167-49274-14620-55383-23033-26960-31585-34411-6450559377-01535-25859-02729-42340-44002-31180-19826-51572-37426-25833-53451-02530-20898-18863-41455-2922813667-31335-59199-04825-64974-59539
Matlab 7 (R14) 注 册 码 4 ： 14-31062-57999-64507-28421-43456-35967-55178-20933-12777-3402644684-07146-17266-64175-62985-50264-38373-35045-48372-03550-51628-06609-24618-64094-55458-49747-0484824494-63995-46820-01807-20764-37086
Matlab 7 (R14) 注 册 码 5 ： 14-17107-22787-29968-14354-40195-52542-37833-61505-03296-5754210587-61927-24639-23185-47049-43791-51460-27360-26190-65454-27576-01212-32058-00132-19787-28479-0872821126-59415-29872-44199-41000-65357
Mathworks Matlab V7.0 R14 PLP
14-35392-57842-08484-51918-65127-20615-58251-25863-41734-36749-62468-18588-174
16-34304-09016-01360-60742-14636-39593-62798-29971-50921-64950-39730-63578-402
65-12488-31210-33908-42704-31275-23962-29047
14-13531-19296-24560-20147-44308-03958-00844-31365-30221-49822-53101-49371-404
64-62160-05002-40310-37359-48633-64466-15884-57760-64649-27202-63216-21440-581
83-16381-31228-38995-55241-00649-11266-22286
14-13079-17698-23560-45609-07352-05352-05350-56924-06021-29849-49518-65460-213
29-16030-08711-42918-36765-10752-22284-40230-60254-23812-15890-10926-28557-333
10-64529-52031-44719-51491-31733-52511-04580
14-33171-17783-64592-35456-50316-07400-44955-36962-37980-13959-26362-17710-640
56-38199-63996-17554-20508-45692-11950-26271-04118-15594-56475-57425-15435-478
45-62908-17443-47971-65086-33513-16377-01657
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14-24872-36179-32308-22113-61986-33825-00564-45479-60426-10395-51330-19488-622
01-37785-19497-45389-18974-51073-03706-04875-59691-49786-28969-00719-61582-144
02-53787-33213-56814-33775-57022-14254-56498
70/70