How Nature Works : The Science of Self-Organized Criticality by Per Bak

1. Self-Organized Criticality How Nature Works : The Science of Self-Organized Criticalityby Per Bak

2. Self-Organized Criticality (自組織臨界現象)一種相變現象 • 相變與臨界現象是同一種物質在因應不同外在變因，如溫度、壓力等參數的不同，而表現的不同巨觀行為。如水的固、液、汽三相變化，以及導體與陶瓷材料在低溫時所出現的超導現象，都是屬於相變現象的範圍。更廣義的延伸，凡一系統在特定參數條件下有非連續性變化，如蛋自質折疊過程中的二態變化。或是一些非線性系統的自組臨界現象(Self-Organized Criticality)，皆可稱為相變現象。

3. Self-Organized Criticality (自組織臨界現象) In physics, a critical point is a point at which a system changes radically 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. • Bak, P., Tang, C. & Wiesenfeld, K. Self-organized criticality: An explanation of 1/f noise, Phys. Rev. Lett., 59, 381-384, 1987. • Bak, P. & Chen, K. Self-organized criticality. Scientific American, 264, 46-53, 1991.

4. Self-organized critical phenomena, by contrast, is exhibited by driven systems which reach a critical state by theirintrinsic 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. This is the (self-organized) critical slope.

5. Critical states of a system are signaled by a power-law distribution in some observable. In the case of a solid-liquid transition, one can measure the heat-capacity of the system. In the case of sand-piles, one can measure the distribution of avalanche sizes. In the present case of internet access, curiosity is measured. The analogy with sand piles is clear: a grain dropped onto the pile corresponds to an initial access to the document. The size of an avalanche corresponds to depth of reading of a document. In order to maintain a critical slope in a sand pile in a finite geometry, sand is removed at the edges of the pile. One can think of the sand pile as sitting on a table. Sand falls off as it reaches the edge of the table. The same process could be operating in the case of hypertext access to a document: once readers have achieved a certain depth in the document, they may decide that the document is sufficiently useful to them that they should obtain a hardcopy. At that point, they will stop issuing http requests and then issue a ftp request to retrieve the full document.

6. Self-Organized Criticality • http://www.cmth.bnl.gov/~maslov/soc.htm • Self-Organized Criticality (SOC) is a concept introduced by Per Bak, Chao Tang, and Kurt Wiesenfeld in 1987.

7. The Bak-Tang-Wiesenfeld (BTW) sandpile model

8. Starting with a flat surface Z(x,y) = 0 for all x and y. • Add a grain of sand: Z(x,y) = Z(x,y) + 1 . • And avalanche if Z(x,y) > Zc:

9. 240 × 240 pixel

11. D(s) s

12. Power law

13. Power law • http://en.wikipedia.org/wiki/Power_law • A power law is any polynomial relationship that exhibits the property of scaleinvariance. The most common power laws relate two variables and have the form • where a and k are constants. Here, k is typically called the scaling exponent.

14. Scale invariance • The main property of power laws that makes them interesting is their scale invariance. Given a relation , or, indeed any homogeneous polynomial, scaling the argument x by a constant factor causes only a proportionate scaling of the function itself. That is,

15. To determine if this picture is correct, we must correlate http and ftp accesses. This can be done as follows: for each ftp download, we search the http log to find a corresponding session. If there is one, we measure the curiosity exhibited in that session. Unfortunately, the small number of such events (AL-SIM:127, CA-FAQ:214) renders the experiment inconclusive. Nonetheless, comparison of figure 5 (exponential fit) with figure 6 (power-law fit), suggests that a power law may account better for the data than an exponential, at least in the case of CA-FAQ.

16. 帕雷托法則(80/20法則 ) • 帕雷托法則（Pareto principle），也稱為80/20法則 • 這個法則最初是義大利經濟學家維弗雷多·帕雷托（Vilfredo Pareto）在1906年對義大利20%的人口擁有80%的財產的觀察而得出的，後來管理學思想家約瑟夫·朱蘭（Joseph M. Juran）和其他人把它概括為帕雷托法則。 • 若進一步推算，以掌握了80%財富的人作統計，會發現4%的人口（20% × 20%）掌握了社會64%(80% × 80%)的財富。這一猜想說明絕大部分的產量或結果取決於一小部分的投入和勞動。在商業活動中，80%的銷量來自與20%的客戶。（80%的80%是64%，20%的20%是4%，意味著這是64/4法則）。

17. 80/20法則 作者：李察‧寇區/著 出版社：大塊文化 出版日期：2005年04月25日

18. 在許多的實際經驗與現象，80％的結果是由20％的原因所決定的。在許多的實際經驗與現象，80％的結果是由20％的原因所決定的。 • 80％的成果來自於20％的努力 。 • 一個公司的營運成果中的80％由20％的各部門員工所創造。而主要創造業績的業務部門，當中有20％的業務人員能拿到佔業務部門總業績數字80％的訂單。 • 學校裡有20％的學生成績在平均80分以上。 • 20％的重點客戶提供80％的銷售額。 • 人的一天只有20％的時間，約莫5小時以內的生產力與學習力最高。 • 20％的網友貢獻產出80％的網路文件資源。 • 20％的人口掌握了80％的財富。 • 職業運動聯盟20％的運動員擁有80％的薪水。 • 成績在前面20％的運動員有80％的機率能拿到獎牌。 • 20％的罪犯引發了80％的犯罪。

19. 善用80/20法則 • 「關鍵少數法則」（Law of the Vital Few）、最省力法則（Principle of Least Effort）和「不平衡原則」（Principle of Imbalance） • 把關鍵的20％做到最好「把錢花在刀口上，讓最佳人才掌握最佳機會，」這是傑克‧威爾許（Jack Welch）描述自己在擔任奇異（GE）執行長時所扮演的角色。換言之，80/20法則可做為組織在配置資源時的重要依據。

20. 贏家的80/20法則 • 贏家總有許多80/20法則華倫‧巴菲特（Warren Buffet）只投資自己熟悉的事業，不但買得少、也很少賣。他形容自己的投資哲學是：「近乎懶惰。」《80/20法則》的作者李察‧寇區（Richard Koch）將這種少做多得的情形，稱為「有生產力的懶惰」（productive laziness）。

21. Exponential growth • http://en.wikipedia.org/wiki/Exponential_law Exponential growth        Linear growth        Cubic growth

22. The Long Tail • http://en.wikipedia.org/wiki/Long_Tail • The phrase The Long Tail was first coined by Chris Andersonin an October 2004Wired magazine article to describe certain business and economic models such as Amazon.com or Netflix. • The term long tail is also generally used in statistics, often applied in relation to wealth distributions or vocabulary use.

23. 長尾理論 • http://www.books.com.tw/exep/prod/booksfile.php?item=0010341673 • 長尾理論─打破80/20法則的新經濟學 • 企業界向來奉80/20法則為鐵律，認為80%的業績來自20%的產品；企業看重的是曲線左端的少數暢銷商品，曲線右端的多數商品，則被認為不具銷售力。但本書指出，網際網路的崛起已打破這項鐵律，99% 的產品都有機會銷售，「長尾」商品將鹹魚翻身。 暢銷

24. 長尾理論是許多企業成功的秘訣 • Google的主要利潤不是來自大型企業的廣告，而是小公司（廣告的長尾）的廣告； • eBay的獲利主要也來自長尾的利基商品，例如典藏款汽車、高價精美的高爾夫球桿等。 • 一家大型書店通常可擺放十萬本書，但亞馬遜網路書店的書籍銷售額中，有四分之一來自排名十萬以後的書籍。這些「冷門」書籍的銷售比例正以高速成長，預估未來可占整體書市的一半。

25. Some Ideas Bak Offers Regarding Economics • Bak uses a metaphor and the concept of long-term equilibrium to describe how traditional economists model economics. In this metaphor, economic flow and the economic agents are compared with water and reservoirs, respectively. The economic flow then will correspond to water flowing continuously and linearly through the reservoirs in such a manner that all reservoirs obtain the best value they can (with accumulation of water corresponding to economic satisfaction) - achieving some sort of stability equivalent to Nash equilibrium.

26. He rejects this traditional view, considering it simplistic, and presents his idea that the dynamics of economic flows is more like the dynamics observed in (and in his models of) sand piles, as changes are not linear and continuous but rather non-linear and discrete. The forces which each individual agent (grain) exercises over the others plays an important role in the dynamics of the system. He considers that there is friction in the economic flow and that agents are not perfectly rational. He believes that friction prevents (long-term) equilibrium from being reached and that fluctuations in economics are of a different nature than those notions the traditional economists propose. He refers to empirical data to support his suggestion that economic systems would be better modelled as critically self-organised systems. For example, he discusses results obtained by Benoit Mandelbrot, in which the percentage of monthly variation in the price of cotton versus the number of occurrences of such percentages over several months follows a power law distribution.

27. Bak also hypothesises that the dynamics of an economic system should be somewhat similar to that shown by the evolution model described above, where agents (consumers, producers, traders, thieves) interact with each other in accordance with the set of options they have, exploiting such options in order to increase their 'happiness'. These ideas depict a co-evolution model where the more successful agents will survive while the least fitted ones will not or will be forced to mutate by changing their strategy.

28. 網路(NET)

29. Réka Albert, Albert-László Barabási, Hawoong Jeong, and Ginestra BianconiPower-law distribution of the World Wide WebScience 287 2115a (2000) http://www.nd.edu/~alb/

30. 無比例(scale free) 網路 • Barabási說：“事情變得很明顯--我們所考察的情況比隨機網路所描述的要複雜”-沒有鐘形曲線。網路是具有許多個有少量連線的網站、少量具有中等數量連線的網站和為數極少的具有大量聯線的網站。這所產生的是一條不斷遞減的曲線，其特徵是物裡學家所說的一種冪次法則。鐘形曲線的連線平均數或比例不見了。 • Barabási宣稱，全球資訊網是一個無比例(scale free)網路。

31. 小世界 • Barabási說：“這一分布所表明的事實是，網路的結構被少數連線極多的網站所主宰。” 他把這些網站稱為『集散點』--典型的實例有雅虎和Napster --它們之所以發展狀大，是因此它們提供了獲得我們想要獲得的訊息的捷徑。這一結構的一個令人好奇的性質是，從一個網站到達全球資訊網上的任何另一個網站只需點擊很少的次數。他說：『從一個網頁到達任何另外一個平均只需點擊19次。』這表明，全球資訊網是一種類型的小世界。

32. 富者愈富 • 熱門的網站會愈來愈熱門 • 1999年勞倫斯及吉爾斯發現仍有16%的網頁沒有被任何搜尋引擎所含蓋 • 大部分的搜尋引擎是根據網頁的熱門程度作「索引」的，愈熱門的網頁，愈先編入來愈索引 • 開設新的網站加入一些鏈結，加的自然是熱門的網站

33. 網路包圍世界

34. 我們被網絡包圍著：社會和職業的網絡。生態系統是網路，甚至我們的身體以致細菌也是由化學物質的網路維持生命。我們被網絡包圍著：社會和職業的網絡。生態系統是網路，甚至我們的身體以致細菌也是由化學物質的網路維持生命。

35. Internet From Y. Tu, “How robust is the Internet?” , Nature 406, 353 (2000)

36. Protein binding network Binding network: enzymes bind to their substrates in a metabolic network and to other proteins to form complexes

37. Some scale-free networks may appear similar In both networks the degree distribution is scale-free P(k)~ k-with ~2.2-2.5

38. 食物網 • 複雜的生態系，綿密的食物網 • 天生我才必有用。 • 當生物物種愈多時，食物網較容易維持平衡與穩定。 • 當食物網中有生物消失時，其他生物也會受到影響。

39. Self-Organized Criticality & Earthquakes

40. Interacting Earthquake Fault Systems: Cellular Automata and beyond... D. Weatherley QUAKES & AccESS 3rd ACES Working Group Meeting Brisbane, Aust. 5th June, 2003.

41. Scope of the Problem Complicated interactions between faults due to stress transfer during Eqs Nonlinear Rheology X Earthquake Fault systems are COMPLEX: • Many degrees of freedom • Strongly coupled spatial and temporal scales • Nonlinear dynamical equations & constitutive laws • Multi-physics: mechanical, chemical, thermal, fluids, (EM?) Multi-Fractal fault heirarchy

42. Accelerating Moment Release Bufe & Varnes, 1993 Power-law Cumulative moment N(M) ~ M-b Number of Eqs, N(M) 1920 1940 1960 1980 Year EQ magnitude, M

43. Archetypical Earthquake Model: Burridge-Knopoff Block-Slider (Figure thanks to J.Rundle, ICCS 2003 presentation)

44. Per Bak's Sand-pile Automaton • Rectangular grid of sites • Each site may support a maximum of 4 grains of sand • Sand is added to sites at random • Sites with 4 grains avalanche i.e the sand cascades to the nearest neighbouring sites • Redistribution of sand can trigger neighbouring sites to fail which in turn may trigger failure of their neighbours -> avalanches may be any size between one site and the entire grid

45. What has the BK model taught us? The Block-Slider model can reproduce the power-law earthquake size-distribution without prescribing any power-law correlations/structure. Power-law distributions are a natural consequence of the dynamics of systems with: • Large numbers of elements (DOFs) • Nonlinear interactions between elements • External loading of elements • Energy dissipation during interaction cascades This conclusion was drawn by Per Bak et al by studying an analogous model, the so-called Sandpile Automaton. Per Bak proposed the concept of self-organised criticality as a description of the dynamics of such systems.

46. Thermodynamic Criticality & Self-Organised Criticality THERMODYNAMIC CRITICALITY • Occurs when thermodynamic systems are driven through a phase transition by varying properties such as temperature, pressure etc. • Characterised by a sudden change is macroscopic properties of the system • As a critical point is approached, long-range spatial and temporal correlations emerge →power-laws • Thanks to mean-field theory etc. thermodynamic criticality is relatively well understood and the values of various measurable quantities (e.g power-law exponents) can be predicted SELF-ORGANISED CRITICALITY • Certain classes of systems do not require “tuning” to go critical • Criticality represents an attractor for the dynamics of said systems • SOC is elegent because it can explain observations of power-law correlations in natural systems without needing to hypothesize the existence of a “god-like” system-tuner who turns the knobs to cause criticality

48. Financial Earthquakes, Aftershocks And Scaling In Emerging Stock Markets • http://www.comdig.org/article.php?id_article=14912 • http://www.itpa.lt/~kvantas/BK/04/EconoPhys/PAFinanc.pdf

49. Stock trade patterns could predict financialearthquakes • http://web.mit.edu/newsoffice/2003/powerlaw.html • "We have found that the artificial world of the financial markets follows a pattern similar to one found in our natural world," said Gabaix. "Trading on the stock market has a lot of randomness, but at the end of the day you find that a pattern emerges that matches power-law patterns found empirically in data from systems as diverse as earthquakes and human language."

50. 穿透迷霧看網路效應 專家學者說話的特色在於，景氣的時候他們語調鏗鏘，對任何火上烹油的誇張現象都有一番解釋；不景氣的時候他們同樣語調鏗鏘，還是對任何淒涼蕭條的失意產業都有一番解釋。有些屬於後見之明，有些屬於人云亦云，但也有一些，像柏克來加大資訊管理學院院長Hal Varian最近的評論，就從網路股從炫爛歸於平淡的歷程中，提煉出兩個本質問題，沒有花俏的預測，但具有咀嚼的價值。在B2C、B2B、基礎設備產業或多或少都受到質疑之際，回溯本質對迷惑的人們來說，自有提神醒腦的作用。 1999網路幸福年已逝，就連讓它從雲端跌進谷底的2000年都進入尾聲，Hal Varian此時提出疑問︰究竟讓整個1999年舉世皆high的網際網路，有哪些真正的價值？他沒有預設一個答案，但舉出兩大線索，請大家從組織效應（Network effect）與上鎖能力（lock up）思考起。 「很多投資人都誤解了網際網路的這兩大趨動力。」Hal Varian認為，組織效應和上鎖能力是判斷網路經濟之下商業模式的兩大指標。所謂組織效應，即是相加大於總和，「如果全世界只有十個人用傳真機，只能說傳真機是項近乎無用的發明；但是當一億人都有傳真機，因而構成一個密實的網路之時，它的價值就會暴增。」Hal Varian說。 在一個組織效應強大的市場裡，作一個先驅型、迅速擴張的重量級玩家，在吞吃市場佔有率的同時，組織效應也將呈幾何級數成長，於是就容易產生「贏家通吃」（winner-take-all）的結構，由一、兩個大廠佔領整個市場，其他只能揀它們吃剩的碎屑，微軟和eBay正是典型的例子。