Transcript PowerPoint 簡報

Self-Organized
Criticality
How Nature Works : The Science of Self-Organized Criticality
by Per Bak
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。
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.
1.
2.
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.
• 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. This is the (selforganized) critical slope.
• Critical states of a system are signaled by a power-law
distribution in some observable. In the case of a solidliquid 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.
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.
The Bak-Tang-Wiesenfeld
(BTW) sandpile model
• 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 :
Z ( x, y)  Z ( x, y)  4
Z ( x  1, y)  Z ( x  1, y)  1
Z ( x, y  1)  Z ( x, y  1)  1
• 240 × 240 pixel
D(s)
s
Power law
log D(s)  k log s
D(s)  s
k
Power law
• http://en.wikipedia.org/wiki/Power_law
• A power law is any polynomial relationship that
exhibits the property of scale invariance. The
most common power laws relate two variables and
have the form
f ( x)  axk  o( x k )
• where a and k are constants. Here, k is typically
called the scaling exponent.
• Scale invariance
• The main property of power laws that makes them
interesting is their scale invariance. Given a
relation f ( x)  axk , or, indeed any homogeneous
polynomial, scaling the argument x by a constant
factor causes only a proportionate scaling of the
function itself. That is,
f (cx)  a(cx)  c f ( x)  f ( x)
k
k
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.
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）。
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
• 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。
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。
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）。
Exponential growth
• http://en.wikipedia.org/wiki/Exponential_law
x
2
3
x
Exponential growth
Linear growth
Cubic growth
The Long Tail
• http://en.wikipedia.org/wiki/Long_Tail
• The phrase The Long Tail was first coined by
Chris Anderson in an October 2004 Wired
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.
長尾理論
• http://www.books.com.tw/exep/prod/booksfile.php?item=001
0341673
• 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
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。
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.
– 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 selforganised 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.
– 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.
網路(NET)
Réka Albert, Albert-László Barabási, Hawoong Jeong,
and Ginestra Bianconi
Power-law distribution of the World Wide Web
Science 287 2115a (2000)
http://www.nd.edu/~alb/
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。
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。
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
網路包圍世界
• 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。
Internet
From Y. Tu,
“How robust is the
Internet?” ,
Nature 406, 353
(2000)
Protein binding network
Binding network: enzymes bind to their substrates in a metabolic network and
to other proteins to form complexes
Some scale-free networks
may appear similar
In both networks the degree distribution is scale-free P(k)~ k- with
~2.2-2.5
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。
Self-Organized Criticality &
Earthquakes
Interacting Earthquake Fault
Systems:
Cellular Automata and beyond...
D. Weatherley
QUAKES & AccESS
3rd ACES Working Group Meeting
Brisbane, Aust. 5th June, 2003.
Scope of the Problem
Complicated interactions
between faults due to stress
transfer during Eqs
X
Multi-Fractal fault heirarchy
Nonlinear Rheology
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?)
Accelerating Moment
Release
Number of Eqs, N(M)
Power-law
Cumulative moment
Bufe & Varnes, 1993
N(M) ~ M-b
1920
EQ magnitude, M
1940
1960
Year
1980
Archetypical Earthquake Model:
Burridge-Knopoff Block-Slider
(Figure thanks to J.Rundle, ICCS 2003 presentation)
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
●
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.
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
Financial Earthquakes, Aftershocks And Scaling In
Emerging Stock Markets
• http://www.comdig.org/article.php?id_articl
e=14912
• http://www.itpa.lt/~kvantas/BK/04/EconoPh
ys/PAFinanc.pdf
Stock trade patterns could predict
financial earthquakes
• 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."
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。
這樣的道理投資人都懂，但下一步就有很多人想偏。
原因在於，並非所有產業都會出現組織效應，因此現
在的市場贏家並不一定能夠靠著組織效應爬上唯我獨
尊的地位。「譬如戴爾電腦是一家公認的好公司，也
具有領導地位。但是你喜歡戴爾電腦並不代表我也會
喜歡，或者我也非用它不可。」戴爾電腦唯有靠不懈
的客戶服務、有效率的存貨管理、耐用的品質來壟固
它的江山，它的市場佔有率並不能帶來重大的加值利
益。戴爾的例子正揭示著，「整個電腦製造業，或者
說，任何生產製造標準化產品的廠商，包括昇陽、惠
普等都是如此。他們都是很成功的公司，但投資人不
應用組織效應理論誇張他們的價值，而是應該用衡量
汽車廠商的標準衡量他們。」
另一個值得思考的指標是上鎖能力，也就是說，當消費者見異思遷的時候，他∕她必須付出相當的代價
，這層束縛因此讓消費者打消了出走念頭。美國線上就是這麼個能夠把消費者鎖住的廠商，想想看，
想要脫離美國線上，改用其他ISP的消費者，有多少麻煩事得處理︰必須通知所有親朋好友更改電子郵
件信箱、學會使用新的界面、同時不能再上熟悉的聊天室。正是這些麻煩事幫助AOL守住了它的2300
萬用戶。
AOL能，並不代表其他人也能。「投資人在此處又陷入了上鎖能力的迷思，以為所有的網路技術都能
自動創造上鎖效應」，Hal Varian說，「譬如說雅虎，以搜尋引擎起家的時候其實就沒有鎖住客戶的能
力，於是它才不斷鋪展服務路線，發送免費電子郵箱、開發聯絡簿（address book）、行事曆」，最
後發展出全套的個人數位助理服務，終於以黏性取勝，完成網路帝國的佈局。
此處又繞回「黏性」的老話題了，黏性很重要，早已是人盡皆知的道理，Hal Varian不厭其詳，再推演
一遍網路股價值與黏性的關係，正因為他仍然相信，一個黏得住訪客的網站就是好網站，無論市場對
內容網站、入門網站、網店所有這些與B2C有直接關連的字眼有多麼冷淡。「傳統評估股價的方式完
全無法套用在dot-com身上，在1999年，評價網路股的唯一工具就是使用人數，經過教訓與修正之後，
投資人認同，訪客的忠誠度也應列入考量。於是，現在的規則應該是，網路的收益來自網站在三個方
面的得分數︰不重覆訪客的人數、訪客停留時間、以及訪客回訪機率。」
很多人拿網際網路來和人類文明史上的其他科技大躍進相比，譬如電話、火車、電視等發明，甚至與
開啟一整個工業時代的工業革命來相提並論，前者是因為它們共同具有通訊、運輸、傳播的功能，後
者則因網路被認為同樣具有改進人類生產效能的能力。Hal Varian的看法屬於前者，但他更認為，同樣
都是通路，網路更具有前所未有的容量︰「1890年代是美國史上出現最多鐵路公司的一段時期，但過
度投資帶來過度供給，於是那個年代同時也是美國史上最多鐵路公司倒閉的時期。歷史可能重演，（
dot-com倒閉潮－這是不是正在閱讀的你所想到的呢？）但投資一條鐵路和投資一段資訊高速公路會有
相同的下場嗎？當然不會。鐵路只能讓火車在上面跑，只能運送人和貨物，網路卻能運送資料、聲音
、影像、交易。如果鐵路也能有這麼多種用途，鐵路公司就不會在1890年大量倒閉。」
認同Hal Varian也好，質疑他的邏輯也罷，你不能不同意，我們可以從這裡開始思考。（
[email protected], 11/30/2000）
HIGHLY OPTIMIZED TOLERANCE.
Many natural and man-made systems exhibit power-law statistics. That is, when you plot
the likelihood of an event (e.g., sizes of forest fires, power outages, and web file transfers,
or losses due to hurricanes, floods, earthquakes, and man-made disasters) as a function of
size the resulting graph will fall off proportionally to the size of the event raised to some
exponent. Interactions or phenomena at many size scales (from very small to very large)
contribute to the overall state of these systems.
One theory which tries to explain all this is "self organized criticality." Jean Carlson of UC
Santa Barbara ([email protected]) and John Doyle of Caltech
([email protected]) now propose another theory, called highly optimized tolerance
(HOT), which they believe does a better job of accounting for the tendency in
interconnected systems to gain a measure of robustness against uncertainties in one area by
becoming more sensitive elsewhere. As with energy conservation or the inexorable increase
in entropy, efforts to violate the robustness principle will fail. Especially in biological
evolution or in engineering, this means that a system might obtain robustness against
common and designed-for uncertainties and yet be hypersensitive to design flaws or rare
events.
For example, organisms and ecosystems exhibit remarkable robustness to large variations in
temperature, moisture, nutrients, and predation, but can be catastrophically sensitive to tiny
perturbations, such as a genetic mutation, an exotic species, or a novel virus. Engineers
deliberately design systems to be robust to common uncertainties. Cost and performance
tradeoffs force an acceptance of some hypersensitivity to (one hopes) rare perturbations.
In evolved or designed systems, this tradeoff leads to the "robust, yet fragile" characteristic
of complexity, one feature of which is power laws. Doyle and Carlson have been exploring
the application of their theory to a number of biological and engineering problems with the
help of experts in those fields. (Carlson, Doyle, Physical Review Letters, 13 March 2000;
Select Article; a longer version appears in Physical Review E, August 1999.)
[SciScape] 複雜系統之強健性與指數定律
複雜系統之強健性與指數定律 許多自然界或人為的系統都能遵守統計上的指數定律 power
law, 比 如說當事件和事件規模具有函數關係時, 事件將隨著事件規模的增 加而呈指數式的遞
減. 這些事件包含森林大火之範圍大小, 停電時間 , 颶風所造成的損失, 透過網路傳送的檔案,
地震等等. 而指數律正是 Self organized criticality 及edge of chaos 狀態的明顯特徵之一. 加州
大學 Jean Carlson 以及加州理工 John Doyle 提出 highly optimized tolerance 理論, 他們宣稱,
對於由許多子系統所連結成的複雜系統, 不管是自然演進還是人為設計, 當該系統可以有效
的容忍某些不確 定因素時 (具強健性), 那麼這系統對於其它未被考慮到的不確定因 素將變得
更敏感, 也就是說, 強健性和敏感度具有相互遞換的效果, 其中, 這不確定因素包含系統內部
的不確定因素以及外在環境的干 擾. 就如同能量守恆與熵, 如果試圖去打破這強健性原則, 都
會面臨 失敗. 以生態系統為例, 如果生態可以容忍氣溫變化, 濕度, 養分等巨幅變 化, 那麼這
生態卻可能無法容忍一些意料之外的小干擾, 如基因突變 , 外來族群遷入, 或新的病毒. 這些
干擾可能會造成生態巨大的改變. 因此當一具有指數律的複雜系統處於 highly optimized
tolerance 狀態 , 強健性原則將導致該系統 “強健, 但易被破壞”. 研究者在文獻中提 出幾個
生物學及工程例子來支持他們的論點, 並提供相關領域參考.
--論文出處: J. M. Carlson and John Doyle, Highly Optimized Tolerance: Robustness and Design
in Complex Systems, Physical Review Letters -- March 13, 2000 -- Volume 84, Issue 11, pp. 25292532