overview of complex systems - university of vermontoverview course information major centers...

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Overview of Complex SystemsPrinciples of Complex Systems

Course CSYS/MATH 300, Fall, 2009

Prof. Peter Dodds

Dept. of Mathematics & StatisticsCenter for Complex Systems :: Vermont Advanced Computing Center

University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

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OutlineCourse Information

Major CentersResourcesProjectsTopics

FundamentalsComplexityEmergenceSelf-OrganizationModelingStatistical MechanicsUniversalitySymmetry BreakingThe big theoryTools and TechniquesMeasures of Complexity

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Basics:

I Instructor: Prof. Peter DoddsI Lecture room and meeting times:

307 Lafayette, Tuesday and Thursday, 10:00 am to11:30 pm

I Office: 203 Lord House, 16 Colchester AvenueI E-mail: [email protected] Website:

http://www.uvm.edu/ pdodds/teaching/2009-08UVM-300/ (�)I Suggested Texts:

I “Critical Phenomena in Natural Sciences: Chaos,Fractals, Selforganization and Disorder: Conceptsand Tools” by Didier Sornette [12].

I “Critical Mass: How One Thing Leads to Another” byPhilip Ball [3]

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Admin:

I CSYS/MATH 300 is one of two core requirements forUVM’sCertificate of Graduate Study in Complex Systems (�).

I Five course requirement.

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Admin:

Paper products:

1. Outline

Office hours:I Tuesday: 2:30 pm to 4:30 pm

Thursday: 11:30 am to 12:30 pmRm 203, Math Building

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Admin:

Paper products:

1. Outline

Office hours:I Tuesday: 2:30 pm to 4:30 pm

Thursday: 11:30 am to 12:30 pmRm 203, Math Building

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Grading breakdown:

I Projects/talks (55%)—Students will work onsemester-long projects. Students will develop aproposal in the first few weeks of the course whichwill be discussed with the instructor for approval.Details: 15% for the first talk, 20% for the final talk,and 20% for the written project.

I Assignments (40%)—All assignments will be ofequal weight and there will be three or four of them.

I General attendance/Class participation (5%)

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How grading works:

Questions are worth 3 points according to thefollowing scale:

I 3 = correct or very nearly so.I 2 = acceptable but needs some revisions.I 1 = needs major revisions.I 0 = way off.

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Schedule:Week # (dates) Tuesday Thursday1 (9/1, 9/3) guest lecture:

Josh Bongardlecture

2 (9/8, 9/10) lecture lecture3 (9/15, 9/17) lecture lecture4 (9/22, 9/24) Project

presentationsProjectpresentations

5 (9/28, 10/1) lecture lecture6 (10/6, 10/8) lecture lecture7 (10/13, 10/15) lecture lecture8 (10/20, 10/22) lecture lecture9 (10/27, 10/29) lecture lecture10 (11/3, 11/5) lecture lecture11 (11/10, 11/12) lecture lecture12 (11/17, 11/19) lecture lecture13 (11/24, 11/26) Thanksgiving Thanksgiving14 (12/1, 12/3) lecture lecture15 (12/8, 12/10) Project

PresentationsProjectPresentations

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Important dates:

1. Classes run from Monday, August 31 to Wednesday,December 9.

2. Add/Drop, Audit, Pass/No Pass deadline—Monday,September 14.

3. Last day to withdraw—Friday, November 6.4. Reading and exam period—Thursday, December 10

to Friday, December 18.

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More stuff:

Do check your zoo account for updates regarding thecourse.

Academic assistance: Anyone who requires assistance inany way (as per the ACCESS program or due to athleticendeavors), please see or contact me as soon aspossible.

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References

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Centers

I Santa Fe Institute (SFI)I New England Complex Systems Institute (NECSI)I Michigan’s Center for the Study of Complex Systems

(CSCS (�))I Northwestern Institute on Complex Systems

(NICO (�))I Also: Indiana, Davis, Brandeis, University of Illinois,

Duke, Warsaw, Melbourne, ...,I UVM’s Complex System Center (�)

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References

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Books:

I “Modeling Complex Systems” by Nino Boccara [6]

I “Critical Phenomena in Natural Sciences” by DidierSornette [12]

I “Complex Adaptive Systems: An Introduction toComputational Models of Social Life,” by John Millerand Scott Page [10]

I “Micromotives and Macrobehavior” by ThomasSchelling [11]

I “Social Network Analysis” by Stanley Wassermanand Katherine Faust [14]

I “Handbook of Graphs and Networks” by StefanBornholdt and Hans Georg Schuster [7]

I “Dynamics of Complex Systems” by YaneerBar-Yam [4]

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Useful Resources:

I Complexity Digest:http://www.comdig.org (�)

I Cosma Shalizi’s notebooks:http://www.cscs.umich.edu/ crshalizi/notebooks/ (�)

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Projects

I Semester-long projects.I Develop proposal in first few weeks.I May range from novel research to investigation of an

established area of complex systems.I We’ll go through a list of possible projects soon.

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Projects

The narrative hierarchy—explaining things on manyscales:

I 1 to 3 word encapsulation, a soundbite,I a sentence/title,I a few sentences,I a paragraph,I a short paper,I a long paper,I a chapter,I a book,I . . .

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Topics:

Measures of complexity

Scaling phenomena

I Zipf’s lawI Non-Gaussian statistics and power law distributionsI Sample mechanisms for power law distributionsI Organisms and organizationsI Scaling of social phenomena: crime, creativity, and

consumption.I Renormalization techniques

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Topics:

Multiscale complex systems

I Hierarchies and scalingI ModularityI Form and context in design

Complexity in abstract models

I The game of lifeI Cellular automataI Chaos and order—creation and maintenance

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Topics:

Integrity of complex systems

I Generic failure mechanismsI Network robustnessI Highly optimized tolerance: Robustness and fragilityI Normal accidents and high reliability theory

Complex networks

I Small-world networksI Scale-free networks

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Topics:

Collective behavior and contagion in social systems

I Percolation and phase transitionsI Disease spreading modelsI Schelling’s model of segregationI Granovetter’s model of imitationI Contagion on networksI Herding phenomenaI CooperationI Wars and conflicts

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Topics:

Large-scale Social patterns

I Movement of individuals

Collective decision making

I Theories of social choiceI The role of randomness and chanceI Systems of votingI JuriesI Success inequality: superstardom

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Topics:

InformationI Search in networked systems (e.g., the WWW, social

systems)I Search on scale-free networksI Knowledge trees, metadata and tagging

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Definitions

Complex: (Latin = with + fold/weave (com + plex))

Adjective:

1. Made up of multiple parts; intricate or detailed.2. Not simple or straightforward.

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Definitions

Possible properties of a Complex System:

I Many interacting agents or entitiesI Relationships are nonlinearI Presence of feedbackI Complex systems are open (out of equilibrium)I Presence of memoryI Modular/multiscale/hierarchical structureI Evidence of emergence propertiesI Evidence of self-organization

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Examples

Examples of Complex Systems:

I human societiesI cellsI organismsI ant coloniesI weather systemsI ecosystems

I animal societiesI disease ecologiesI brainsI social insectsI geophysical systemsI the world wide web

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Examples

Relevant fields:

I PhysicsI EconomicsI SociologyI PsychologyI Information

Sciences

I CognitiveSciences

I BiologyI EcologyI GeociencesI Geography

I MedicalSciences

I SystemsEngineering

I ComputerScience

I . . .

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Definitions

Complicated versus Complex.

I Complicated: Mechanical watches, airplanes, ...I Engineered systems can be made to be highly robust

but not adaptable.I But engineered systems can become complex

(power grid, planes).I They can also fail spectacularly.I Explicit distinction: Complex Adaptive Systems.

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Definitions

Nino Boccara in Modeling Complex Systems:[6] “... there is no universally accepted definition of acomplex system ... most researchers would describe asystem of connected agents that exhibits an emergentglobal behavior not imposed by a central controller, butresulting from the interactions between the agents.”

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Definitions

The Wikipedia on Complex Systems:“Complexity science is not a single theory: itencompasses more than one theoretical framework andis highly interdisciplinary, seeking the answers to somefundamental questions about living, adaptable,changeable systems.”

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Philip Ball in Critical Mass:[3] “...complexity theory seeks to understand how orderand stability arise from the interactions of manycomponents according to a few simple rules.”

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Definitions

Cosma Shalizi:“The "sciences of complexity" are very much a potpourri,and while the name has some justification—chaoticmotion seems more complicated than harmonicoscillation, for instance—I think the fact that it is moredignified than "neat nonlinear nonsense" has not beenthe least reason for its success.—That opinion wasn’texactly changed by working at the Santa Fe Institute forfive years.”

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Definitions

Steve Strogatz in Sync:

“... every decade or so, a grandiose theory comes along,bearing similar aspirations and often brandishing anominous-sounding C-name. In the 1960s it wascybernetics. In the ’70s it was catastrophe theory. Thencame chaos theory in the ’80s and complexity theory inthe ’90s.”

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Outreach

08/27/2007 09:17 PMComplexity Society

of 1http://www.complexity-society.com/

MembershipTo join TCS apply here.

Complexity Society NewsletterThe August 2007 edition is nowavailable.

Complexity DigestThe current issue of ComplexityDigest 2007.29 is now availableon-line.

Recent Event:Summer School in ComplexityScience organised by ImperialCollege, London,Wye College, Kent, UK.8–17th July 2007.

Forthcoming Events:ECCS’07 European Conferenceon Complex Systems,Dresden,Germany.1-5th October 2007.

New PaperThe Fractal Imagination: NewResources for ConceptualisingCreativity.

Journal"Emergence: Complexity &Organization (ECO)", A journalof research, theory and practiceon Organisations as complexsystems.

HOME ABOUTUS CONTACTS NEWS &

EVENTS LINKS FAQs JOURNAL& PAPERS

SITEMAP

Welcome to theCOMPLEXITY SOCIETY

"The Application of Complexity Science to Human Affairs"

The Complexity Society provides a focal point for people in theUK interested in complexity. It is a community that usescomplexity science to rethink and reinterpret all aspects of theworld in which we live and work.

Its core values are OPENNESS, EQUALITY and DIVERSITY.

Open to all, open to ideas, open in process and activities Equality, egalitarian, non-hierarchical, participative Diverse, connecting and embracing a wide range of views,respecting differences

The society objectives are to promote the theory of complexity ineducation, government, the health service and business as wellas the beneficial application of complexity in a wide variety ofsocial, economic, scientific and technological contexts such assources of competitive advantage, business clusters andknowledge management.

Complexity includes ideas such as complex adaptive systems,self-organisation, co-evolution, agent based computer models,chaos, networks, emergence and fractals.

Membership is open to all and current members include peoplefrom universities, business, and government fundedorganisations.

Page last updated: 13 August, 2007

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Outreach

“The society objectives are to promote the theory ofcomplexity in education, government, the health serviceand business as well as the beneficial application ofcomplexity in a wide variety of social, economic, scientificand technological contexts such as sources ofcompetitive advantage, business clusters and knowledgemanagement.”

“Complexity includes ideas such as complex adaptivesystems, self-organisation, co-evolution, agent basedcomputer models, chaos, networks, emergence, andfractals.”

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Outreach

“The society objectives are to promote the theory ofcomplexity in education, government, the health serviceand business as well as the beneficial application ofcomplexity in a wide variety of social, economic, scientificand technological contexts such as sources ofcompetitive advantage, business clusters and knowledgemanagement.”

“Complexity includes ideas such as complex adaptivesystems, self-organisation, co-evolution, agent basedcomputer models, chaos, networks, emergence, andfractals.”

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Definitions

The Wikipedia on Emergence:

“In philosophy, systems theory and the sciences,emergence refers to the way complex systems andpatterns arise out of a multiplicity of relatively simpleinteractions.

... emergence is central to the physics ofcomplex systems and yet very controversial.”

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Definitions

The Wikipedia on Emergence:

“In philosophy, systems theory and the sciences,emergence refers to the way complex systems andpatterns arise out of a multiplicity of relatively simpleinteractions. ... emergence is central to the physics ofcomplex systems and yet very controversial.”

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Emergence:

Examples:

I Fundamental particles ⇒ Life, the Universe, andEverything

I Genes ⇒ OrganismsI Brains ⇒ ThoughtsI Fireflies ⇒ Synchronized Flashes [videos]I People ⇒ World Wide WebI People ⇒ Behavior in games not specified by rules

(e.g., bluffing in poker)I People ⇒ Religion

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Emergence

Thomas Schelling (Economist/Nobelist):

[youtube] (�)

I “Micromotives andMacrobehavior” [11]

I SegregationI Wearing hockey helmetI Seating choices

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Emergence

Friedrich Hayek (�)(Economist/Philospher/Nobelist):

I Markets, legal systems, political systems areemergent and not designed.

I ‘Taxis’ = made order (by God, Sovereign,Government, ...)

I ‘Cosmos’ = grown orderI Archetypal limits of hierarchical and decentralized

structures.I Hierarchies arise once problems are solved.I Decentralized structures help solve problems.I Dewey Decimal System versus tagging.

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Emergence

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Emergence

James Coleman in Foundations of Social Theory :Weber

CapitalismProtestantReligiousDoctrine

EconomicBehavior

Values

Societal level

Individual level

Coleman

I Understand macrophenomena arises frommicrobehavior which in turn depends onmacrophenomena. [8]

I More on Coleman here (�).

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Emergence

James Coleman in Foundations of Social Theory :Weber

CapitalismProtestantReligiousDoctrine

EconomicBehavior

Values

Societal level

Individual level

Coleman

I Understand macrophenomena arises frommicrobehavior which in turn depends onmacrophenomena. [8]

I More on Coleman here (�).

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Emergence

Higher complexity:

I Many system scales (or levels)that interact with each other.

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Emergence

Even mathematics: [9]

Gödel’s Theorem (roughly):we can’t prove every theorem that’s true.

Suggests a strong form of emergence:

Some phenomena cannot be formally deduced fromelementary aspects of a system.

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Emergence

Even mathematics: [9]

Gödel’s Theorem (roughly):we can’t prove every theorem that’s true.

Suggests a strong form of emergence:

Some phenomena cannot be formally deduced fromelementary aspects of a system.

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Emergence

The idea of emergence is rather old...

“The whole is more than the sum of its parts” –Aristotle

Philosopher G. H. Lewes firstused the word explicity in 1875.

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Emergence

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Definitions

There appears to be two types of emergence:

Weak emergence:System-level phenomena is different from that of itsconstituent parts yet can be connected theoretically.

Strong emergence:System-level phenomena fundamentally cannot bededuced from how parts interact.

(Strong emergence is what Mark Bedau calls magic...)

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Definitions

Strong emergence:System-level phenomena fundamentally cannot bededuced from how parts interact.(Strong emergence is what Mark Bedau calls magic...)

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Definitions

Complex Systems enthusiasts often decry reductionistapproaches . . .

But reductionism seems to be misunderstood.

Reductionist techniques can explain weak emergence(e.g., phase transitions).

‘A Miracle Occurs’ explains strong emergence.

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The emergence of taste:

I Molecules ⇒ Ingredients ⇒ TasteI See Michael Pollan’s article on nutritionism (�) in the

New York Times, January 28, 2007.

nytimes.com

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Reductionism

Reductionism and food:I Pollan: “even the simplest food is a hopelessly

complex thing to study, a virtual wilderness ofchemical compounds, many of which exist incomplex and dynamic relation to one another...”

I “So ... break the thing down into its component partsand study those one by one, even if that meansignoring complex interactions and contexts, as wellas the fact that the whole may be more than, or justdifferent from, the sum of its parts. This is what wemean by reductionist science.”

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Reductionism

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Reductionism

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Reductionism

I “people don’t eat nutrients, they eat foods, and foodscan behave very differently than the nutrients theycontain.”

I Studies suggest diets high in fruits and vegetableshelp prevent cancer.

I So... find the nutrients responsible and eat more ofthem

I But “in the case of beta carotene ingested as asupplement, scientists have discovered that itactually increases the risk of certain cancers. Bigoops.”

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Reductionism

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Reductionism

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Reductionism

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Reductionism

Thyme’s known antioxidants:4-Terpineol, alanine, anethole, apigenin,ascorbic acid, beta carotene, caffeicacid, camphene, carvacrol, chlorogenicacid, chrysoeriol, eriodictyol, eugenol,ferulic acid, gallic acid,gamma-terpinene isochlorogenic acid,isoeugenol, isothymonin, kaempferol,labiatic acid, lauric acid, linalyl acetate,luteolin, methionine, myrcene, myristicacid, naringenin, oleanolic acid,p-coumoric acid, p-hydroxy-benzoicacid, palmitic acid, rosmarinic acid,selenium, tannin, thymol, tryptophan,ursolic acid, vanillic acid.

[cnn.com]

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Reductionism

“It would be great to know how this all works, but in themeantime we can enjoy thyme in the knowledge that itprobably doesn’t do any harm (since people have beeneating it forever) and that it may actually do some good(since people have been eating it forever) and that even ifit does nothing, we like the way it tastes.”

Gulf between theory and practice: baseball andbumblebees.

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Reductionism

“It would be great to know how this all works, but in themeantime we can enjoy thyme in the knowledge that itprobably doesn’t do any harm (since people have beeneating it forever) and that it may actually do some good(since people have been eating it forever) and that even ifit does nothing, we like the way it tastes.”

Gulf between theory and practice: baseball andbumblebees.

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Definitions

Self-Organization

“Self-organization is a process in which the internalorganization of a system, normally an open system,increases in complexity without being guided or managedby an outside source.”(also: Self-assembly)

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Definitions

Emergence but no Self-Organization?

H20 molecules ⇒ Water

Random walks ⇒ Normal distributions

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Self-organization but no Emergence?

Water above and near the freezing point.

Emergence may be limited to a low scale of a system.

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Economics

Eric Beinhocker (The Origin of Wealth): [5]

Dynamic:

I Complexity Economics: Open, dynamic, non-linearsystems, far from equilibrium

I Traditional Economics: Closed, static, linear systemsin equilibrium

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Economics

Agents:

I Complexity Economics:Modelled individually; use inductive rules of thumb tomake decisions; have incomplete information; aresubject to errors and biases; learn to adapt over time

I Traditional Economics: Modelled collectively; usecomplex deductive calculations to make decisions;have complete information; make no errors and haveno biases; have no need for learning or adaptation(are already perfect)

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Economics

Networks:I Complexity Economics: Explicitly model bi-lateral

interactions between individual agents; networks ofrelationships change over time

I Traditional Economics: Assume agents only interactindirectly through market mechanisms (e.g. auctions)

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Economics

Emergence:

I Complexity Economics: No distinction betweenmicro/macro economics; macro patterns areemergent result of micro level behaviours andinteractions

I Traditional Economics: Micro-and macroeconomicsremain separate disciplines

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Economics

Evolution:I Complexity Economics:

The evolutionary process of differentiation, selectionand amplification provides the system with noveltyand is responsible for its growth in order andcomplexity

I Traditional Economics:No mechanism for endogenously creating novelty, orgrowth in order and complexity

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Upshot

I The central concepts Complexity and Emergence arenot precisely defined.

I There is as yet no general theory of ComplexSystems.

I But the problems exist...Complex (Adaptive) Systems abound...

I Framing: Thinking about systems is essential today.I We use whatever tools we need.

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Upshot

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Upshot

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Upshot

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Upshot

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Models

Nino Boccara in Modeling Complex Systems:

“Finding the emergent global behavior of a large systemof interacting agents using methods is usually hopeless,and researchers therefore must rely on computer-basedmodels.”

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Approaches

Focus is on dynamical systems models:I differential and difference equation modelsI chaos theoryI cellular automataI networksI power-law distributions

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Models

Philip Ball in Critical Mass:[3] “... very often what passes today for ‘complexityscience’ is really something much older, dressed up infashionable apparel. The main themes in complexitytheory have been studied for more than a hundred yearsby physicists who evolved a tool kit of concepts andtechniques to which complexity studies have barelyadded a handful of new items.”

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Old School

I Statistical Mechanics is “a science of collectivebehavior.”

I Simple rules give rise to collective phenomena.

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Statistical mechanics

The Ising Model (�):

I Idealized model of a ferromagnet.I Each atom is assumed to have a local spin that can

be up or down: Si = ±1.I Spins are assumed arranged on a lattice

(e.g. square lattice in 2-d).I In isolation, spins like to align with each other.I Increasing temperature breaks these alignments.I The drosophila of statistical mechanics.

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Ising model

2-d Ising model simulation:http://www.pha.jhu.edu/ javalab/ising/ising.html (�)

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Phase diagrams

Qualitatively distinct macro states.

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Phase diagrams

Oscillons, bacteria, traffic, snowflakes, ...

Umbanhowar et al., Nature, 1996 [13]

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Phase diagrams

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Phase diagrams

W0 = initial wetness, S0 = initial nutrient supply

http://math.arizona.edu/~lega/HydroBact.html

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Ising model

Analytic issues:

I 1-d: simple (Ising & Lenz, 1925)I 2-d: hard (Onsager, 1944)I 3-d: extremely hard...I 4-d and up: simple.

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Ising model

Analytic issues:

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Ising model

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Statistics

I Origins of Statistical Mechanics are in the studies ofpeople... (Maxwell and co.)

I Now physicists are using their techniques to studyeverything else including people...

I See Philip Ball’s “Critical Mass” [3]

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Statistics

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Statistics

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Universality

Universality:The property that the macroscopic aspects of a systemdo not depend sensitively on the system’s details.

I The Central Limit Theorem.I Lattice gas models of fluid flow.

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Universality

Universality:The property that the macroscopic aspects of a systemdo not depend sensitively on the system’s details.

I The Central Limit Theorem.I Lattice gas models of fluid flow.

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Universality

I Sometimes details don’t matter too much.I Many-to-one mapping from micro to macroI Suggests not all possible behaviors are available

at higher levels of complexity.

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Universality

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Universality

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Fluids

Fluid flow is modeled by the Navier-Stokes equations.

Works for many very different fluids:

I The atmosphere, oceans, blood, galaxies, the earth’smantle...

and ball bearings on lattices...?

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Fluids

Fluid flow is modeled by the Navier-Stokes equations.

Works for many very different fluids:

I The atmosphere, oceans, blood, galaxies, the earth’smantle...

and ball bearings on lattices...?

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Lattice gas models

Collision rules in 2-d on a hexagonal lattice:

Lattice matters...No ‘good’ lattice in 3-d.

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Lattice gas models

Collision rules in 2-d on a hexagonal lattice:

Lattice matters...No ‘good’ lattice in 3-d.

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Symmetry Breaking

Philip Anderson’s paper: “More is Different.”Science (1972). [1]

I Argues against idea that the only real scientists arethose working on the fundamental laws.

I Symmetry breaking ⇒ different laws/rules at differentscales...

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Symmetry Breaking

“Elementary entities of science X obey the laws ofscience Y”

I XI solid state or

many-body physicsI chemistry

I molecular biologyI cell biologyI ·I psychologyI social sciences

I YI elementary particle

physicsI solid state

many-body physicsI chemistryI molecular biologyI ·I physiologyI psychology

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Symmetry Breaking

Anderson:[the more we know about] “fundamental laws, the lessrelevance they seem to have to the very real problems ofthe rest of science.”

Scale and complexity thwart the constructionisthypothesis.

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Symmetry Breaking

Anderson:[the more we know about] “fundamental laws, the lessrelevance they seem to have to the very real problems ofthe rest of science.”

Scale and complexity thwart the constructionisthypothesis.

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Symmetry Breaking

I Page 291–292 of Sornette [12]:Renormalization ⇔ Anderson’s hierarchy.

I But Anderson’s hierarchy is not a simple one: therules change.

I Crucial dichotomy between evolving systemsfollowing stochastic paths that lead toinevitable or particular destinations (states).

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Symmetry Breaking

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More is different:

from http://www.xkcd.com

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A real science of complexity:

A real theory of everything:

1. Is not just about the ridiculously small stuff...2. It’s about the increase of complexity

Symmetry breaking/Accidents of history

vs. Universality

I Second law of thermodynamics: we’re toast in thelong run.

I So how likely is the local complexification of structurewe enjoy?

I Another key: randomness can give order.

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vs. Universality

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vs. Universality

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vs. Universality

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vs. Universality

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vs. Universality

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vs. Universality

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Tools and techniques:

I Differential equations, difference equations, linearalgebra.

I Statistical techniques for comparisons anddescriptions.

I Methods from statistical mechanics and computerscience.

I Computer modeling.

Key advance:

I Representation of complex interaction patterns asdynamic networks.

I The driver: Massive amounts of DataI More later...

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Key advance:

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The absolute basics:

Science in three steps:

1. Find interesting/meaningful/important phenomenainvolving spectacular amounts of data.

2. Describe what you see.3. Explain it.

Beware your assumptionsDon’t use tools/models because they’re there, or becauseeveryone else does...

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How do we measure the complexity of a system?

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(1) Entropy: number of microstates that could underlie aparticular macrostate.

I Used in information theory and statisticalmechanics/thermodynamics.

I Measures how uncertain we are about the details ofa system.

I Problem: Randomness maximizes entropy, perfectorder minimizes.

I Our idea of ‘maximal complexity’ is somewhere inbetween...

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(Aside)

What about entropy and self-organization?

Isn’t entropy supposed to always increase?

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(Aside)

What about entropy and self-organization?

Isn’t entropy supposed to always increase?

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Two ways for order to appear in a system withoutoffending the second law of thermodynamics:

(1) Entropy of the system decreases at the expense ofentropy increasing in the environment.

(2) The system becomes more ordered macroscopicallywhile becoming more disordered microscopically.

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(2) Various kinds of information complexity:

I Roughly, what is the size of a program required toreproduce a string of numbers?

I Again maximized by random strings.I Very hard to measure.

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(3) Variation on (2): what is the size of a programrequired to reproduce members of an ensemble of astring of numbers?

Now: Random strings have very low complexity.

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(3) Variation on (2): what is the size of a programrequired to reproduce members of an ensemble of astring of numbers?

Now: Random strings have very low complexity.

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Large problem: given any one example, how do we knowwhat ensemble it belongs to?

One limited solution: divide the string up intosubsequences to create an ensemble.

See Complexity by Badii & Politi [2]

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So maybe no one true measure of complexity exists.

Cosma Shalizi:

“Every few months seems to produce another paperproposing yet another measure of complexity, generally aquantity which can’t be computed for anything you’dactually care to know about, if at all. These quantities arealmost never related to any other variable, so they formno part of any theory telling us when or how things getcomplex, and are usually just quantification forquantification’s own sweet sake.”

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References I

P. W. Anderson.More is different.Science, 177(4047):393–396, August 1972. pdf (�)

R. Badii and A. Politi.Complexity: Hierarchical structures and scaling inphysics.Cambridge University Press, Cambridge, UK, 1997.

P. Ball.Critical Mass: How One Thing Leads to Another.Farra, Straus, and Giroux, New York, 2004.

Y. Bar-Yam.Dynamics of Complex Systems”.Westview Press, Boulder, CO, 2003.

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References II

E. D. Beinhocker.The Origin of Wealth.Harvard Business School Press, Cambridge, MA,2006.

N. Boccara.Modeling Complex Systems.Springer-Verlag, New York, 2004.

S. Bornholdt and H. G. Schuster, editors.Handbook of Graphs and Networks.Wiley-VCH, Berlin, 2003.

J. S. Coleman.Foundations of Social Theory.Belknap Press, Cambridge, MA, 1994.

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References III

R. Foote.Mathematics and complex systems.Science, 318:410–412, 2007. pdf (�)

J. H. Miller and S. E. Page.Complex Adaptive Systems: An introduction tocomputational models of social life.Princeton University Press, Princeton, NJ, 2007.

T. C. Schelling.Micromotives and Macrobehavior.Norton, New York, 1978.

D. Sornette.Critical Phenomena in Natural Sciences.Springer-Verlag, Berlin, 2nd edition, 2003.

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References IV

P. B. Umbanhowar, F. Melo, and H. L. Swinney.Localized excitations in a vertically vibrated granularlayer.Nature, 382:793–6, 29 August 1996. pdf (�)

S. Wasserman and K. Faust.Social Network Analysis: Methods and Applications.Cambridge University Press, Cambridge, UK, 1994.

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