Overview of Complex Systems - University of VermontReferences Frame 1/108 Overview of Complex...

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Basic DefinitionsComplexity

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Overview of Complex SystemsPrinciples of Complex Systems

Course 300, Fall, 2008

Prof. Peter Dodds

Department of Mathematics & StatisticsUniversity of Vermont

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

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OutlineCourse Information

Major CentersResourcesProjectsTopics

Basic DefinitionsComplexityEmergenceSelf-OrganizationModelingStatistical MechanicsUniversalitySymmetry BreakingTools and TechniquesMeasures of Complexity

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Basics:

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

220 Votey, Tuesday and Thursday, 11:00 am to 12:30pm

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

http://www.uvm.edu/ pdodds/teaching/2008-08UVM-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]

http://www.uvm.edu/~pdodds/teaching/2008-08UVM-300/

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Admin:

Paper products:

1. Outline

Office hours:

I 9:00 am to 10:30 amTuesday and ThursdayRm 203, Math Building

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Admin:

Paper products:

1. Outline

Office hours:I 9:00 am to 10:30 am

Tuesday and ThursdayRm 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 (9/2, 9/4) lecture lecture2 (9/9, 9/11) lecture lecture3 (9/16, 9/18) lecture lecture4 (9/23, 9/25) Project

presentationsProjectpresentations

5 (9/30, 10/2) lecture lecture6 (10/7, 10/9) lecture lecture7 (10/14, 10/16) lecture lecture8 (10/21, 10/23) lecture guest lecture:

Stuart Kauffman9 (10/28, 10/30) lecture lecture10 (11/4, 11/6) lecture lecture11 (11/11, 11/13) lecture lecture12 (11/18, 11/20) lecture lecture13 (11/25, 11/27) Thanksgiving Thanksgiving14 (12/2, 12/4) lecture lecture15 (12/9, 12/11) Project

PresentationsProjectPresentations

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Important dates:

1. Classes run from Tuesday, Septeber 2nd toThursday, December 11.

2. Add/Drop, Audit, Pass/No Pass deadline—Monday,September 15.

3. Last day to withdraw—Friday, October 31.4. Reading and exam period—Friday, December 12th

to Friday, December 19th.

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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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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, ..., UVM (CSC)

http://www.cscs.umich.edu/

http://www.northwestern.edu/nico/

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

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

http://www.cscs.umich.edu/~crshalizi/notebooks/

http://www.comdig.org

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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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Definitions


I Many interacting agents or entities

I 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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Definitions


I Many interacting agents or entitiesI Relationships are nonlinear

I 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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Definitions


I Many interacting agents or entitiesI Relationships are nonlinearI Presence of feedback

I 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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Definitions


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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Definitions


I Many interacting agents or entitiesI Relationships are nonlinearI Presence of feedbackI Complex systems are open (out of equilibrium)I Presence of memory

I Modular/multiscale/hierarchical structureI Evidence of emergence propertiesI Evidence of self-organization

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Definitions


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 structure

I Evidence of emergence propertiesI Evidence of self-organization

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Definitions


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 properties

I Evidence of self-organization

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Definitions


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 robustbut 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


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).


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Definitions



but not adaptable.I But engineered systems can become complex

(power grid, planes).


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Definitions




(power grid, planes).I They can also fail spectacularly.

I Explicit distinction: Complex Adaptive Systems.

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Definitions




(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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Definitions

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

Nonlinear (OED)1. a. Math. and Physics. Not linear; ... involving orpossessing the property that the magnitude of an effector output is not linearly or proportionally related to that ofthe cause or input. First cited use 1844.

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Definitions

Nonlinear (OED)b. colloq. to go non-linear: to lose one’s head; to rave,esp. about a particular obsession. First cited use 1985.

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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

Page 1 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

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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.

©2007 The Complexity Society Privacy Policy Disclaimer

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,wombats, and fractals.”

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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 FlashesI 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

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




I ‘Cosmos’ = grown order

I Archetypal limits of hierarchical and decentralizedstructures.

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

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Emergence





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





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





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





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

Thomas Schelling (Economist/Nobelist):

I “Micromotives and Macrobehavior” [11]

I Segregation, wearing hockey helmet, seating choices

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Emergence

James Coleman in Foundations of Social Theory :Weber

CapitalismProtestantReligiousDoctrine

EconomicBehavior

Values

Societal level

Individual level

Coleman

Understand macrophenomena arises from microbehaviorwhich in turn depends on macrophenomena. [8]

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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

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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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



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

http://www.nytimes.com/2007/01/28/magazine/28nutritionism.t.html

nytimes.com

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Reductionism

Reductionism and food:I Unhappy Meals, Michael Pollan, NY Times, January 2007I “even the simplest food is a hopelessly complex thing

to study, a virtual wilderness of chemicalcompounds, many of which exist in complex anddynamic 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

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

Thyme’s known antioxidants:4-Terpineol, alanine, anethole, apigenin, ascorbic acid,beta carotene, caffeic acid, camphene, carvacrol,chlorogenic acid, chrysoeriol, eriodictyol, eugenol, ferulicacid, gallic acid, gamma-terpinene isochlorogenic acid,isoeugenol, isothymonin, kaempferol, labiatic acid, lauricacid, linalyl acetate, luteolin, methionine, myrcene,myristic acid, naringenin, oleanolic acid, p-coumoric acid,p-hydroxy-benzoic acid, palmitic acid, rosmarinic acid,selenium, tannin, thymol, tryptophan, ursolic acid, vanillicacid.

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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

Yaneer Bar-Yam (founder and head of NECSI) onemergence:

Suggests there are four types of emergence. No, five!

One very weak, one weak, two strong, and a dynamicstrong version.

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Emergence

Example—string of three bits with one odd parity bit:particle), the description of the state of the system is acompilation of the descriptions of the parts. The ensembleof a compilation of the description of the parts is the prod-uct space of the ensemble of the parts. In general this isdifferent from the ensemble of the system itself. This differ-ence can arise because of the interactions between the partsor because the system ensemble itself is affected by itsenvironment. In either case we have an ensemble of theentire system P(s), s ! {si), i ! 1, . . . N, which is not theproduct of the ensembles of the parts P! (s) ! "i psi

(si). Theformer includes interdependence, whereas the latter doesnot. From the point of view of observation, a more carefulconsideration of physics principles suggests that a singlestate does not correspond to a physically observable systemand therefore is not a physically meaningful concept. Phys-ics is only concerned with reproducible experiments andtherefore with experiments that are performed on a systemthat is defined by a preparation process rather than aunique microstate.For quantum theory, the analog of considering the ensem-ble of the system is considering the density matrix. The shiftfrom particle positions and momenta to an ensemble cor-responds to a shift from the wavefunction to a densitymatrix and the response of a system is determined by itsdensity matrix, not a particular wavefunction. In this article,we are not concerned with quantum issues, but rather withthe semi-classical perspective that is generally consideredsufficient to represent macroscopic behavior of systems(when collective quantum phenomena like superfluidity arenot relevant because temperatures are not ultralow) even ifthey have complex structures.

3.2. Type 2 Strong EmergenceWe focus on the properties of ensembles in order to explainthe possibility of strong emergence. System ensemble prop-erties may not be observable in the states of the compo-nents or the ensembles of the components. Specifically, weconstruct a system with a constraint on the whole that doesnot apply to any subsystem. Assume that we have a systemwith N bits. These bits are constrained to have an oddnumber of ON bits. An example of a three bit system isillustrated in Figure 3. If we look at any subsystem of thebits, the subsystem does not have the same constraint.Indeed, any possible arrangement of a subsystem is equallylikely. Still, the constraint on the whole system is a propertyof the system. We see that this property cannot be inferredfrom observations of the components themselves, butrather only by examining the system as a whole.

The key to this observation is the recognition that prop-erties of a system may be constrained in multiple relation-ships. Unlike conventional physical interactions that areonly pairwise, there may be ensemble constraints that donot simplify to pairwise constraints. Pairwise interactionsmay also lead to multiparticle dependencies. However, mul-

ticomponent dependencies can arise from environmentaleffects that are not captured by pairwise interactions. Inparticular there may be constraints that only apply to amacroscopic subsystem or the entirety of the system.Such constraints are only observable through observa-tions of the whole and not through combinations of ob-servations of the components compiled into observationsof the entirety.

Moreover, it is interesting that any of the bits is totallyconstrained by the values of the other bits in the system.Given two bits, that can be set arbitrarily, the third bit mustbe specified so that the sum over the bits is odd. On theother hand, by looking at that bit or any subset of bits, onecannot see this constraint. The reason is that when consid-ered over the ensemble of all possible states of the system,there is no net impact on the ensemble of the individual bit.Does this mean that it is impacted or not? The value of theindividual bit is impacted by the values of the rest of the bitsas far as a single state is concerned but not as far as anensemble is concerned. This is the opposite of what onewould say about the entire system, which is impacted in theensemble picture but not in the state picture.

3.3. Mathematical FormulationPreviously [19], we have described a formalism that cap-tures the multiscale variety of a system. This formalismconsiders the constraints that exist in a system and lead tocollective behaviors. It separates behaviors that correspond

FIGURE 3

Parity bit ensemble for a three-bit system with two possibilitiesrepresented by circles colored black and white. The system allowsfour (out of the usual eight) possible states that can be identified asconstrained to allow only an odd number of white circles. Each bit has50% probability of each possibility, and each pair of bits has 25% ofeach of the four possible states for two bits. Thus observations of anyproper subset cannot reveal the existence of the global constraint.This system can be constructed by allowing all possible arrangementsof any two bits and requiring the third bit (the parity bit) to satisfy theconstraint. All bits satisfy this property and the system states aresymmetric with respect to bit exchange. Remarkably, this implies thatthe state of any bit is completely constrained by the global constraintapplied to the whole system, even though no observation of a bitreveals this. These properties appear to satisfy the conceptual de-scription of strong emergent properties, and they provide for adistinctive multiscale mathematical signature found in Figure 4.

20 C O M P L E X I T Y © 2004 Wiley Periodicals, Inc.

Global constraint on bits not seen in individual bits.

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Emergence

Strong: constraints on the global structure may not beobservable by viewing behavior of individual parts.

Not a pure micro-to-macro story.

Still... seems that analysis of the system is possible bythinking about the parts.

And centralized control is a simple system-level feature.

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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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Definitions

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 well defined.

I There is no general theory of Complex Systems.I But the problems exist...

Complex (Adaptive) Systems abound...I • We use whatever tools we need.

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Upshot


I There is no general theory of Complex Systems.

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

I • We use whatever tools we need.

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Upshot



Complex (Adaptive) Systems abound...

I • We use whatever tools we need.

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Upshot



Complex (Adaptive) Systems abound...I • We use whatever tools we need.

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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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Approaches


Focus is on dynamical systems models:I differential and difference equation models

I chaos theoryI cellular automataI networksI power-law distributions

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Approaches


Focus is on dynamical systems models:I differential and difference equation modelsI chaos theory

I cellular automataI networksI power-law distributions

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Approaches


Focus is on dynamical systems models:I differential and difference equation modelsI chaos theoryI cellular automata

I networksI power-law distributions

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Approaches


Focus is on dynamical systems models:I differential and difference equation modelsI chaos theoryI cellular automataI networks

I power-law distributions

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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 canbe 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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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).


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The Ising Model:


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

(e.g. square lattice in 2-d).


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The Ising Model:



(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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The Ising Model:



(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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The Ising Model:



(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 (�)

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

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:

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:

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:

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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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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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

I Sometimes details don’t matter too much.I Many-to-one mapping from micro to macro

I Suggests not all possible behaviors are availableat higher levels of complexity.

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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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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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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

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

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

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More is different:

from http://www.xkcd.com

http://www.xkcd.com

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Tools

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 (specialized, Swarm, Starlogo,and more...)

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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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Hmmm

(Aside)

What about entropy and self-organization?

Isn’t entropy supposed to always increase?

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Hmmm

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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I Again maximized by random strings.

I Very hard to measure.

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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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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.”

Overview


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Self-Organization

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Symmetry Breaking



References

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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.

http://www.uvm.edu/~pdodds/research/papers/others/1972/anderson1972a.pdf

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References

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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

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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.

http://www.uvm.edu/~pdodds/research/papers/others/2007/foote2007a.pdf

Overview


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Self-Organization

Modeling


Universality

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References

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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.

http://www.uvm.edu/~pdodds/research/papers/others/1996/umbanhowar1996a.pdf

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