Introduction to Complex Systems

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Anxo Sánchez
Grupo Interdisciplinar de Sistemas Complejos
Departamento de Matemáticas
Institute UC3M-BS for Financial Big Data (IFiBiD)
Universidad Carlos III de Madrid
Instituto de Biocomputación y Física de Sistemas Complejos (BIFI)
Universidad de Zaragoza
Introduction to complex systems

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¿What are complex systems?

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❖ Many interacting components / agents

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❖ Emergent collective behavior

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❖ Examples:
❖ Water (molecules vs phases) (more later)
❖ Proteins (aminoacids vs molecule / structure)
❖ Brain (neurons vs intelligence)
❖ Society (people vs institutions /norms)
❖ Biosphere (species vs ecosystem) (more later)

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❖ Examples:
❖ Water (molecules vs phases) (more later)
❖ Proteins (aminoacids vs molecule / structure)
❖ Brain (neurons vs intelligence)
❖ Society (people vs institutions /norms)
❖ Biosphere (species vs ecosystem) (more later)
❖ Frontier between sciences or subjects

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(Physics Nobel Laureate) Phil Anderson, 1972
“More is different” (emergence)

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When there are MORE  
than one simple agent (e.g. molecule)

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those agents may self-organize in collective objects (e.g. cells)

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which have emergent behavior (e.g. life)

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which have emergent behavior (e.g. life)
that  
IS DIFFERENT  
from the behavior of the simple agent (e.g. chemical reactions)

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MICRO: the relevant elementary agents
INTER: their basic, simple interactions
MACRO: the emerging collective objects
Complex Systems Paradigm:

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MICRO: the relevant elementary agents
INTER: their basic, simple interactions
MACRO: the emerging collective objects
orders, transactions
traders
herds,crashes,booms
Economy:

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Intrinsically (3x) interdisciplinary:
❖ MICRO belongs to one science
❖ MACRO to another science
❖ Mechanisms: a third science

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Intrinsically (3x) interdisciplinary:
❖ MICRO belongs to one science
❖ MACRO to another science
❖ Mechanisms: a third science
Decision making, psychology
Financial economics
Statistical mechanics, Physics 
Mathematics
Complex Systems Paradigm: Economy:

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“More is different” (fluctuations)
• Role of fluctuations: many, but not infinite, 
agents / interactions

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“More is different” (fluctuaciones)
• Nonlinear systems with instabilities

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• External influences: noise, disorder

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• External influences: noise, disorder
• Creative effects, e.g., stochastic resonance

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950C
1Kg
1cm
Water level vs. temperature
“More is different” (phase transition)

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950C
1Kg
1cm
970C
1cm
1Kg

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950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg

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950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg
? Extrapolation?

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950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg
1010C
Macroscopic linear
extrapolation
breaks down!
? Extrapolation?

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950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg
1010C
Macroscopic linear
extrapolation
breaks down!
? Extrapolation?
(a single molecule
does not boil)Water level vs. temperature

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Water level:
economic index

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95 97 99
Water level:
economic index

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95 97 99 101
Crash = result of
collective behavior of
individual traders
Water level:
economic index

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Statistical Mechanics  
Phase Transition

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Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
Anderson abstractization

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Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
Biology
Social Science
Brain Science
Economics and
Finance
Business 
AdministrationICT
Semiotics and
Ontology

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Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
Biology
Social Science
Brain Science
Economics and
Finance
Business 
AdministrationICT
Semiotics and
Ontology
Chemicals
E-pages
Neurons
Words
people
Customers
Traders
Cells,life
Meaning
Social groups
WWW
Cognition,
perception
Markets
Herds,
Crashes

“More is different” (frontier science)

Chemicals
Ion channels
Neurons
Brain
Thoughts
Economy, Culture, Social groups

Chemicals
Ion channels
Neurons
Brain
Thoughts
Conceptual boundary between disciplines

Chemicals
Ion channels
Neurons
Brain
Thoughts
It helps to bridge them by addressing  
within a common conceptual framework 
the fundamental problems  
of one of them  
in terms of the collective phenomena of another.
Conceptual boundary between disciplines

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Santa Fe Institute for Complex Systems
“... a private, non-profit, multidisciplinary research and education center”

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“Since its founding in 1984, the Santa Fe Institute (SFI) has devoted itself to fostering a
multidisciplinary scientific research community pursuing frontier science. SFI seeks to
catalyze new research activities and serve as an "institute without walls.”
Topics
• Physics and Computation of Complex Systems
• Human Behavior, Institutions and Social Systems
• Living Systems: Emergence, Hierarchy and Dynamics

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Research projects
• A theory of invention and innovation
• Theory of embodied intelligence
• Biology, behavior, and disease
• Social networks, big data, and physics-powered inference
• Information, thermodynamics, and the evolution of complexity in  
biological systems
• Neighborhoods, slums, & human development
• Emergence of complex societies
• Hidden laws in biological and social systems
• Evolution of complexity on earth
• Cities, scaling, & sustainability

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Institutions in Spain

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Asociación para el estudio de Sistemas Complejos
Sociotecnológicos (COMSOTEC)

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Asociación Madrileña de Ciencias de la Complejidad
(ComplejiMad)

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Foundations
❖ Statistical Mechanics (Boltzmann, Gibbs, 1900)

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Foundations
❖ Nonlinear Science
❖ Chaos (Lorenz, 1963)
❖ Coherents Structures (Fermi, Pasta y Ulam, 1955)
❖ Patterns (Bénard, 1900; Belusov, 1951; Winfree, 1967)
❖ Evolutionary Dynamics (Maynard-Smith, 1974)

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Foundations
❖ Nonlinear Science
❖ Chaos (Lorenz, 1963)
❖ Coherents Structures (Fermi, Pasta y Ulam, 1955)
❖ Patterns (Bénard, 1900; Belusov, 1951; Winfree, 1967)
❖ Evolutionary Dynamics (Maynard-Smith, 1974)
❖ Computation (1990)

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Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear physics
❖Nanotechnology, quantum computation, astronomy,…

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J. Muñoz, R. Cuerno & M. Castro (2006)
❖ Physics
❖Statistical and nonlinear physics
❖Nanotechnology, quantum computation, astronomy,…

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❖Economy
❖Micro vs macro, financial markets, management, …

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E. Moro, Leganés (2006)
❖Economy

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E. Moro, Leganés (2006) P. Richmond, Dublin (2006)
❖Economy

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❖Sociology
❖Norms and institutions, cultural dynamics, cooperation,…

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M. San Miguel, Palma de Mallorca (2005)
❖Sociology

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A. Arenas, Tarragona (2002)M. San Miguel, Palma de Mallorca (2005)
❖Sociology

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❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, …

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❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, ……

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Innovation between Science and Technology

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❖ Bottom-up approximation

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❖ Robust, self-organized systems

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❖ Control of complex systems
❖ Internet
❖ Agent-based software
❖ Design of organization
❖ Risks: spam, SIDA/SARS/Avian flu
❖ New drugs / genetic therapy

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❖ Internet
❖ Basis for new technologies (ICT, transport, …)

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❖ Internet
❖ Basis for new technologies (ICT, transport, …)
❖ ¿A paradigm shift?

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Mathematics
❖ Graph theory (Complex networks)

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Mathematics
❖ Stochastic processes and statistics (Disorder, noise)

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Mathematics
❖ Functional analysis (Phase transitions)

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Mathematics
❖ Control theory and signal theory

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Mathematics
❖ Evolutionary dynamics and game theory

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Mathematics
❖ A “discrete analysis”

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Mathematics
❖ New simulation techniques (Agents, OOP)

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Mathematics
❖ Computational complexity

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Mathematics
❖ Computational complexity
❖ Data mining, data analysis

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The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)

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The Sicomoro course
❖ Sociophysics

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The Sicomoro course
❖ Sociophysics
❖ Econophysics (Bartolo Luque, May 3)

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The Sicomoro course
❖ Sociophysics
❖ Fractals and scale invariance (Bartolo Luque, May 3)

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The Sicomoro course
❖ Sociophysics
❖ The game of evolution (José A. Cuesta, May 9)

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The Sicomoro course
❖ Sociophysics
❖ Genes and human genealogies (Susanna Manrubia, May 9)

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The Sicomoro course
❖ Sociophysics
❖ Complex networks (Javier Galeano, May 17)

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The Sicomoro course
❖ Sociophysics
❖ Complex networks (Javier Galeano, May 17)
❖ Complexity in biology (Ester Lázaro, May 17)

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Examples
❖ Traffic
• Question 1: How are jams formed in highways?
• Question 2: How are jams formed in cities?

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Examples
❖ Traffic
• Question 1: How are jams formed in highways?
• Question 2: How are jams formed in cities?
❖ Opinion formation
• Question: How can a minority win?

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• Important problem
Examples: Traffic

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❖ 82% travellers and 53% commercial transport (Germany)
❖10% asphalt land (Netherlands)
❖ Billions of lost hours (Spain)
Examples: Traffic

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❖ Billions of euros in gas (Europe) 
• Difficult solution
❖ Wrong traditional answer (new roads don’t fix it)
Examples: Traffic

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❖ Billions of euros in gas (Europe) 
• Difficult solution
❖ Wrong traditional answer (new roads don’t fix it)
Wider applicability: other transports, pedestrians, internet,…
Examples: Traffic

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Simulation work needed:
❖ Discrete models suited to simulation and amenable to analytics
(at least to some extent)
❖ Need for predictions: realistic models impossible
❖ Controlled experiments and identification of relevant parameters
❖ Monitor global variables
Examples: Traffic

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1. Acceleration: if possible, increase speed by 1; vmax=5
7.5
2. Braking: slow down to the fastest possible speed
1D Nagel-Schreckenberg model (1992)

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3. Randomization: with probability p, brake (no apparent cause)
4. Motion
Parallel updating (important)

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Empirical findings:
“Fundamental diagram”

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Empirical findings:
Phantom jams

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Simulation results:
Good agreement

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What can be inferred?

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• The NaSch is a good (stylized) description of highway  
traffic
• (Can be extended to more complicated  
situations/geometries)
• Averages are not very relevant
• New interesting magnitudes to monitor: “throughput” vs
“volatility”

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BML Automata (Biham,
Middleton & Levine, 1992)
• Traffic lights
even instants
odd instants
• No overlap
• Parallel update
• Periodic B.C.
City traffic: 2D models

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Aleatoriedad: Probabilidad g de giro, g < 0.5 (BML g=0)
g
1-g
CMMS Automata (Cuesta,
Martínez, Molera
& Sánchez, 1993)

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Main result:
Phase diagram

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• The phase transition picture applies to traffic as well as  
to molecules
• Phase diagram similar to water; g similar to temperature
• Additional info by analytical means (low density limit,  
other approaches)
• Note interaction through excluded volume

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Generalization: particle flow

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Examples: Opinion formation
Question: How can the minority win?

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Serge Galam has proposed a mechanism, social inertia,
that leads to a democratic rejection of social reforms initially
favored by the majority

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Social inertia: Ties favor the “no” option

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! Conservative reaction to the risk of change
! Keep the social “statu quo”

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! Conservative reaction to the risk of change
! Keep the social “statu quo”
(Taken from Maxi San Miguel, IFISC, Mallorca)

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S. Galam: Le Monde, 26 February, 2005

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Binary opinion, either yellow or blue, about reform
Galam´s model

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For Against
Galam´s model

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For Against
Initially, there is a blue minority
Galam´s model

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Social life: Discussion in groups
(e.g., at work, at the bar, at the church,…)
Galam´s model

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Social life: Discussion in groups
(e.g., at work, at the bar, at the church,…)
Example,k
=16
M, maximum
cell size
Cells defined by
their size k
Galam´s model

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Galam´s model
Interaction: Majority convinces minority in a cell

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6
10
Galam´s model

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Every agent
becomes yellow
6
10
Galam´s model

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Galam´s model
Social inertia: Ties resolved in favor of blue

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8
8
Galam´s model
Social inertia: Ties resolved in favor of blue

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Galam´s model
Evolution: Random reshuffling in cells

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Phase diagram: Initial minority vs max cell size
p: initial minority population
Threshold
line
Eur. Phys. J. B 39, 535 (2004)
Galam´s model
M: max
cell size

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• There is a threshold value pc<½ such that for p>pc  
the minority becomes a majority
• For the effect to happen far from ½ M needs to be  
small
• Time to consensus T ~ ln N
• Note that this is a proposal for a mechanism but  
not a proof that this mechanism is the correct one
• Models can be used to verify or falsify intuitions in 
social problems

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By way of conclusion
❖ Complexity Science is here to stay

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❖ Relevant to real life problems of different nature 
(cf. Examples)

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(cf. Examples)
❖ Key concept: Emergence

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(cf. Examples)
❖ Involves many sciences, but strongly based on
Mathematics and Computation

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(cf. Examples)
❖ Requires working and thinking about frontiers

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(cf. Examples)
❖ Requires working and thinking about frontiers
❖ Key for future R+D+i

Introduction to Complex Systems

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